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Volume  111     Number  1 
February  2001 


Proceedings  of  the  of 

Mathematical 

Editor 

Aiiadi  Sitaram 

Indian  Statistical  Institute,  Bangalore 

Associate  Editors 


Kapil  H  Paranjape 

The  Institute  of  Mathematical  Sciences,  Chennai 


J  R  Ramadas 

Tata  Institute  of  Fundamental  Research,  Mumbai 


Editorial  Board 


S  S  Abhyankar 

Purdue  University,  West  Lafayette,  USA 

K  B  Athreya 

Iowa  State  University,  Iowa,  USA 

B  Bagchi 

Indian  Statistical  Institute,  Bangalore 

S  G  Dani 

Tata  Institute  of  Fundamental  Research,  Mumbai 

J  K  Ghosh 

Indian  Statistical  Institute,  Calcutta 

Gadadhar  Misra 

Indian  Statistical  Institute,  Bangalore 

M  V  Mori 

University  of  Chicago,  Chicago,  USA 

D  Prasad 

MRI  for  Math,  and  Math.  Phys.,  Allahabad 

Phoolan  Prasad 

Indian  Institute  of  Science,  Bangalore 

M  S  Raghunathan 

Tata  Institute  of  Fundamental  Research,  Mumbai 


BVRao 

Indian  Statistical  Institute,  Calcutta 

C  S  Seshadri 

Chennai  Mathematical  Institute,  Chennai 

K  B  Sinha 

Indian  Statistical  Institute,  Calcutta 

R  Sridharan 

Tata  Institute  of  Fundamental  Research,  Mumbai 

V  S  Sunder 

The  Institute  of  Mathematical  Sciences,  Chennai 

M  Vanninathan 

TIFR  Centre,  Indian  Institute  of  Science,  Bangalore 

V  S  Varadarajan 

University  of  California,  Los  Angeles,  USA 

S  R  S  Varadhan 

Courant  Institute  of  Mathematical  Sciences,  USA 

K  S  Yajnik 

Bangalore 


Editor  of  Publications 

N  Mukunda 

Indian  Institute  of  Science,  Bangalore 


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©  2000  by  the  Indian  Academy  of  Sciences.  All  rights  reserved 

The  "Notes  on  the  preparation  of  papers"  are  printed  in  the  last  issue  of  every  volume. 


Proceedings  of  the  Indian  Academy  of  Sciences 
Mathematical  Sciences 


Volume  111 
2001 


Published  by  the  Indian  Academy  of  Sciences 
Bangalore  560  080 


Proceedings  of  the  Indian  Academy  of  Sciences 
Mathematical  Sciences 

Editor 

ASSadi  Sitaram 

Indian  Statistical  Institute,  Bangalore 

Associate  Editors 


Kapil  H  Paranjape 

The  Institute  of  Mathematical  Sciences,  Chennai 


T  R  Ramadas 

Tata  Institute  of  Fundamental  Research,  Mumbai 


Editorial  Board 


S  S  Abhyankar 

Purdue  University,  West  Lafayette,  USA 

K  B  Athreya 

Iowa  State  University,  Iowa,  USA 

B  Bagchi 

Indian  Statistical  Institute,  Bangalore 

S  G  Dani 

Tata  Institute  of  Fundamental  Research,  Mumbai 

J  K  Ghosh 

Indian  Statistical  Institute,  Calcutta 

Gadadhar  Misra 

Indian  Statistical  Institute,  Bangalore 

M  V  Nori 

University  of  Chicago,  Chicago,  USA 

Dipendra  Prasad 

Harish-Chandra  Research  Institute,  Allahabad 

Phoolan  Prasad 

Indian  Institute  of  Science,  Bangalore 

M  S  Raghunathan 

Tata  Institute  of  Fundamental  Research,  Mumbai 


B  VRao 

Indian  Statistical  Institute,  Calcutta 

C  S  Seshadri 

Chennai  Mathematical  Institute,  Chennai 

K  B  Sinha 

Indian  Statistical  Institute,  Calcutta 

R  Sridharan 

Tata  Institute  of  Fundamental  Research,  Mumbai 

V  S  Sunder 

The  Institute  of  Mathematical  Sciences,  Chennai 

M  Vanninathan 

TIFR  Centre,  Indian  Institute  of  Science,  Bangalore 

V  S  Varadarajan 

University  of  California,  Los  Angeles,  USA 

S  R  S  Varadhan 

Courant  Institute  of  Mathematical  Sciences,  USA 

K  S  Yajnik 

Bangalore 


Editor  of  Publications 

N  Mukunda 

Indian  Institute  of  Science,  Bangalore 


Annual  Subscription  Rates 

All  countries  except  India  US$  1 00 

(Price  includes  AIR  MAIL  charges) 

India  Rs.  200 

Annual  subscriptions  are  available  for  Individuals  for  India  and  abroad  at  the  concessional  rates  of  Rs.  125/- 
and  $  30  respectively. 

All  correspondence  regarding  subscription  should  be  addressed  to  The  Circulation  Department  of  the 
Academy. 

Editorial  Office 


Indian  Academy  of  Sciences,  C  V  Raman  Avenue, 
P.B.  No.  8005,  Bangalore  560  080,  India 


Telephone:  80-334  2546,  80-334  2943 
Telefax:  91 -80-334  6094 
Email:  mathsci@ias.ernet.in 
Website:  www.ias.ac.in/mathsci/ 


©2001  by  the  Indian  Academy  of  Sciences.  All  rights  reserved 

The  "Notes  on  the  preparation  of  papers"  are  printed  in  the  last  issue  of  every  volume. 


Proceedings  of  the  Indian  Academy  of  Sciences 

Mathematical  Sciences 

Volume  111,  2001 

VOLUME  CONTENTS 

Number  1,  February  2001 

Algebraic  stacks Tomds  L  Gomez  1 

Variational  formulae  for  Fuchsian  groups  over  families  of  algebraic  curves 

Dakshini  Bhattacharyya  33 

Limits  of  commutative  triangular  systems  on  locally  compact  groups 

Riddhi  Shah          49 

Topological  *-algebras  with  C*-enveloping  algebras  II S  J  Bhatt          65 

On  the  equisummability  of  Hermite  and  Fourier  expansions 


E  K  Narayanan  and  S  Thangavelu  95 


Periodic  and  boundary  value  problems  for  second  order  differential  equations 
Nikolaos  S  Papageorgiou  and  Francesca  Papalini         107 

Boundary  controllability  of  integrodifferential  systems  in  Banach  spaces 

K  Balachandran  and  E  R  Anandhi         127 

Errata 

Steady-state  response  of  a  micropolar  generalized  thermoelastic  half-space  to 

the  moving  mechanical/thermal  loads  ....  Rajneesh  Kumar  and  Sunita  Deswal         137 

Number  2,  May  2001 

Descent  principle  in  modular  Galois  theory 

Shreeram  S  Abhyankar  and  Pradipkumar  H  Keskar         139 

Obstructions  to  Clifford  system  extensions  of  algebras 

Antonio  M  Cegarra  and  Antonio  R  Garzon         151 

The  multiplication  map  for  global  sections  of  line  bundles  and  rank  1  torsion  free 
sheaves  on  curves E  Ballico         163 

Boundedness  results  for  periodic  points  on  algebraic  varieties 

Najmuddin  Fakhruddin         173 

Spectra  of  Anderson  type  models  with  decaying  randomness 

M  Krishna  and  K  B  Sinha         179 

Multipliers  for  the  absolute  Euler  summability  of  Fourier  series  .  . .  Prem  Chandra         203 
On  a  Tauberian  theorem  of  Hardy  and  Littlewood T  Pali         221 


ii  Volume  contents 

Proximinal  subspaces  of  finite  codimension  in  direct  sum  spaces  . . .  V  Indumathi        229 

Common  fixed  points  for  weakly  compatible  maps 

Renu  Chugh  and  Sanjay  Kumar         24 1 

Number  3,  August  2001 

On  totally  reducible  binary  forms:  I C  Hooley        249 

Stability  of  Picard  bundle  over  moduli  space  of  stable  vector  bundles  of  rank  two 

over  a  curve Indranil  Biswas  and  Tomds  L  Gomez         263 

Principal  G-bundles  on  nodal  curves Usha  N  Bhosle         271 

Uncertainty  principles  on  two  step  nilpotent  Lie  groups S  K  Ray        293 

On  property  (J3)  in  Banach  lattices,  Calderon-Lozanowskii  and  Orlicz-Lorentz 

spaces Pawet  Kolwicz        319 

On  oscillation  and  asymptotic  behaviour  of  solutions  of  forced  first  order  neutral 
differential  equations N  Parhi  and  R  N  Rath        337 

Monotone  iterative  technique  for  impulsive  delay  differential  equations 

Baoqiang  Van  and  Xilin  Fu         35 1 

On  initial  conditions  for  a  boundary  stabilized  hybrid  Euler-Bemoulli  beam  .... 
Sujit  K  Bose         365 

Number  4,  November  2001 

Cyclic  codes  of  length  T Manju  Pruthi         371 

Unitary  tridiagonalization  in  M(4,  C) Vishwambhar  Pati         381 

On  Ricci  curvature  of  C-totally  real  submanifolds  in  Sasakian  space  forms 

Liu  Ximin         399 

A  variational  proof  for  the  existence  of  a  conformal  metric  with  preassigned 

negative  Gaussian  curvature  for  compact  Riemann  surfaces  of  genus  >  1 

Rukmini  Dey       407 

Homogeneous  operators  and  project! ve  representations  of  the  Mobius  group: 

A  survey Bhaskar  Bagchi.and  Gadadhar  Misra         415 

Multiwavelet  packets  and  frame  packets  of  L2(RJ) Biswaranjan  Behera         439 

A  variational  principle  for  vector  equilibrium  problems K  R  Kazmi        465 

On  a  generalized  Hankel  type  convolution  of  generalized  functions 

S  P  Malgonde  and  G  S  Gaikawad        471 

Nonlinear  elliptic  differential  equations  with  multivalued  nonlinearities 

Antonella  Fiacca,  Nikolaos  Matzakos,  Nikolaos  S  Papageorgiou  and 

Raffaella  Servadei        489 

Subject  Index 509 

Author  Index 513 


Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  Ill,  No.  1,  February  2001,  pp.  1-31. 
(G)  Printed  in  India 


Algebraic  stacks 


TOMAS  L  GOMEZ 

Tata  Institute  of  Fundamental  Research,  Homi  Bhabha  Road,  Mumbai  400  005,  India 
Email:  tomas@math.tifr.res.in 

MS  received  16  February  2000;  revised  24  August  2000 

Abstract.  This  is  an  expository  article  on  the  theory  of  algebraic  stacks.  After 
introducing  the  general  theory,  we  concentrate  in  the  example  of  the  moduli  stack  of 
vector  bundles,  giving  a  detailed  comparison  with  the  moduli  scheme  obtained  via 
geometric  invariant  theory. 

Keywords.     2  categories;  algebraic  stacks;  moduli  spaces;  vector  bundles. 

1.  Introduction 

The  concept  of  algebraic  stack  is  a  generalization  of  the  concept  of  scheme,  in  the  same 
sense  that  the  concept  of  scheme  is  a  generalization  of  the  concept  of  projective  variety.  In 
many  moduli  problems,  the  functor  that  we  want  to  study  is  not  representable  by  a  scheme. 
In  other  words,  there  is  no  fine  moduli  space.  Usually  this  is  because  the  objects  that  we 
want  to  parametrize  have  automorphisms.  But  if  we  enlarge  the  category  of  schemes 
(following  ideas  that  go  back  to  Grothendieck  and  Giraud  [Gi],  and  were  developed  by 
Deligne,  Mumford  and  Artin  [DM,  Ar2])  and  consider  algebraic  stacks,  then  we  can  cons- 
truct the  'moduli  stack',  that  captures  all  the  information  that  we  would  like  in  a  fine 
moduli  space.  For  other  sources  on  stacks,  see  [E,  La,  LaM,  Vi]. 

The  idea  of  enlarging  the  category  of  algebraic  varieties  to  study  moduli  problems  is 
not  new.  In  fact  Weil  invented  the  concept  of  abstract  variety  to  give  an  algebraic  cons- 
truction of  the  Jacobian  of  a  curve. 

These  notes  are  an  introduction  to  the  theory  of  algebraic  stacks.  I  have  tried  to 
emphasize  ideas  and  concepts  through  examples  instead  of  detailed  proofs  (I  give 
references  where  these  can  be  found).  In  particular,  §3  is  a  detailed  comparison  between 
the  moduli  scheme  and  the  moduli  stack  of  vector  bundles. 

First  I  will  give  a  quick  introduction  in  subsection  1.1,  just  to  give  some  motivations 
and  get  a  flavor  of  the  theory  of  algebraic  stacks. 

Section  2  has  a  more  detailed  exposition.  There  are  mainly  two  ways  of  introducing 
stacks.  We  can  think  of  them  as  2-functors  (I  learnt  this  approach  from  Nitsure  and 
Sorger,  cf.  subsection  2.1),  or  as  categories  fibered  on  groupoids.  (This  is  the  approach 
used  in  the  references,  cf.  subsection  2.2.)  From  the  first  point  of  view  it  is  easier  to  see  in 
which  sense  stacks  are  generalizations  of  schemes,  and  the  definition  looks  more  natural, 
so  conceptually  it  seems  more  satisfactory.  But  since  the  references  use  categories  fibered 
on  groupoids,  after  we  present  both  points  of  view,  we  will  mainly  use  the  second. 

The  concept  of  stack  is  merely  a  categorical  concept.  To  do  geometry  we  have  to 
add  some  conditions,  and  then  we  get  the  concept  of  algebraic  stack.  This  is  done  in 
subsection  2.3. 

1 


2  Tomds  L  Gomez 

In  subsection  2.4  we  introduce  a  third  point  of  view  to  understand  stacks:  as  groupoid 
spaces. 

In  subsection  2.5  we  define  for  algebraic  stacks  many  of  the  geometric  properties  that 
are  defined  for  schemes  (smoothness,  irreducibility,  separatedness,  properness,  etc. .  .)•  In 
subsection  2.6  we  introduce  the  concept  of  point  and  dimension  of  an  algebraic  stack,  and 
in  subsection  2.7  we  define  sheaves  on  algebraic  stacks. 

In  §3  we  study  in  detail  the  example  of  the  moduli  of  vector  bundles  on  a  scheme  X, 
comparing  the  moduli  stack  with  the  moduli  scheme. 

Prerequisites.  In  the  examples,  I  assume  that  the  reader  has  some  familiarity  with  the 
theory  of  moduli  spaces  of  vector  bundles.  A  good  source  for  this  material  is  [HL].  The 
necessary  background  on  Grothendieck  topologies,  sheaves  and  algebraic  spaces  is  in 
Appendix  A,  and  the  notions  related  to  the  theory  of  2-categories  are  explained  in 
Appendix  B. 

1.1  Quick  introduction  to  algebraic  stacks 

We  will  start  with  an  example:  vector  bundles  (with  fixed  prescribed  Chern  classes  and 
rank)  on  a  projective  scheme  X  over  an  algebraically  closed  field  k.  What  is  the  moduli 
stack  MX  of  vector  bundles  on  X?  I  do  not  know  a  short  answer  to  this,  but  instead  it  is 
easy  to  define  what  is  a  morphism  from  a  scheme  B  to  the  moduli  stack  MX-  It  is  just  a 
family  of  vector  bundles  parametrized  by  B.  More  precisely,  it  is  a  vector  bundle  V  on 
B  x  X  (hence  flat  over  B)  such  that  the  restriction  to  the  slices  b  x  X  have  prescribed 
Chern  classes  and  rank.  In  other  words,  MX  has  the  property  that  we  expect  from  a  fine 
moduli  space:  the  set  of  morphisms  Hom(B,Aix)  is  equal  to  the  set  of  families 
parametrized  by  B. 

We  will  say  that  a  diagram 


(1) 


is  commutative  if  the  vector  bundle  V  on  B  x  X  corresponding  to  g  is  isomorphic  to  the 
vector  bundle  (/  x  id*)*  V,  where  V  is  the  vector  bundle  corresponding  to  gf.  Note  that 
in  general,  if  L  is  a  line  bundle  on  B,  then  V  and  V  ®p*BL  won't  be  isomorphic,  and  then 
the  corresponding  morphisms  from  B  to  MX  will  be  different,  as  opposed  to  what 
happens  with  moduli  schemes. 

A  fc-point  in  the  stack  MX  is  a  morphism  u  :  Spec  k  — *  MX,  in  other  words,  it  is  a 
vector  bundle  V  on  X,  and  we  say  that  two  points  are  isomorphic  if  they  correspond  to 
isomorphic  vector  bundles.  But  we  should  not  think  of  MX  just  as  a  set  of  points,  it 
should  be  thought  of  as  a-category.  The  objects  of  MX  are  points1 ,  i.e.  vector  bundles  on 
X,  and  a  morphism  in  MX  is  an  isomorphism  of  vector  bundles.  This  is  the  main 
difference  between  a  scheme  and  an  algebraic  stack:  the  points  of  a  scheme  form  a  set, 
whereas  the  points  of  a  stack  form  a  category,  in  fact  a  groupoid  (i.e.  a  category  in  which 
all  morphisms  are  isomorphisms).  Each  point  comes  with  a  group  of  automorphisms. 
Roughly  speaking,  a  scheme  (or  more  generally,  an  algebraic  space  [Arl,  K])  can  be 


lfro  be  precise,  we  should  consider  also  B-valued  points,  for  any  scheme  5,  but  we  will  only 
consider  fc-valued  points  for  the  moment. 


Algebraic  stacks  3 

thought  of  as  an  algebraic  stack  in  which  these  groups  of  automorphisms  are  all  trivial. 
If  p  is  the  £-point  in  MX  corresponding  to  a  vector  bundle  V  on  X,  then  the  group 
of  automorphisms  associated  to  p  is  the  group  of  vector  bundle  automorphisms  of  V.  This 
is  why  algebraic  stacks  are  well  suited  to  serve  as  moduli  of  objects  that  have 
automorphisms. 

An  algebraic  stack  has  an  atlas.  This  is  a  scheme  U  and  a  (representable)  surjective 
morphism  u  :  U  — »  MX  (with  some  other  properties).  As  we  have  seen,  such  a  morphism 
u  is  equivalent  to  a  family  of  vector  bundles  parametrized  by  U.  The  precise  definition  of 
representable  surjective  morphism  of  stacks  will  be  given  in  §2.  In  this  situation  it  implies 
that  for  every  vector  bundle  V  over  X  there  is  at  least  one  point  in  U  whose  corresponding 
vector  bundle  is  isomorphic  to  V.  The  existence  of  an  atlas  for  an  algebraic  stack  is  the 
analog  of  the  fact  that  for  a  scheme  B  there  is  always  an  affine  scheme  U  and  a  surjective 
morphism  U  — >  B  (if  {£/,  — >  B}  is  a  covering  of  B  by  affine  subschemes,  take  U  to  be  the 
disjoint  union  £J  £//).  Many  local  properties  (smooth,  normal,  reduced. . .)  can  be  studied 
by  looking  at  the  atlas  U.  It  is  true  that  in  some  sense  an  algebraic  stack  looks,  locally, 
like  a  scheme,  but  we  shouldn't  take  this  too  far.  For  instance  the  atlas  of  the  classifying 
stack  EG  (parametrizing  principal  G-bundles,  cf.  Example  2.18)  is  just  a  single  point.  The 
dimension  of  an  algebraic  stack  MX  will  be  defined  as  the  dimension  of  U  minus  the 
relative  dimension  of  the  morphism  u.  The  dimension  of  an  algebraic  stack  can  be 
negative  (for  instance,  dim(J9G)  =  -dim(G)). 

We  will  see  that  many  geometric  concepts  that  appear  in  the  theory  of  schemes  have  an 
analog  in  the  theory  of  algebraic  stacks.  For  instance,  one  can  define  coherent  sheaves  on 
them.  We  will  give  a  precise  definition  in  §2,  but  the  idea  is  that  a  coherent  sheaf  L  on  an 
algebraic  stack  MX  is  a  functor  that,  for  each  morphism  g  :  B  — »  MX,  gives  a  coherent 
sheaf  LB  on  J5,  and  for  each  commutative  diagram  like  (1),  gives  an  isomorphism  between 
/*L#'  and  LB.  The  coherent  sheaf  LB  should  be  thought  of  as  the  pullback  'g*L'  of  L  under 
g  (the  compatibility  condition  for  commutative  diagrams  is  just  the  condition  that 
(g1  °  f)*L  should  be  isomorphic  to  f*g'*L). 

Let's  look  at  another  example:  the  moduli  quotient  (Example  2.18).  Let  G  be  an  affine 
algebraic  group  acting  on  X.  For  simplicity,  assume  that  there  is  a  normal  subgroup  H  of 
G  that  acts  trivially  on  X,  and  that  G  =  G/H  is  an  affine  group  acting  freely  on  X  and 
furthermore  there  is  a  quotient  by  this  action  X  — »  B  and  this  quotient  is  a  principal  G- 
bundle.  We  call  B  =  X/G  the  quotient  scheme.  Each  point  corresponds  to  a  G-orbit  of 
the  action.  But  note  that  B  is  also  equal  to  the  quotient  X/G,  because  H  acts  trivially  and 
then  G-orbits  are  the  same  thing  as  G-orbits.  We  can  say  that  the  quotient  scheme 
'forgets'  H. 

One  can  also  define  the  quotient  stack  [X/G].  Roughly  speaking,  a  point  p  of  [X/G] 
again  corresponds  to  a  G-orbit  of  the  action,  but  now  each  point  comes  with  an 
automorphism  group:  given  a  point  p  in  [X/G],  choose  a  point  x  G  X  in  the  orbit 
corresponding  to  p.  The  automorphism  group  attached  to  p  is  the  stabilizer  Gx  of  x.  With 
the  assumptions  that  we  have  made  on  the  action  of  G,  the  automorphism  group  of  any 
point  is  always  H.  Then  the  quotient  stack  [X/G]  is  not  a  scheme,  since  the  automorphism 
groups  are  not  trivial.  The  action  of  H  is  trivial,  but  the  moduli  stack  still  'remembers' 
that  there  was  an  action  by  H.  Observe  that  the  stack  [X/G]  is  not  isomorphic  to  the  stack 
[X/G]  (as  opposed  to  what  happens  with  the  quotient  schemes).  Since  the  action  of  G  is 
free  on  X,  the  automorphism  group  corresponding  to  each  point  of  [X/G]  is  trivial,  and  it 
can  be  shown  that,  with  the  assumptions  that  we  made,  [X/G]  is  represented  by  the 
scheme  B  (this  terminology  will  be  made  precise  in  §2). 


Tomds  L  Gomez 


2.  Stacks 


2.1  Stacks  as  2-functors:  Sheaves  of  sets 

Given  a  scheme  M  over  a  base  scheme  S,  we  define  its  (contravariant)  functor  of  points 

Homs(-3M) 

Hom5(-,M)  :     (Sch/S)     — »•          (Sets) 

5          F— >    Homs(£,M) 

where  (Sch/5)  is  the  category  of  S-schemes,  5  is  an  S-scheme,  and  Homs(#,  M)  is  the  set 
of  5-scheme  morphisms.  If  we  give  (Sch/5)  the  Zariski  (or  etale,  or  fppf)  topology, 
M  =  Hom5(-,M)  is  a  sheaf  (see  Appendix  A  for  the  definition  of  topologies  and  sheaves 
on  categories).  Furthermore,  given  schemes  M  and  N  there  is  a  bijection  (given  by 
Yoneda  Lemma)  between  the  set  of  morphisms  of  schemes  Hom5(M,  N)  and  the  set  of 
natural  transformations  between  the  associated  functors  M  and  N,  hence  the  category  of 
schemes  is  a  full  subcategory  of  the  category  of  sheaves  on  (Sch/5). 

A  sheaf  of  sets  on  (Sch/5)  with  a  given  topology  is  called  a  space2  with  respect  to  that 
topology  (this  is  the  definition  given  in  ([La],  0)). 

Then  schemes  can  be  thought  of  as  sheaves  of  sets.  Moduli  problems  can  usually  be 
described  by  functors.  We  say  that  a  sheaf  of  sets  F  is  representable  by  a  scheme  M  if  F  is 
isomorphic  to  the  functor  of  points  Hom5(-,M).  The  scheme  M  is  then  called  the  fine 
moduli  scheme.  Roughly  speaking,  this  means  that  there  is  a  one  to  one  correspondence 
between  families  of  objects  parametrized  by  a  scheme  B  and  morphisms  from  B  to  M . 

Example  2.1  (Vector  bundles).  Let  X  be  a  projective  scheme  over  an  algebraically  closed 
field  k.  We  define  the  moduli  functor  M^  of  vector  bundles  of  fixed  r$nk  r  and  Chern 
classes  c/  by  sending  the  scheme  B  to  the  set  M^(#)  of  isomorphism  classes  of  vector 
bundles  on  B  x  X  (hence  flat  over  B)  with  rank  r  and  whose  restriction  to  the  slices 
{b}  x  X  have  Chern  classes  c/.  These  vector  bundles  should  be  thought  of  as  families 
of  vector  bundles  parametrized  by  B.  A  morphism  /  :  Bf  — >  B  is  sent  to  frfx(f)  = 
/*  :  MX(#)  ~^  MX(^')»  tne  maP  °f  sets  induced  by  the  pullback.  Usually  we  will  also  fix  a 
polarization  H  in  X  and  restrict  our  attention  to  stable  or  semistable  vector  bundles  with 
respect  to  this  polarization  (see  [HL]  for  definitions),  and  then  we  consider  the 
corresponding  functors  M£  and  M£s. 

Example  2.2  (Curves).  The  moduli  functor  Mg  of  smooth  curves  of  genus  g  over  a 
Noetherian  base  5  is  the  functor  that  sends  each  scheme  B  to  the  set  Mg(B)  of 
isomorphism  classes  of  smooth  and  proper  morphisms  C  —»  B  (where  C  is  an  S-scheme) 
whose  fibers  are  geometrically  connected  curves  of  genus  g.  Each  morphism  /  :  Bf  — *  B 
is  sent  to  the  map  of  sets  induced  by  the  pullback  /*. 

None  of  these  examples  are  sheaves  (then  none  of  these  are  representable),  because  of 
the  presence  of  automorphisms.  They  are  just  presheaves  (=  functors).  For  instance,  given 
a  curve  C  over  S  with  nontrivial  automorphisms,  it  is  possible  to  construct  a  family 
f  :C-+B  such  that  every  fiber  of  /  is  isomorphic  to  C,  but  C  is  not  isomorphic  to  B  x  C 
(see  [E]).  This  implies  that  Mg  does  not  satisfy  the  monopresheaf  axiom. 


v  -Note  that  the  concept  of  space  is  just  a  categorical  concept.  To  do  geometry  we  need  to  add  some 

N  algebraic  and  technical  conditions  (existence  of  an  atlas,  quasi-separatedness,. . .).  After  we  add 

these  conditions  (see  Definitions  4.3  or  4.4),  we  have  an  algebraic  space. 


Algebraic  stacks  5 

This  can  be  solved  by  taking  the  sheaf  associated  to  the  presheaf  (sheafification).  In  the 
examples,  this  amounts  to  change  isomorphism  classes  of  families  to  equivalence  classes 
of  families,  declaring  two  families  to  be  equivalent  if  they  are  locally  (using  the  etale 
topology  over  the  parametrizing  scheme  B)  isomorphic.  In  the  case  of  vector  bundles,  this 
is  the  reason  why  one  usually  considers  two  vector  bundles  V  and  V'  on  X  x  B  equivalent 
if  V  =  V  <8>  p^L  for  some  line  bundle  L  on  B.  The  functor  obtained  with  this  equivalence 
relation  is  denoted  M_x  (and  analogously  for  M?x  and  M£). 

Note  that  if  two  families  V  and  V'  are  equivalent  in  this  sense,  then  they  are  locally 
isomorphic.  The  converse  is  only  true  if  the  vector  bundles  are  simple  (only  automor- 
phisms are  scalar  multiplications).  This  will  happen,  for  instance,  if  we  are  considering 
the  functor  M^  of  stable  vector  bundles,  since  stable  vector  bundles  are  simple.  In  general, 
if  we  want  the  functor  to  be  a  sheaf,  we  have  to  use  a  weaker  notion  of  equivalence,  but 
this  is  not  done  because  for  other  reasons  there  is  only  hope  of  obtaining  a  fine  moduli 
space  if  we  restrict  our  attention  to  stable  vector  bundles. 

Once  this  modification  is  made,  there  are  some  situations  in  which  these  examples  are 
representable  (for  instance,  stable  vector  bundles  on  curves  with  coprime  rank  and 
degree),  but  in  general  they  will  still  not  be  representable,  because  in  general  we  do  not 
have  a  universal  family: 

DEFINITION  2.3  (Universal  family) 

Let  F  be  a  representable  functor,  and  let  <p  :  F  —*  Horns  (— ,X)  be  the  isomorphism.  The 
object  of  F(X)  corresponding  to  the  element  idx  of  Horns  (X,X)  is  called  the  universal 
family. 

Example  2.4  (Vector  bundles).  If  V  is  a  universal  vector  bundle  (over  M  x  X,  where  M  is 
the  fine  moduli  space),  it  has  the  property  that  for  any  family  W  of  vector  bundles  (i.e.  W 
is  a  vector  bundle  over  B  x  X  for  some  parameter  scheme  B)  there  exists  a  morphism 
/  :  B  — »  M  such  that  (/  x  idj)*V  is  equivalent  to  W. 

In  other  words,  the  functor  M%  is  represented  by  the  scheme  M  iff  there  exists  a 
universal  vector  bundle  on  M  x  X. 

When  a  moduli  functor  F  is  not  representable  and  then  there  is  no  scheme  X  whose 
functor  of  points  is  isomorphic  to  F,  one  can  still  try  to  find  a  scheme  X  whose  functor  of 
points  is  an  approximation  to  F  in  some  sense.  There  are  two  different  notions: 

DEFINITION  2:5  (Corepresents)  ([S],  p.  60),  ([HL],  Definition  2.2.1) 

We  say  that  a  scheme  M  corepresents  the  functor  F  if  there  is  a  natural  transformation  of 
functors  (f> :  F  — >•  Horns  (—,M)  such  that 

•  Given  another  scheme  N  and  a  natural  transformation  i/j :  F  — >  Homs(— ,  N),  there  is  a 
unique  natural  transformation  77 :  Homs(— ,M)  — >  Homs(— , N)  with  ^  =  77  o  <p. 

F,^ 

^       > 


Hom5(-,M) 


This  characterizes  M  up  to  unique  isomorphism.  Let  (Sch/S)r  be  the  functor  category, 
whose  objects  are  contravariant  functors  from  (Sch/S)  to  (Sets)  and  whose  morphisms 


6  Tomds  L  Gomez 

are  natural  transformation  of  functors.  Then  M  represents  F  iff  Horns ( 7,  M)  = 
Hom(Sch/iSy(y,F)  for  all  schemes  7,  where  y  is  the  functor  represented  by  Y.  On  the 
other  hand,  one  can  check  that  M  corepresents  F  iff  Hom$(M,  Y)  =  Horri(Sch/5y  (F,  y)  for 
all  schemes  Y.  If  M  represents  F,  then  it  corepresents  it,  but  the  converse  is  not  true.  From 
now  on  we  will  denote  a  scheme  and  the  functor  that  it  represents  by  the  same  letter. 

DEFINITION  2.6  (Coarse  moduli) 

A  scheme  M  is  called  a  coarse  moduli  scheme  if  it  corepresents  F  and  furthermore 

•  for  any  algebraically  closed  field  fc,  the  map  $(fc)  :  F(Specfc)  —» Hom$(SpecA:,M)  is 
bijective. 

If  M  corepresents  F  (in  particular,  if  M  is  a  coarse  moduli  space),  given  a  family  of 
objects  parametrized  by  B  we  get  a  morphism  from  B  to  M,  but  we  don't  require  the 
converse  to  be  true,  i.e.  not  all  morphisms  are  induced  by  families. 

Example  2.7  (Vector  bundles).  There  is  a  scheme  Afff  that  corepresents  Mf  (see  [HL]).  It 
fails  to  be  a  coarse  moduli  scheme  because  its  closed  points  are  in  one  to  one 
correspondence  with  ^-equivalence  classes  of  vector  bundles,  and  not  with  isomorphism 
classes  of  vector  bundles.  Of  course,  this  can  be  solved  'by  hand'  by  modifying  the 
functor  and  considering  two  vector  bundles  equivalent  if  they  are  S-equivalent.  Once  this 
modification  is  done,  M£  is  a  coarse  moduli  space. 

But  in  general  Mf  doesn't  represent  the  moduli  functor  Mf .  The  reason  for  this  is  that 
vector  bundles  have  always  nontrivial  automorphisms  (multiplication  by  scalar),  but  the 
moduli  functor  does  not  record  information  about  automorphisms:  recall  that  to  a  scheme 
B  it  associates  just  the  set  of  equivalence  classes  of  vector  bundles.  To  record  the 
automorphisms  of  these  vector  bundles,  we  define 

MX  :     (Sch/5)     — »     (groupoids) 
B         .— >       MX(B), 

where  Mx(B)  is  the  category  whose  objects  are  vector  bundles  V  on  B  x  X  of  rank  r 
and  with  fixed  Chern  classes  (note  that  the  objects  are  vector  bundles,  not  isomor- 
phism classes  of  vector  bundles),  and  whose  morphisms  are  vector  bundle  isomorphisms 
(note  that  we  use  isomorphisms  of  vector  bundles,  not  5-equivalence  nor  equivalence 
classes  as  before).  This  defines  a  2-functor  between  the  2-category  associated  to  (Sch/5) 
and  the  2-category  (groupoids)  (for  the  definition  of  2-categories  and  2-functors,  see 
Appendix  B). 

DEFINITION  2.8 

Let  (groupoids)  be  the  2-category  whose  objects  are  groupoids,  1 -morphisms  are  functors 
between  groupoids,  and  2-morphisms  are  natural  transformation  between  these  functors. 
A  presheaf  in  groupoids  (also  called  quasi-functor)  is  a  contravariant  2-functor  F  from 
(Sch/5)  to  (groupoids).  For  each  scheme  B  we  have  a  groupoid  F(B)  and  for  each 
morphism  /  :  Bf  -+  B  we  have  a  functor  JF(/)  :  F(S)  ->  F(ff)  that  is  denoted  by  /* 
(usually  it  is  actually  defined  by  a  pull-back). 

Example  2.9  (Vector  bundles)  ([La],  1.3.4).  MX  is  a  presheaf.  For  each  object  B  of 
(Sch/5)  it  gives  the  groupoid  MX(B)  defined  in  Example  2.7.  For  each  1-morphism 


Algebraic  stacks  7 

f  :  Bf  ->  B  it  gives  the  functor  F(f)  =  /*  :  MX(B)  ->  MX(B')  given  by  pull-back,  and 
for  every  diagram 

Bff^Bf-^B  (2) 

it  gives  a  natural  transformation  of  functors  (a  2-isomorphism)  cgj  :  g*  o  /*  —  »  (fog)*. 
This  is  the  only  subtle  point.  First  recall  that  the  pullback  /*  V  of  a  vector  bundle  (or  more 
generally,  any  fiber  product)  is  not  uniquely  defined:  it  is  only  defined  up  to  unique 
isomorphism.  First  choose  once  and  for  all  a  pullback  f*V  for  each  /  and  V.  Then,  given 
a  diagram  like  2,  in  principle  g*(f*V)  and  (/  o  g)*  V  are  not  the  same,  but  (because  both 
solve  the  same  universal  problem)  there  is  a  canonical  isomorphism  (the  unique 
isomorphism  of  the  universal  problem)  g*(f*V)  —  »  (f  o  g)*V  between  them,  and  this 
defines  the  natural  transformation  of  functors  egj  :  g*  o  /*  —  >  (/  o  #)*.  By  a  slight  abuse 
of  language,  usually  we  will  not  write  explicitly  these  isomorphisms  e^/,  and  we  will 
write  g*  o  f*  =  (f  o  g)*.  Since  they  are  uniquely  defined  this  will  cause  no  ambiguity. 

Example  2.10  (Stable  curves)  ([DM],  Definition  1.1).  Let  B  be  an  S-scheme.  Let  g  >  2. 
A  stable  curve  of  genus  g  over  B  is  a  proper  and  flat  morphism  TT  :  C  —  *  B  whose 
geometric  fibers  are  reduced,  connected  and  one-dimensional  schemes  Q,  such  that 

1.  The  only  singularities  of  Q,  are  ordinary  double  points. 

2.  If  E  is  a  non-singular  rational  component  of  Q,,  then  E  meets  the  other  components  of 
Cb  in  at  least  3  points. 

3. 


Condition  2  is  imposed  so  that  the  automorphism  group  of  Q,  is  finite.  A  stable  curve 
over  B  should  be  thought  of  as  a  family  of  stable  curves  (over  5)  parametrized  by  B. 

For  each  object  B  of  (Sch/S),  let  Mg(B)  be  the  groupoid  whose  objects  are  stable 
curves  over  B  and  whose  (iso)morphisms  are  Cartesian  diagrams 


W  ^~    JTL. 

\         \ 

B~B 


For_each  mjorphism  /  :  B'  -»  B  of  (Sch/S),  we  define  the  pullback  functor 
/*  :  Mg(B)  ->  Mg(B')>  sending  an  object  X  ->  B  to  f*X  -*  Bf  (and  a  morphism 
(p  :  X\  -»  X2  of  curves  over  B  to  f*(p  '-f*Xi  —  >/*X2).  A™1  finally*  for  each  diagram 


we  have  to  give  a  natural  transformation  of  functors  (i.e.  a  2-isomorphism  in  (groupoids)) 
%/  :  £*  °  /*  ">  (/  °  #)*•  As  in  the  case  of  vector  bundles,  this  is  defined  by  first  choosing 
once  an  for  all  a  pullback  f*X  for  each  curve  X  and  morphism  /,  and  then  egj  is  given  by 
the  canonical  isomorphism  between  g*(/*X)  and  (fog)*X.  Since  this  isomorphism  is 
canonical,  by  a  slight  abuse  of  language  we  usually  write  g*  o  /*  =  (fog)*. 

Now  we  will  define  the  concept  of  stack.  First  we  have  to  choose  a  Grothendieck 
topology  on  (Sch/S),  either  the  etale  or  the  fppf  topology.  Later  on,  when  we  define 
algebraic  stack,  the  etale  topology  will  lead  to  the  definition  of  a  Deligne-Mumford  stack 
([DM,  Vi,  E]),  and  the  fppf  to  an  Artin  stack  ([La]).  For  the  moment  we  will  give  a  unified 
description. 


8  Tomds  L  Gomez 

In  the  following  definition,  to  simplify  notation  we  denote  by  XI;  the  pullback  f*X 
where  ff  :  Ui  -»  U  and  X  is  an  object  of  F(U),  and  by  X/  tj  the  pullback  /^X/  where 
ftj,i  '.  Ut  Xu  Uj  — >  £/i  andX;  is  an  object  of  J-(Ui).  We  will  also  use  the  obvious  variations 
of  this  convention,  and  will  simplify  the  notation  using  Remark  5.3. 

DEFINITION  2. 1 1  (Stack) 

A  stack  is  a  sheaf  of  groupoids,  i.e.  a  2-functor  (=  presheaf)  that  satisfies  the  following 
sheaf  axioms.  Let  {£//  -+  U}ieJ  be  a  covering  of  U  in  the  site  (Sch/5).  Then 

1.  Glueing  of  morphisms.  If  X  and  Y  are  two  objects  of  f(U)9  and  &  :  X\t  — >  Y\t  are 
morphisms  such  that  (pi\^  =  c^/l^-,  then  there  exists  a  morphism  77 :  X  -»  7  such  that 

'nli  =  <#• 

2.  Monopresheaf.  If  X  and  F  are  two  objects  of  f(U)9  and  (p  :  X  — >  F,  -0  :  X  —»  F  are 
morphisms  such  that  (p\t  =  ^\-9  then  (p  —  ?/>. 

3.  Glueing  of  objects.  If  X;  are  objects  of  F(Ui)  and  </?//  :  X;-|^-  — >  X/L  are  morphisms 
satisfying  the  cocycle  condition  vfy'li./*  °  ^lo*  =  ^*lo'k'  t^ien  ^ere  exists  an  object  X 

of  /"(£/)  and  c^j  :  X|z-  -^X/  such  that  </?#  o  ^|. .  =  <p;-  tj. 

At  first  sight  this  might  seem  very  complicated,  but  if  we  check  in  a  particular  example 
we  will  see  that  it  is  a  very  natural  definition: 

Example  2.12  (Stable  curves).  It  is  easy  to  check  that  the  presheaf  Mg  defined  in  2.10  is 
a  stack  (all  properties  hold  because  of  descent  theory).  We  take  the  etale  topology  on 
(Sch/51)  (we  will  see  that  the  reason  for  this  is  that  the  automorphism  group  of  a  stable 
curve  is  finite).  Let  {[//  -+  U}i€J  be  a  cover  of  U.  Item  1  says  that  if  we  have  two  curves 
X  and  Y  over  U,  and  we  have  isomorphisms  <#  :  X|-  ->  Y\t  on  the  restriction  for  each  £/,-, 
then  these  isomorphisms  glue  to  give  an  isomorphism  77 :  X  — >  Y  over  U  if  the  restrictions 
to  the  intersections  y?;|2..  and  <p7-|. .  coincide. 

Item  2  says  that  two  morphisms  of  curves  over  U  coincide  if  the  restrictions  to  all  I/,- 
coincide. 

Finally,  item  3  says  that  if  we  have  curves  Xi  over  Ui  and  we  are  given  isomorphisms 
iptj  over  the  intersections  U^  then  we  can  glue  the  curves  to  get  a  curve  over  U  if  the 
isomorphisms  satisfy  the  cocycle  condition. 

Example  2.13  (Vector  bundles).  It  is  also  easy  to  check  that  the  presheaf  of  vector 
bundles  MX  is  a  sheaf.  In  this  case  we  take  the  fppf  topology  on  (Sch/S)  (we  will  see  that 
the  reason  for  this  choice  is  that  the  automorphism  group  of  a  vector  bundle  is  not  finite, 
because  it  includes  multiplication  by  scalars). 

Let  us  stop  for  a  moment  and  look  at  how  we  have  enlarged  the  category  of  schemes  by 
defining  the  category  of  stacks.  We  can  draw  the  following  diagram 

Algebraic  Stacks  — ^Stacks  — » Presheaves  of  groupoids 

/  T  T  T 

Sch/5 — *  Algebraic  Spaces — ^Spaces — ^Presheaves  of  sets 

where  A  -»  B  means  that  the  category  A  is  a  subcategory  B.  Recall  that  a  presheaf  of  sets 
is  just  a  functor  from  (Sch/5)  to  the  category  (Sets),  a  presheaf  of  groupoids  is  just  a  2- 
functor  to  the  2-category  (groupoids).  A  sheaf  (for  example  a  space  or  a  stack)  is  a 


Algebraic  stacks  9 

presheaf  that  satisfies  the  sheaf  axioms  (these  axioms  are  slightly  different  in  the  context 
of  categories  or  2-categories),  and  if  this  sheaf  satisfies  some  geometric  conditions  (that 
we  have  not  yet  specified),  we  will  have  an  algebraic  stack  or  algebraic  space. 

2.2  Stacks  as  categories:  Groupoids 

There  is  an  alternative  way  of  defining  a  stack.  From  this  point  of  view  a  stack  will  be  a 
category,  instead  of  a  functor. 

DEFINITION  2.14 

A  category  over  (Sch/5)  is  a  category  T  and  a  covariant  functor  p-p  :  T  — >  (Sch/5) 
(called  the  structure  functor).  If  X  is  an  object  (resp.  0  is  a  morphism)  of  T,  and 
pf(X)  =  B  (resp.  pp((t>)  =  /),  then  we  say  that  X  lies  over  B  (resp.  <j>  lies  over  /). 

DEFINITION  2.15  (Groupoid) 

A  category  f  over  (Sch/5)  is  called  a  category  fibered  on  groupoids  (or  just  groupoid)  if 

1.  For  every  /  :  Bf  — >  B  in  (Sch/5)  and  every  object  X  with  p?(X)  =  B9  there  exists  at 
least  one  object  Xe  and  a  morphism  </> :  X'  — >  X  such  that  p^(Xr)  =  Bf  and p^((f>)  =  /. 


X 


u 

rw •* 


2.  For  every  diagram 


(where  p^(Xi]  =  #/,  pr(</>)  =  /, 
with  i/j  =  <f)  o  y?  and  PF(V)  =  f. 


=  /  o  /'),  there  exists  a  unique  < 


Condition  2  implies  that  the  object  Xf  whose  existence  is  asserted  in  condition  1  is 
unique  up  to  canonical  isomorphism.  For  each  X  and  /  we  choose  once  and  for  all  such 
an  X'  and  call  it  f*X.  Another  consequence  of  condition  2  is  that  </>  is  an  isomorphism  if 
and  only  if  p?((j))  =  /  is  an  isomorphism. 

Let  B  be  an  object  of  (Sch/5).  We  define  F(B),  the  fiber  of  f  over  B,  to  be  the  sub- 
category  of  F  whose  objects  lie  over  B  and  whose  morphisms  lie  over  id#.  It  is  a  groupoid. 

The  association  B  — >  ^(B)  in  fact  defines  a  presheaf  of  groupoids  (note  that  the  2- 
isomorphisms  e/^  required  in  the  definition  of  presheaf  of  groupoids  are  well  defined 
thanks  to  condition  2).  Conversely,  given  a  presheaf  of  groupoids  Q  on  (Sch/5),  we  can 
define  the  category  T  whose  objects  are  pairs  (B,  X)  where  B  is  an  object  of  (Sch/5)  and 


10  Tomds  L  Gomez 

X  is  an  object  of  Q(B),  and  whose  morphisms  (ff,X*)  -»  (B,X)  are  pairs  (/,  a)  where 
/  :  Bf  — >  B  is  a  morphism  in  (Sch/5)  and  a  :  f*X  —>  Xf  is  an  isomorphism,  where 
f*  =  <?(/)•  This  gives  the  relationship  between  both  points  of  view.  Since  we  have  a 
canonical  one-to-one  relationship  between  presheaves  of  groupoids  and  groupoids  over  5, 
by  a  slight  abuse  of  language,  we  denote  both  by  the  same  letter. 

Example  2.16  (Vector  bundles).  The  groupoid  of  vector  bundles  MX  on  a  scheme  X  is 
the  category  whose  objects  are  vector  bundles  over  B  x  X  (for  B  a  scheme),  and  whose 
morphisms  are  isomorphisms 


where  V  (resp.  V)  is  a  vector  bundle  over  B  x  X  (resp.  B'  x  X)  and  f :  B'  ->  B  is  a 
morphism  of  schemes.  The  structure  functor  sends  a  vector  bundle  over  B  x  X  to  the 
scheme  B,  and  a  morphism  <p  to  the  corresponding  morphism  of  schemes  /. 

Example  2.17  (Stable  curves)  ([DM],  Definition  1.1).  We  define  Alg,  the  groupoid  over  S 
whose  objects  are  stable  curves  over  B  of  genus  g  (see  Definition  2.10),  and  whose 
morphisms  are  Cartesian  diagrams 

v/  ^   V 

Ui 

The  structure  functor  sends  a  curve  over  B  to  the  scheme  B,  and  a  morphism  as  in  (3)  to  /. 

Example  2.18  (Quotient  by  group  action)  ([La],  1,3.2),  ([DM],  Example  4.8),  ([E], 
Example  2.2).  Let  X  be  an  S-scheme  (assume  all  schemes  are  Noetherian),  and  G  an 
affme  flat  group  5-scheme  acting  on  the  right  on  X.  We  define  the  groupoid  [X/G]  whose 
objects  are  principal  G-bundles  TT  :  E  — *  B  together  with  a  G-equivariant  morphism 
/:£—>•  X.  A  morphism  is  Cartesian  diagram 

E»    *>  »  E 

"I  .  *1  (4) 

such  that  /op=/'. 

The  structure  functor  sends  an  object  (TT  :  E  — >  #,/  :  E  -»  X)  to  the  scheme  B,  and  a 
morphism  as  in  (4)  to  g. 

DEFINITION  2.19  (Stack) 

A  stack  is  a  groupoid  that  satisfies 

1.  (Prestack).  For  all  scheme  B  and  pair  of  objects  X,  Y  of  T  over  jB,  the  contravariant 
functor 

Iso*(X,F):         (Sch/B)         — »  (Sets) 


is  a  sheaf  on  the  site  (Sch/B). 
2.  Descent  data  is  effective  (this  is  just  condition  3  in  the  Definition  2.11  of  sheaf). 


Algebraic  stacks  1 1 

Example  2.20.  If  G  is  smooth  and  affine,  the  groupoidJX/G]  is  a  stack  ([La],  2.4.2),  ; 

([Vi],  Example  7.17),  ([E],  Proposition  2.2).  Then  also  Mg  (cf.  Example  2.17)  is  a  stack,  ; 

because  it  is  isomorphic  to  a  quotient  stack  of  a  subscheme  of  a  Hilbert  scheme  by  j 

PGL(N)  ([E],  Theorem  3.2),  [DM].  The  groupoid  MX  defined  in  Example  2.16  is  also  a  I 

stack  ([La],  2.4.4).  [f 

From  now  on  we  will  mainly  use  this  approach.  Now  we  will  give  some  definitions  for  p , 

stacks.  !, 


Morphisms  of  stacks.  A  morphism  of  stacks  /  :  F  — >  Q  is  a  functor  between  the  cate- 
gories, such  that  pg  o  /  =  pp.  A  commutative  diagram  of  stacks  is  a  diagram 


such  that  a  :  g  o  /  —  >  h  is  an  isomorphism  of  functors.  If  /  is  an  equivalence  of  cate- 
gories, then  we  say  that  the  stacks  F  and  Q  are  isomorphic.  We  denote  by  Horns  (.T7,  Q) 
the  category  whose  objects  are  morphisms  of  stacks  and  whose  morphisms  are  natural 
transformations  . 

Stack  associated  to  a  scheme.  Given  a  scheme  U  over  5,  consider  the  category  (Sch/U). 
Define  the  functor  pv  :  (Sch/U)  ->  (Sch/S)  which  sends  the  (/-scheme  /  :  B  -»  U  to  the 
composition  B  —  >  U  —>  5,  and  sends  the  £/-morphism  (Bf  —  »  17)  —  >  (B  -»  U)  to  the  5- 
morphism  (Bf  —  >  5)  ->  (B  -»  5).  Then  (Sch/U)  becomes  a  stack.  Usually  we  denote  this 
stack  also  by  U.  From  the  point  of  view  of  2-functors,  the  stack  associated  to  U  is  the  2- 
functor  that  for  each  scheme  B  gives  the  category  whose  objects  are  the  elements  of  the 
set  Hom$(#,  U),  and  whose  only  morphisms  are  identities. 

We  say  that  a  stack  is  represented  by  a  scheme  U  when  it  is  isomorphic  to  the  stack 
associated  to  U.  We  have  the  following  very  useful  lemmas: 

Lemma  2.21.  If  a  stack  has  an  object  with  an  automorphism  other  that  the  identity,  then 
the  stack  cannot  be  represented  by  a  scheme. 

Proof.  In  the  definition  of  stack  associated  with  a  scheme  we  see  that  the  only  auto- 
morphisms are  identities.  D 

Lemma  2.22  ([Vi],  7.10).  Let  F  be  a  stack  and  U  a  scheme.  The  functor 


that  sends  a  morphism  of  stacks  f  :  U  —  >  J-  t  o  f(idu)  is  an  equivalence  of  categories. 

Proof.  Follows  from  Yoneda  lemma.  D 

This  useful  observation  that  we  will  use  very  often  means  that  an  object  of  .F'that  lies 
over  U  is  equivalent  to  a  morphism  (of  stacks)  from  U  to  T. 

Fiber  product.  Given  two  morphisms  f\  :  f\  -»  Q,  /2  :  ^"2  -+  G,  we  define  a  new  stack 
f\^Q^2  (with  projections  to  T\  and  J"2)  a&  follows.  The  objects  are  triples  (Xi,X2,  a) 
where  X\  and  X2  are  objects  of  'T\  and  TI  that  lie  over  the  same  scheme  U,  and 
a  :fi(Xi)  -+fi(X2)  is  an  isomorphism  in  G  (equivalently,  pg(a)  =  id^/).  A  morphism 


R 


12  Tomds  L  Gomez 

from  (Xi  ,X2,a)  to  (Y\  ,  Y2,  0)  is  a  pair  (<£i  ,  <fe)  of  morphisms  <£/  :  X,-  ->  ft  that  lie  over 
the  same  morphism  of  schemes  f  :  U  —*  V,  and  such  that  /?  o  /i(0i)  =/2(</>2)  °  #•  The 
fiber  product  satisfies  the  usual  universal  property. 

Representability.  A  stack  X  is  said  to  be  representable  by  an  algebraic  space  (resp. 
scheme)  if  there  is  an  algebraic  space  (resp.  scheme)  X  such  that  the  stack  associated  to  X 
is  isomorphic  to  X.  If  T'  is  a  property  of  algebraic  spaces  (resp.  schemes)  and  X  is  a 
representable  stack,  we  will  say  that  X  has  'P'  iff  X  has  'P'. 

A  morphism  of  stacks  /  :  T  —  »  £7  is  said  to  be  representable  if  for  all  objects  U  in 
(Sch/S)  and  morphisms  U  -*  5,  the  fiber  product  stack  U  XgF  is  representable  by  an 
algebraic  space.  Let  P  be  a  property  of  morphisms  of  schemes  that  is  local  in  nature  on  the 
target  for  the  topology  chosen  on  (Sch/S)  (etale  or  fppf),  and  it  is  stable  under  arbitrary 
base  change.  For  instance:  separated,  quasi-compact,  unramified,  flat,  smooth,  etale,  sur- 
jective,  finite  type,  locally  of  finite  type,  ____  Then,  for  a  representable  morphism  /,  we  say 
that  /  has  P  if  for  every  £7  ->  ft  the  puUback  U  xg  f  -+  U  has  P  ([La],  p.  17,  [DM],  p.  98). 

Diagonal.  Let  Ajr  :  f  —  >  f  xs  T  be  the  obvious  diagonal  morphism.  A  morphism  from 
a  scheme  U  to  f  xs  T  is  equivalent  to  two  objects  X\,  Xi  of  F(U).  Taking  the  fiber 
product  of  these  we  have 


- 


hence  the  group  of  automorphisms  of  an  object  is  encoded  in  the  diagonal  morphism. 

PROPOSITION  2.23  ([La],  Corollary  2.12),  ([Vi],  Proposition  7.13) 
The  following  are  equivalent 

1.  The  morphism  Ajr  is  representable. 

2.  The  stack  Isoj/pfi,^)  is  representable  for  all  U,  X\  and  X^. 

3.  For  all  scheme  U,  every  morphism  U  -+  f  is  representable. 

4.  For  all  schemes  U,  V  and  morphisms  U  -»  T  and  V  —  >  T,  the  fiber  product  U  x?V 
is  representable. 

Proof.  The  implications  1  <£>  2  and  3  <S>  4  follow  easily  from  the  definitions. 

(1  =^  4)  Assume  that  A^  is  representable.  We  have  to  show  that  U  XJT  V  is  representable 

for  any  /  :  U  -»  J7  and  ^  :  V  ->  J7.  Check  that  the  following  diagram  is  Cartesian 


Then  [/  x^  V  is  representable. 

(1  4=  4)  First  note  that  the  Cartesian  diagram  defined  by  h:  U  -+  F  xsF  and 

factors  as  follows: 


Algebraic  stacks  13 

The  outer  (big)  rectangle  and  the  right  square  are  Cartesian,  so  the  left  square  is  also 
Cartesian.  By  hypothesis  U  x^  U  is  representable,  then  U  XFXSF  f  is  also 
representable.  D 

2.3  Algebraic  stacks 

Now  we  will  define  the  notion  of  algebraic  stack.  As  we  have  said,  first  we  have  to  choose 
a  topology  on  (Sch/S).  Depending  of  whether  we  choose  the  etale  or  fppf  topology,  we 
get  different  notions. 

DEFINITION  2.24  (Deligne-Mumford  stack) 

Let  (Sch/5)  be  the  category  of  S-schemes  with  the  etale  topology.  Let  f  be  a  stack.  Assume 

1.  Quasi-separatedness.  The  diagonal  A^-  is  representable,  quasi-compact  and  separated. 

2.  There  exists  a  scheme  U  (called  atlas)  and  an  etale  surjective  morphism  u  :  U  — >  T . 

Then  we  say  that  T  is  a  Deligne-Mumford  stack. 

The  morphism  of  stacks  u  is  representable  because  of  Proposition  2.23  and  the  fact  that 
the  diagonal  A^  is  representable.  Then  the  notion  of  etale  is  well  defined  for  u.  In  [DM] 
this  was  called  an  algebraic  stack.  In  the  literature,  algebraic  stack  usually  refers  to  Artin 
stack  (that  we  will  define  later).  To  avoid  confusion,  we  will  use  'algebraic  stack'  only 
when  we  refer  in  general  to  both  notions,  and  we  will  use  'Deligne-Mumford'  or  'Artin' 
stack  when  we  want  to  be  specific. 

Note  that  the  definition  of  Deligne-Mumford  stack  is  the  same  as  the  definition  of 
algebraic  space,  but  in  the  context  of  stacks  instead  of  spaces.  Following  the  terminology 
used  in  scheme  theory,  a  stack  such  that  the  diagonal  A^-  is  quasi-compact  and  separated 
is  called  quasi-separated.  We  always  assume  this  technical  condition,  as  it  is  usually  done 
both  with  schemes  and  algebraic  spaces. 

Sometimes  it  is  difficult  to  find  explicitly  an  etale  atlas,  and  the  following  proposition 
is  useful. 

PROPOSITION  2.25  ([DM],  Theorem  4.21),  [E] 
Let  jF  be  a  stack  over  the  etale  site  (Sch/S).  Assume 

1 .  The  diagonal  A^-  is  representable,  quasi-compact,  separated  and  unramified. 

2.  There  exists  a  scheme  U  of  finite  type  over  S  and  a  smooth  surjective  morphism 
u-.U-^F. 

Then  T  is  a  Deligne-Mumford  stack. 
Now  we  define  the  analog  for  the  fppf  topology  [Ar2]. 

DEFINITION  2.26  (Artin  stack) 

Let  (Sch/S)  be  the  category  of  5-schemes  with  the  fppf  topology.  Let  T  be  a  stack. 
Assume 

1.  Quasi-separatedness.  The  diagonal  A^  is  representable,  quasi-compact  and  separated. 

2.  There  exists  a  scheme  U  (called  atlas)  and  a  smooth  (hence  locally  of  finite  type)  and 
surjective  morphism  u  :  U  — >  F. 

Then  we  say  that  T  is  an  Artin  stack. 


1  14  Tomds  L  Gomez 

For  propositions  analogous  to  proposition  2.25,  see  [La,  4]. 


PROPOSITION  2.27  ([Vi],  Proposition  7.15),  ([La],  Lemma  3.3) 

If  jF  is  a  Deligne-Mumford  (resp.  Artin)  stack,  then  the  diagonal  A^-  is  unramified  (resp. 
finite  type). 

Recall  that  Ajr  is  unramified  (resp.  finite  type)  if  for  every  scheme  B  and  objects  X9  Y 
of  F(B),  the  morphism  Iso5(X,  Y)  —  >  B  is  unramified  (resp.  finite  type).  If  B  =  Spec  S 
and  X  =  F,  then  this  means  that  the  automorphism  group  of  X  is  discrete  and  reduced  for 
a  Deligne-Mumford  stack,  and  it  is  of  finite  type  for  an  Artin  stack. 

Example  2.28  (Vector  bundles).  The  stack  MX  is  an  Artin  stack,  locally  of  finite  type 
([La],  4.14.2.1).  The  atlas  is  constructed  as  follows:  Let  P^c.  be  the  Hilbert  polynomial 
corresponding  to  locally  free  sheaves  on  X  with  rank  r  and  Chern  classes  c,-.  Let  Quot 
(O(-m)®N,pHc.)  be  the  Quot  scheme  parametrizing  quotients  of  sheaves  on  X, 

0(-mfN  -*  V,  (5) 

where  V  is  a  coherent  sheaf  on  X  with  Hilbert  polynomial  P^c..  Let  RN^m  be  the  sub- 
scheme  corresponding  to  quotients  (5)  such  that  V  is  a  vector  bundle  with  Hp(V(m))  =  0 
for  p  >  0  and  the  morphism  (5)  induces  an  isomorphism  on  global  sections 


The  scheme  RN^  has  a  universal  vector  bundle,  induced  from  the  universal  bundle  of  the 
Quot  scheme,  and  then  there  is  a  morphism  M#jm  :  /?#j/n  —  >•  MX-  Since  //  is  ample,  for 
every  vector  bundle  V,  there  exist  integers  N  and  m  such  that  /?Ar>m  has  a  point  whose 
corresponding  quotient  is  V,  and  then  if  we  take  the  infinite  disjoint  union  of  these 
morphisms  we  get  a  surjective  morphism 


It  can  be  shown  that  this  morphism  is  smooth,  and  then  it  gives  an  atlas.  Each  scheme 
RNim  is  of  finite  type,  so  the  union  is  locally  of  finite  type,  which  in  turn  implies  that  the 
stack  MX  is  locally  of  finite  type. 

Example  2.29  (Quotient  by  group  action).  The  stack  [X/G]  is  an  Artin  stack  ([La], 
4.14.1.1).  If  G  is  smooth,  an  atlas  is  defined  as  follows  (for  more  general  G,  see  ([La], 
4.14.1.1)):  Take  the  trivial  principal  G-bundle  X  x  G  over  X,  and  let  the  map 
/  :  X  x  G  —  »>  X  be  the  action  of  the  group.  This  defines  an  object  of  [X/G](X)9  and  by 
Lemma  2.22,  it  defines  a  morphism  u  :  X  -*  [X/G].  It  is  representable,  because  if  B  is  a 
scheme  and  g  :  B  —  >  [X/G]  is  the  morphism  corresponding  to  a  principal  G-bundle  E  over 
B  with  an  equivariant  morphism  /  :  E  -~+  X,  then  B  X[X/G]  X  is  isomorphic  to  the  scheme 
E,  and  in  fact  we  have  a  Cartesian  diagram 


The  morphism  u  is  surjective  and  smooth  because  TT  is  surjective  and  smooth  for  every  g 
(if  G  is  not  smooth,  but  only  separated,  flat  and  of  finite  presentation,  then  u  is  not  an 


Algebraic  stacks  15 

atlas,  but  if  we  apply  Artin's  theorem  ([Ar2],  Theorem  6.1),  ([La],  Theorem  4.1),  we 
conclude  that  there  is  a  smooth  atlas). 

If  either  G  is  etale  over  S  ([DM],  Example  4.8)  or  the  stabilizers  of  the  geometric 
points  of  X  are  finite  and  reduced  ([VI],  Example  7.17),  then  [X/G]  is  a  Deligne-Mumford 
stack. 

Note  that  if  the  action  is  not  free,  then  [X/G]  is  not  representable  by  Lemma  2.21.  On 
the  other  hand,  if  there  is  a  scheme  Y  such  that  X  —  >  Y  is  a  principal  G-bundle,  then  [X/G] 
is  represented  by  Y. 

Let  G  be  a  reductive  group  acting  on  X.  Let  H  be  an  ample  line  bundle  on  X,  and 
assume  that  the  action  is  polarized.  Let  Xs  and  Xss  be  the  subschemes  of  stable  and 
semistable  points.  Let  Y  =  X//G  be  the  GIT  quotient.  Recall  that  there  is  a  good  quotient 
Xss  —  *  7,  and  that  the  restriction  to  the  stable  part  Xs  —  >  Y  is  a  principal  bundle.  There  is  a 
natural  morphism  [XSS/G]  —  >  XSS//G.  By  the  previous  remark,  the  restriction  [X*/G]  —  » 
F  is  an  isomorphism  of  stacks. 

If  X  =  5  (with  trivial  action  of  G  on  5),  then  [S/G]  is  denoted  BG,  the  classifying 
groupoid  of  principal  G-bundles. 

Example  2.30  (Stable  curves).  The  stack  ^Mg  is  a  Deligne-Mumford  stack  ([DM], 
Proposition  5.1),  [E].  The  idea  of  the  proof  is  to  show  that  Mg  is  the  quotient  stack 
\Hg/PGL(N)}  of  a  scheme  H8  by  a  smooth  group  PGL(N).  This  gives  a  smooth  atlas. 
Then  one  shows  that  the  diagonal  is  unramified,  and  finally  we  apply  Proposition  2.25. 

2.4  Algebraic  stacks  as  groupoid  spaces 

We  will  introduce  a  third  equivalent  definition  of  stack.  First  consider  a  category  C.  Let  U 
be  the  set  of  objects  and  R  the  set  of  morphisms.  The  axioms  of  a  category  give  us  four 
maps  of  sets 

R=tU-^R    Rxs^tR-^R, 

where  5-  and  t  give  the  source  and  target  for  each  morphism,  e  gives  the  identity  mor- 
phism, and  m  is  composition  of  morphisms.  If  the  category  is  a  groupoid  then  we  have  a 
fifth  morphism 


that  gives  the  inverse.  These  maps  satisfy 


.  soe  =  to  e  =     t 

2.  Associativity,  m  o  (m  x  id/?)  =  m  o  (id/?  x  m). 

3.  Identity.  Both  compositions 

idRxe 

R  =  R  x   v  U=UXujR  —  —  t  R  X     UitR-3i+R 

exidR 

are  equal  to  the  identity  map  on  R. 

4.  Inverse.  mo(ix  id/?)  =  e  o  s,  m  o  (id/?  x  i)  =  e  o  f  . 

DEFINITION  2.31  (Groupoid  space)  ([La],  1.3.3),  ([DM],  pp.  668-669) 

A  groupoid  space  is  a  pair  of  spaces  (sheaves  of  sets)  U,  R,  with  five  morphisms  5,  f,  e,  m, 
i  with  the  same  properties  as  above. 


16  Tomds  L  Gomez 

DEFINITION  2.32  ([La],  1.3.3). 

Given  a  groupoid  space,  define  the  groupoid  over  (Sch/5)  as  the  category  [P,  U}'  over 
(Sch/5)  whose  objects  over  the  scheme  B  are  elements  of  the  set  U(B)  and  whose 
morphisms  over  B  are  elements  of  the  set  R(B).  Given  /  :  B1  — •>  B  we  define  a  functor 
/*  :  [R,  U]'(B)  ->  [R,  U]'(ff)  using  the  maps  U(B)  ->  U(Bf)  and  R(B)  -»  #(£')• 

The  groupoid  [R,U]f  is  in  general  only  a  prestack.  We  denote  by  [ft,  17]  the  associated 
stack.  The  stack  [/?,  C7]  can  be  thought  of  as  the  sheaf  associated  to  the  presheaf  of 
groupoids  B*-+[R,  U]'(B)  ([La],  2.4.3). 

Example  2.33  (Quotient  by  group  action).  Let  X  be  a  scheme  and  G  an  affine  group 
scheme.  We  denote  by  the  same  letters  the  associated  spaces  (functors  of  points).  We  take 
U  =  X  and  R  =  X  x  G.  Using  the  group  action  we  can  define  the  five  morphisms  (t  is  the 
action  of  the  group,  s  =  p\,  m  is  the  product  in  the  group,  e  is  defined  with  the  identity  of 
G,  and  i  with  the  inverse). 

The  objects  of  [X  x  G,X}f(B)  are  morphisms  /  :  B  —»  X.  Equivalently,  they  are  trivial 
principal  G-bundles  B  x  G  over  B  and  a  map  B  x  G  — >  X  defined  as  the  composition  of 
the  action  of  G  and  /.  The  stack  [X  x  G,X]  is  isomorphic  to  [X/G], 

Example  2.34  (Algebraic  stacks').  Let  R,  U  be  a  groupoid  space  such  that  R  and  U  are 
algebraic  spaces,  locally  of  finite  presentation  (equivalently  locally  of  finite  type  if  S  is 
noetherian).  Assume  that  the  morphisms  s,  t  are  fiat,  and  that  S  =  (.s1,  t)  :  R  — >  U  xs  U  is 
separated  and  quasi-compact.  Then  [R,  U]  is  an  Artin  stack,  locally  of  finite  type  ([La], 
Corollary  4.7). 

In  fact,  any  Artin  stack  F  can  be  defined  in  this  fashion.  The  algebraic  space  U  will  be 
the  atlas  of  F,  and  we  set  R  =  U  x  jr  U.  The  morphisms  s  and  t  are  the  two  projections,  i 
exchanges  the  factors,  e  is  the  diagonal,  and  m  is  defined  by  projection  to  the  first  and 
third  factor. 

Let  S  :  R  -+  U  x5  £7  be  an  equivalence  relation  in  the  category  of  spaces.  One  can  define 
a  groupoid  space,  and  [/?,  U]  is  to  be  thought  of  as  the  stack-theoretic  quotient  of  this 
equivalence  relation,  as  opposed  to  the  quotient  space,  used  for  instance  to  define  algebraic 
spaces  (for  more  details  and  the  definition  of  equivalence  relation  see  appendix  A). 

2.5  Properties  of  algebraic  stacks 

So  far  we  have  only  defined  scheme-theoretic  properties  for  representable  stacks  and 
morphisms.  We  can  define  some  properties  for  arbitrary  algebraic  stacks  (and  morphisms 
among  them)  using  the  atlas. 

Let  P  be  a  property  of  schemes,  local  in  nature  for  the  smooth  (resp.  etale)  topology. 
For  example:  regular,  normal,  reduced,  of  characteristic  /?,...  Then  we  say  that  an  Artin 
(resp.  Deligne-Mumford)  stack  has  P  iff  the  atlas  has  P  ([La],  p.  25),  ([DM],  p.  100). 

Let  P  be  a  property  of  morphisms  of  schemes,  local  on  source  and  target  for  the  smooth 
(resp.  etale)  topology,  i.e.  for  any  commutative  diagram 

Xt tY'xyX^—^X 


with  p  and  g  smooth  (resp.  etale)  and  surjective,  /  has  P  iff  /"  has  P.  For  example:  flat, 
smooth,   locally  of  finite  type, For  the  etale  topology  we  also  have:   etale, 


Algebraic  stacks  17 

unramified,. . ..  Then  if  /  :  X  —>  y  is  a  morphism  of  Artin  (resp.  Deligne-Mumford) 
stacks,  we  say  that  /  has  P  iff  for  one  (and  then  for  all)  commutative  diagram  of  stacks 


where  X',  Y'  are  schemes  and  p,  g  are  smooth  (resp.  etale)  and  surjective,  f"  has  P  ([La], 
pp.  27-29). 
For  Deligne-Mumford  stacks  it  is  enough  to  find  a  commutative  diagram 


where  p  and  g  are  etale  and  surjective  and  /"  has  P.  Then  it  follows  that  /  has  P  ([DM], 
p.  100). 

Other  notions  are  defined  as  follows. 

DEFINITION  2.35  (Substack)  ([La],  Definition  2.5),  ([DM],  p.  102). 
A  stack  £  is  a  substack  of  T  if  it  is  a  full  subcategory  of  T  and 

1.  If  an  object  X  of  J-  is  in  £ ,  then  all  isomorphic  objects  are  also  in  £. 

2.  For  all  morphisms  of  schemes  /  :  U  ->  V,  if  X  is  in  £(V),  then  f*X  is  in  £ (U). 

3.  Let  {Ui  ->  U}  be  a  cover  of  U  in  the  site  (Sch/S).  Then  X  is  in  £  iff  X\t  is  in  £  for  all  L 

DEFINITION  2.36  ([La],  Definition  2.13) 

A  substack  £  of  f  is  called  open  (resp.  closed,  resp.  locally  closed)  if  the  inclusion 
morphism  £  — >  f  is,  representable  and  it  is  an  open  immersion  (resp.  closed  immersion, 
resp.  locally  closed  immersion). 

DEFINITION  2.37  (Irreducibility)  ([La],  Definition  3.10),  ([DM],  p.  102) 

An  algebraic  stack  T  is  irreducible  if  it  is  not  the  union  of  two  distinct  and  nonempty 
proper  closed  substacks. 

DEFINITION  2.38  (Separatedness)  ([La],  Definition  3.17),  ([DM],  Definition  4.7) 

An  algebraic  stack  T  is  separated,  if  the  (representable)  diagonal  morphism  A^  is  uni- 
versally closed  (and  hence  proper,  because  it  is  automatically  separated  and  of  finite 
type). 

A  morphism  /  :  F  —>  Q  of  algebraic  stacks  is  separated  if  for  all  U  — >  Q  with  U  affine, 
U  XQ  f  is  a  separated  (algebraic)  stack. 

For  Deligne-Mumford  stacks,  A^  is  universally  closed  iff  it  is  finite.  There  is  a  valuative 
criterion  of  separatedness,  similar  to  the  criterion  for  schemes.  Recall  that  by  Yoneda 
lemma  (Lemma  2.22),  a  morphism  /  :  U  -»  f  between  a  scheme  and  a  stack  is  equivalent 
to  an  object  in  ^(U).  Then  we  will  say  that  a  is  an  isomorphism  between  two  morphisms 
/i 5/2  -  U  —»  T  when  a  is  an  isomorphism  between  the  corresponding  objects  of 


18  Tomds  L  Gomez 

PROPOSITION  2.39  (Valuative  criterion  of  separatedness  (stacks))  ([La],  Proposition 
3.19),  ([DM],  Theorem  4.18) 

An  algebraic  stack  f  is  separated  (over  5)  if  and  only  if  the  following  holds.  Let  A  be  a 
valuation  ring  with  fraction  field  K.  Let  g\  :  Spec  A  -*  f  and  g2  :  Spec  A  — >  T  be  two 
morphisms  such  that: 

1-   fpr°8l  =  fpr°82> 

2.  There  exists  an  isomorphism  a  :  gi|spec£  ~~* 


SpecK- 
then  there  exists  an  isomorphism  (in  fact  unique)  a  :  gi  — >g2  that  extends  a,  i.e.  <5|Specj^  =  a. 

Remark  2.40.  It  is  enough  to  consider  complete  valuation  rings  A  with  algebraically 
closed  residue  field  ([La],  3.20.1).  If  furthermore  S  is  locally  Noetherian  and  T  is  locally 
of  finite  type,  it  is  enough  to  consider  discrete  valuation  rings  A  ([La],  3.20.2). 

Example  2.41 .  The  stack  BG  will  not  be  separated  if  G  is  not  proper  over  S  ([La],  3.20.3), 
and  since  we  assumed  G  to  be  affine,  this  will  not  happen  if  it  is  not  finite. 

In  general  the  moduli  stack  of  vector  bundles  MX  is  not  separated.  It  is  easy  to  find 
families  of  vector  bundles  that  contradict  the  criterion. 

The  stack  of  stable  curves  Mg  is  separated  ([DM],  Proposition  5.1). 

The  criterion  for  morphisms  is  more  involved  because  we  are  working  with  stacks  and 
we  have  to  keep  track  of  the  isomorphisms. 

PROPOSITION  2.42  (Valuative  criterion  of  separatedness  (morphisms))  ([La],  Proposi- 
tion 3.19) 

A  morphism  of  algebraic  stacks  f  :  f  — *  Q  is  separated  if  and  only  if  the  following 
holds.  Let  A  be  a  valuation  ring  with  fraction  field  K.  Let  gi  :  Spec  A  — >•  T  and 
g2  :  Spec  A  — >  T  be  two  morphisms  such  that: 

1.  There  exists  an  isomorphism  0  :f  o  gi  — >/  o  g2* 

2.  There  exists  an  isomorphism  a  :  ( 

3.  f(a)  = 


Then  there  exists  an  isomorphism  (in  fact  unique)  a  :  gi  -»  g2  that  extends  a,  i.e. 
/(a)  =  /?. 


Remark  2.40  is  also  true  in  this  case. 

DEFINITION  2.43  ([La],  Definition  3.21),  ([DM],  Definition  4.11) 

An  algebraic  stack  T  is  proper  (over  5)  if  it  is  separated  and  of  finite  type,  and  if  there  is  a 
scheme  X  proper  over  S  and  a  (representable)  surjective  morphism  X  —  »  f. 

A  morphism  T  —  »  Q  is  proper  if  for  any  affine  scheme  U  and  morphism  U  ->  Q,  the 
fiber  product  U  Xg  T  is  proper  over  U. 

For  properness  we  only  have  a  satisfactory  criterion  for  stacks  (see  ([La],  Proposition 
3.23  and  Conjecture  3.25)  for  a  generalization  for  morphisms). 


Algebraic  stacks  19 

PROPOSITION  2.44  (Vaiuative  criterion  of  properness)  ([La],  Proposition  3.23),  ([DM], 
Theorem  4.19) 

Let  T  be  a  separated  algebraic  stack  (over  S).  It  is  proper  (over  S)  if  and  only  if  the 
following  condition  holds.  Let  A  be  a  valuation  ring  with  fraction  field  K.  For  any 
commutative  diagram 


Spec  K-+-*-  Spec  A 


there  exists  a  finite  field  extension  K'  ofK  such  that  g  extends  to  Spec  (A'),  where  A'  is  the 
integral  closure  of  A  in  K1. 


Example  2.45  (Stable  curves).  The  Deligne-Mumford  stack  of  stable  curves  Mg  is 
proper  ([DM],  Theorem  5.2). 

2.6  Points  and  dimension 

We  will  introduce  the  concept  of  point  of  an  algebraic  stack  and  dimension  of  a  stack  at  a 
point.  The  reference  for  this  is  ([La],  Chapter  5). 

DEFINITION  2.46 

Let  JF  be  an  algebraic  stack  over  5.  The  set  of  points  of  T  is  the  set  of  equivalence  classes 
of  pairs  (£,*),  with  K  a  field  over  S  (i.e.  a  field  with  a  morphism  of  schemes  SpecAT  — >  S) 
and  x  :  SpzcK  — >  T  a  morphism  of  stacks  over  S.  Two  pairs  (K^xf)  and  (K",x"}  are 
equivalent  if  there  is  a  field  K  extension  of  Kr  and  K"  and  a  commutative  diagram 

Spectf 


Given  a  morphism  f  —  »  Q  of  algebraic  stacks  and  a  point  of  f^  we  define  the  image  of 
that  point  in  Q  by  composition. 

Every  point  of  an  algebraic  stack  is  the  image  of  a  point  of  an  atlas.  To  see  this,  given  a 
point  represented  by  Specjfif  —  »  T  and  an  atlas  X  —  »•  T,  take  any  point  SpedT  —  * 
X  XJT  Spec^C.  The  image  of  this  point  in  X  maps  to  the  given  point. 

To  define  the  concept  of  dimension,  recall  that  if  X  and  Y  are  locally  Noetherian 
schemes  and  /  :  X  —>  Y  is  flat,  then  for  any  point  x  G  X  we  have 


with  dimjc(/)  =  dim^X/^)),  where  Xy  is  the  fiber  of  /  over  y. 


20  Tomds  L  Gomez 

DEFINITION  2.47 

Let  /  :  F  —  »  Q  be  a  representable  morphism,  locally  of  finite  type,  between  two  algebraic 
spaces.  Let  £  be  a  point  of  f  '.  Let  Y  be  an  atlas  of  Q.  Take  a  point  jc  in  the  algebraic  space 
Y  Xg  F  that  maps  to  f, 


Y  -  *5 
and  define  the  dimension  of  the  morphism  /  at  the  point  £  as 


It  can  be  shown  that  this  definition  is  independent  of  the  choices  made. 

DEFINITION  2.48 

Let  f  be  a  locally  Noetherian  algebraic  stack  and  £  a  point  of  JF.  Let  w  :  X  —  >  J"  be  an 
atlas,  and  x  a  point  of  X  mapping  to  £.  We  define  the  dimension  of  J*  at  the  point  £  as 


The  dimension  of  T  is  defined  as 


Again,  this  is  independent  of  the  choices  made. 

Example  2.49  (Quotient  by  group  action).  Let  X  be  a  smooth  scheme  of  dimension 
dim(X)  and  G  a  smooth  group  of  dimension  dim(G)  acting  on  X.  Let  [X/G]  be  the 
quotient  stack  defined  in  Example  2.18.  Using  the  atlas  defined  in  Example  2.29,  we 
see  that 

dimpf/G]  =  dim(X)  -  dim(G). 

Note  that  we  have  not  made  any  assumption  on  the  action.  In  particular,  the  action  could 
be  trivial.  The  dimension  of  an  algebraic  stack  can  then  be  negative.  For  instance,  the 
dimension  of  the  classifying  stack  BG  defined  in  Example  2.18  has  dimension 
dim(BG)  =  -dim(G). 

2.7  Quasi-coherent  sheaves  on  stacks 

DEFINITION  2.50  ([Vi],  Definition  7.18),  ([La],  Definition  6.11,  Proposition  6.16).  A 
quasi-coherent  sheaf  S  on  an  algebraic  stack  T  is  the  following  set  of  data: 

1.  For  each  morphism  X  -+  tF  where  X  is  a  scheme,  a  quasi-coherent  sheaf  Sx  on  X. 

2.  For  each  commutative  diagram 


Algebraic  stacks  21 

/•v 

an  isomorphism  y>/  :  Sx-^/**?*',  satisfying  the  cocycle  condition,  i.e.  for  any  com- 
mutative diagram 


(6) 


we  have  ^0/  =  y?/  o  f*<pg. 

We  say  that  <5  is  coherent  (resp.  finite  type,  finite  presentation,  locally  free)  if  Sx  is 
coherent  (resp.  finite  type,  finite  presentation,  locally  free)  for  all  X. 

A  morphism  of  quasi-coherent  sheaves  h  :  S  —*  S'  is  a  collection  of  morphisms  of 
sheaves  hx  :  Sx  —  »•  S'x  compatible  with  the  isomorphisms  <p 

Remark  2.51.  Since  a  sheaf  on  a  scheme  can  be  obtained  by  glueing  the  restriction  to  an 
affme  cover,  it  is  enough  to  consider  affine  schemes. 

Example  2.52  (Structure  sheaf).  Let  T  be  an  algebraic  stack.  The  structure  sheaf  O?  is 
defined  by  taking  (O?)x  —  Ox- 

Example  2.53  (Sheaf  of  differentials).  Let  T  be  a  Deligne-Mumford  stack.  To  define  the 
sheaf  of  differentials  £V,  if  U  —  >  jF  is  an  etale  morphism  we  set  (fl^)v  =  fit/,  the  sheaf 
of  differentials  of  the  scheme  U.  If  V  —  >  T  is  another  etale  morphism  and  we  have  a 
commutative  diagram 


then  /  has  to  be  etale,  there  is  a  canonical  isomorphism  (pf  :  QU/S  ~>/*fiy/5»  a^d  these 
canonical  isomorphisms  satisfy  the  cocycle  condition. 

Once  we  have  defined  (JV)^  for  etale  morphisms  U  —  >  F,  we  can  extend  the  defi- 
nition for  any  morphism  X  —>  F  with  X  an  arbitrary  scheme  as  follows:  take  an  (etale) 
atlas  U  =  U  Ui  ;  —>  J7.  Consider  the  composition  morphism 


and  define  (tor)Xxfu  ===  ^2^^-  The  cocycle  condition  for  Jlj/,  and  etale  descent  implies 
that  (Qf)xxrU  descends  to  give  a  sheaf  (£V)X  on  X.  It  is  easy  to  check  that  this  doesn't 
depend  on  the  atlas  U  used,  and  that  given  a  commutative  diagram  like  (6),  there  are 
canonical  isomorphisms  (p  satisfying  the  cocycle  condition. 

Example  2.54  (Universal  vector  bundle).  Let  MX  be  the  moduli  stack  of  vector  bundles 
on  a  scheme  X  defined  in  2.9.  The  universal  vector  bundle  V  on  MX  x  X  is  defined  as 
follows: 

Let  U  be  a  scheme  and  /  =  (/1,/a)  :  U  -+  A4*  x  X  a  morphism.  By  Lemma  2.22,  the 
morphism  fi  :  U  -»  .M*  is  equivalent  to  a  vector  bundle  W  on  [/  x  X.  We  define  Vfy  as 
/V,  where  /  =  (id^,  f2)  :  U  ~+  U  x  X.  Let 

U' 


22  Tomds  L  Gomez 

be  a  commutative  diagram.  Recall  that  this  means  that  there  is  an  isomorphism  a  :  f  o  g 
— >/',  and  looking  at  the  projection  to  MX  we  have  an  isomorphism  ai  :f\  og  -»/{. 
Using  Lemma  2.22,  f\og  and  /{  correspond  respectively  to  the  vector  bundles 
(g  x  id#)*W  and  Wf  on  Uf  x  X,  and  (again  by  Lemma  2.22)  a\  gives  an  isomorphism 
between  them.  It  is  easy  to  check  that  these  isomorphisms  satisfy  the  cocycle  condition 
for  diagrams  of  the  form  (6). 

3.  Vector  bundles:  Moduli  stack  vs.  moduli  scheme 

In  this  section  we  will  compare,  in  the  context  of  vector  bundles,  the  new  approach  of 
stacks  versus  the  standard  approach  of  moduli  schemes  via  geometric  invariant  theory 
(GIT)  (for  background  on  moduli  schemes  of  vector  bundles,  see  [HL]). 

Fix  a  scheme  X  over  ,  a  positive  integer  r  and  classes  a  €  H2i(X).  All  vector  bundles 
over  X  in  this  section  will  have  rank  r  and  Chern  classes  c*.  We  will  also  consider  vector 
bundles  on  products  B  x  X  where  B  is  a  scheme.  We  will  always  assume  that  these  vector 
bundles  are  flat  over  5,  and  that  the  restriction  to  the  slices  {p}  x  X  are  vector  bundles 
with  rank  r  and  Chern  classes  c/.  Fix  also  a  polarization  on  X.  All  references  to  stability  or 
semistability  of  vector  bundles  will  mean  Gieseker  stability  with  respect  to  this  fixed 
polarization. 

Recall  that  the  functor  Mfx  (resp.  Mf )  is  the  functor  from  (Sch/5)  to  (Sets)  that  for 
each  scheme  B  gives  the  set  of  equivalence  classes  of  vector  bundles  over  B  x  X,  flat  over 
B  and  such  that  the  restrictions  V\b  to  the  slices  p  x  X  are  stable  (resp.  semistable)  vector 
bundles  with  fixed  rank  and  Chern  classes,  where  two  vector  bundles  V  and  Vf  on  B  x  X 
are  considered  equivalent  if  there  is  a  line  bundle  L  on  B  such  that  V  is  isomorphic  to 
V'®p*BL. 

Theorem  3.1.  There  are  schemes  Msx  and  Mx,  called  moduli  schemes,  corepresenting 
the  functors  M sx  and  M?x. 

The  moduli  scheme  Mx  is  constructed  using  the  Quot  schemes  introduced  in  Example 
2.28  (for  a  detailed  exposition  of  the  construction,  see  [HL]).  Since  the  set  of  semistable 
vector  bundles  is  bounded,  we  can  choose  once  and  for  all  N  and  m  (depending  only  on 
the  Chern  classes  and  rank)  with  the  property  that  for  any  semistable  vector  bundle  V 
there  is  a  point  in  R  =  RN,m  whose  corresponding  quotient  is  isomorphic  to  V. 

The  scheme  R  parametrizes  vector  bundles  V  on  X  together  with  a  basis  of  H°(V(m)} 
(up  to  multiplication  by  scalar).  Recall  that  N  =  ft°(V(m)).  There  is  an  action  of  GL(N) 
on  R,  corresponding  to  change  of  basis  but  since  two  basis  that  only  differ  by  a  scalar  give 
the  same  point  on  R,  this  GL(N)  action  factors  through  PGL(N).  Then  the  moduli  scheme 
Mf  is  defined  as  the  GIT  quotient  R//PGL(N). 

The  closed  points  of  Mx  correspond  to  S-equivalence  classes  of  vector  bundles,  so  if 
there  is  a  strictly  semistable  vector  bundle,  the  functor  M_x  is  not  representable. 

Now  we  will  compare  this  scheme  with  the  moduli  stack  MX  defined  on  Example  2.9. 
We  will  also  consider  the  moduli  stack  Mx  defined  in  the  same  way,  but  with  the  extra 
requirement  that  the  vector  bundles  should  be  stable.  The  moduli  stack  Mx  is  a  substack 
(Definition  2.35)  of  MX-  The  following  are  some  of  the  differences  between  the  moduli 
scheme  and  the  moduli  stack: 

1.  The  stack  MX  parametrizes  all  vector  bundles,  but  the  scheme  Mx  only  parametrizes 
semistable  vector  bundles. 


Algebraic  stacks  23 

2.  From  the  point  of  view  of  the  scheme  Mx,  we  identify  two  vector  bundles  on  X 
(i.e.  they  give  the  same  closed  point  on  Msx)  if  they  are  S-equivalent.  On  the  other 
hand,  from  the  point  of  view  of  the  moduli  stack,  two  vector  bundles  are  identified 
(i.e.  give  isomorphic  objects  on  jWx(Spec  k))  only  if  they  are  isomorphic  as  vector 
bundles. 

3.  Let  V  and  Vr  be  two  families  of  vector  bundles  parametrized  by  a  scheme  B,  i.e.  two 
vector  bundles  (flat  over  B)  on  B  x  X.  If  there  is  a  line  bundle  L  on  B  such  that  V  is 
isomorphic  to  V  (g)  p$L,  then  from  the  point  of  view  of  the  moduli  scheme,  V  and  V 
are  identified  as  being  the  same  family.  On  the  other  hand,  from  the  point  of  view  of 
the  moduli  stack,  V  and  V  are  identified  only  if  they  are  isomorphic  as  vector  bundles 
on  B  x  X. 

4.  The  subscheme  Mx  corresponding  to  stable  vector  bundles  is  sometimes  represen- 
table  by  a  scheme,  but  the  moduli  stack  Mx  is  never  representable  by  a  scheme.  To 
see  this,  note  that  any  vector  bundle  has  automorphisms  different  from  the  identity 
(multiplication  by  scalars)  and  apply  Lemma  2.21. 

Now  we  will  restrict  our  attention  to  stable  bundles,  i.e.  to  the  scheme  Mx  and  the  stack 
Mx.  For  stable  bundles  the  notions  of  S-equi  valence  and  isomorphism  coincide,  so  the 
points  of  Mx  correspond  to  isomorphism  classes  of  vector  bundles.  Consider  Rs  C  R,  the 
subscheme  corresponding  to  stable  bundles.  There  is  a  map  TT  :  Rs  — >  Mx  =  RS/PGL(N), 
and  TT  is  in  fact  a  principal  PGL(N) -bundle  (this  is  a  consequence  of  Luna's  etale  slice 
theorem). 

Remark  3.2  (Universal  bundle  on  moduli  scheme).  The  scheme  Mx  represents  the 
functor  M*  if  there  is  a  universal  family.  Recall  that  a  universal  family  for  this  functor  is  a 
vector  bundle  E  on  Mx  x  X  such  that  the  isomorphism  class  of  E\pxX  is  the  isomorphism 
class  corresponding  to  the  point/?  E  Mx,  and  for  any  family  of  vector  bundles  V  on  B  x  X 
there  is  a  morphism  / :  B  -»  Mx  and  a  line  bundle  L  on  B  such  that  V  ®  p*BL  is 
isomorphic  to  (/  x  idx)*E.  Note  that  if  E  is  a  universal  family,  then  E®p*M,  L  will  also 
be  a  universal  family  for  any  line  bundle  L  on  Mx. 

The  universal  bundle  for  the  Quot  scheme  gives  a  universal  family  V  on  Rs  x  X,  but 
this  family  does  not  always  descend  to  give  a  universal  family  on  the  quotient  M^. 

Let  X  A  Y  be  a  principal  G-bundle.  A  vector  bundle  V  on  X  descends  to  Y  if  the  action 
of  G  on  X  can  be  lifted  to  V.  In  our  case,  if  certain  numerical  criterion  involving  r  and  c/ 
is  satisfied  (if  X  is  a  smooth  curve  this  criterion  is  gcd(r,  c\)  —  1),  then  we  can  find  a  line 
bundle  L  on  Rs  such  that  the  PGL(N)  action  on  Rs  can  be  lifted  to  V  ®  p^L,  and  then  this 
vector  bundle  descends  to  give  a  universal  family  on  Mx  x  X.  But  in  general  the  best  that 
we  can  get  is  a  universal  family  on  an  etale  cover  of  Mx. 

Recall  from  Example  2.29  that  there  is  a  morphism  [R?*lPGL(N)]  ->  Msx ,  and  that  the 
morphism  \R?lPGL(N}]  — >  Mx  is  an  isomorphism  of  stacks. 

PROPOSITION  3.3 

There  is  a  commutative  diagram  of  stacks 

[R*/GL(N)} ! »  [R*/PGL(N)] 

*\* 


24  Tom  as  L  Gomez 

where  g  and  h  are  isomorphisms  of  stacks,  but  q  and  </?  are  not.  If  we  change  'stable'  by 
'semistable'  we  still  have  a  commutative  diagram,  but  the  corresponding  morphism  hss  is 
not  an  isomorphism  of  stacks. 

Proof.  The  morphism  (p  is  the  composition  of  the  natural  morphism  Msx  -»  M?x  (sending 
each  category  to  the  set  of  isomorphism  classes  of  objects)  and  the  morphism  MX  —*•  Msx 
given  by  the  fact  that  the  scheme  Msx  =  RS//PGL(N)  corepresents  the  functor  MJ. 

The  morphism  h  was  constructed  in  Example  2.18. 

The  key  ingredient  needed  to  define  g  is  the  fact  that  the  GL(N)  action  on  the  Quot 
scheme  lifts  to  the  universal  bundle,  i.e.  the  universal  bundle  on  the  Quot  scheme  has  a 
GL(N)  -linearization.  Let 


B 

be  an  object  of  [RSS/GL(N)].  Since  Rss  is  a  subscheme  of  a  Quot  scheme,  by  restriction  we 
have  a  universal  bundle  on  Rss  x  X,  and  this  universal  bundle  has  a  GL(N) -linearization. 
Let  E  be  the  vector  bundle  on  B  x  X  defined  by  the  pullback  of  this  universal  bundle. 
Since  /  is  GL(A/r)-equivariant,  E  is  also  GL(N] -linearized.  Since  5xX— »#xXisa 
principal  bundle,  the  vector  bundle  E  descends  to  give  a  vector  bundle  E  on  B  x  X,  i.e.  an 
object  of  MX-  Let 

/ 


--B 

be  a  morphism  in  [RSS/GL(N)].  Consider  the  vector  bundles  E  and  E7  defined  as  before. 
Since  /'  o  <j)  —  f,  we  get  an  isomorphism  of  E  with  (0  x  id)*#'.  Furthermore  this 
isomorphism  is  GL(Af)-equi variant,  and  then  it  descends  to  give  an  isomorphism  of  the 
vector  bundles  E  and  E1  on  B  x  X,  and  we  get  a  morphism  in  M%. 

To  prove  that  this  gives  an  equivalence  of  categories,  we  construct  a  functor  g  from 
MX  to  [RSS/GL(N)}.  Given  a  vector  bundle  E  on  B  x  X,  let  q  :  B  -*  B  be  the  GL(N)- 
principal  bundle  associated  with  the  vector  bundle  ps*E  on  B.  Let  E  =  (q  x  id)*£  be  the 
pullback  of  E  to  B  x  X.  It  has  a  canonical  GL(N)  -linearization  because  it  is  defined  as  a 
pullback  by  a  principal  GL(N) -bundle.  The  vector  bundle  p^JE  is  canonically  isomorphic 
to  the  trivial  bundle  O%,  and  this  isomorphism  is  GL(N)-equivariant,  so  we  get  an 
equivariant  morphism  B  — »  Rss,  and  hence  an  object  of  [RSS/GL(N)]. 

If  we  have  an  isomorphism  between  two  vector  bundles  E  and  E'  on  B  x  X,  it  is  easy  to 
check  that  it  induces  an  isomorphism  between  the  associated  objects  of  [RSS/GL(N)]. 

It  is  easy  to  check  that  there  are  natural  isomorphisms  of  functors  g  o  g  ^  id  and 
g  o  g  ^  id,  and  then  g  is  an  equivalence  of  categories. 

The  morphism  q  is  defined  using  the  following  lemma,  with  G  =  GL(N),  H  the 
subgroup  consisting  of  scalar  multiples  of  the  identity,  G  =  PGL(N)  and  Y=R*5.  D 

Lemma  3.4.  Let  Y  be  an  S~ scheme  and  G  an  affine  flat  group  S-scheme,  acting  on  Y  on 
the  right.  Let  H  be  a  normal  closed  subgroup  of  G.  Assume  that  G  =  G/H  is  affine.  IfH 


Algebraic  stacks  25 

acts  trivially  on  Y,  then  there  is  a  morphism  of  stacks 

[Y/G]  ->  (Y/G}. 
If  H  is  nontrivial,  then  this  morphism  is  not  faithful,  so  it  is  not  an  isomorphism. 

Proof.  Let 


B 
be  an  object  of  [Y/G].  There  is  a  scheme  E/H  such  that  TT  factors 


To  construct  E/H,  note  that  there  is  a  local  etale  cover  £//  of  B  and  isomorphisms 
(f>i  :  7r~l(Ui)  —  >  Ui  x  G,  with  transition  functions  i/>o  =  </>/  °  0/"1-  Since  these  isomorph- 
isms are  G-equivariant,  they  descend  to  give  isomorphisms  ^,7  •  Uj  x  G/H  —  »  Ut  x  G//f  , 
and  using  these  transition  functions  we  get  E/H.  This  construction  shows  that  TT'  is  a 
principal  G-bundle.  Furthermore,  #  is  also  a  principal  //-bundle  ([HL],  Example  4.2.4), 
and  in  particular  it  is  a  categorical  quotient. 

Since  /  is  //-invariant,  there  is  a  morphism/  :  E/H  —  >  F,  and  this  gives  an  object  of 
(Yffl. 

If  we  have  a  morphism  in  [Y/G]9  given  by  a  morphism  g  \  E  -*  E'  of  principal  G- 
bundles  over  B,  it  15  easy  to  see  that  it  descends  (since  g  is  equivariant)  to  a  morphism 
g  :  E/H  -»  E'  IE,  giving  a  morphism  in  [Y/G]. 

This  morphism  is  not  faithful,  since  the  automorphism  E-^  E  given  by  multiplication 
on  the  right  by  a  nontrivial  element  z  £  H  is  sent  to  the  identity  automorphism 
E/H  -+  E/H,  and  then  Hom(£,  E)  ->  Hom(E/#,  £/#  )  is  not  injective.  D 

If  X  is  a  smooth  curve,  then  it  can  be  shown  that  MX  is  a  smooth  stack  of  dimension 

—  1),  where  r  is  the  rank  and  g  is  the  genus  of  X.  In  particular,  the  open  substack 
is  also  smooth  of  dimension  ^(g  —  1),  but  the  moduli  scheme  M£  is  of  dimension 

—  1)  -h  1  and  might  not  be  smooth.  Proposition  3.3  explains  the  difference  in  the 
dimensions  (at  least  on  the  smooth  part):  we  obtain  the  moduli  stack  by  taking  the 
quotient  by  the  group  GL(N),  of  dimension  N2,  but  the  moduli  scheme  is  obtained  by  a 
quotient  by  the  group  PGL(N)9  of  dimension  N2  -  1.  The  moduli  scheme  M£  is  not 
smooth  in  general  because  in  the  strictly  semistable  part  of  Rss  the  action  of  PGL(N)  is 
not  free.  On  the  other  hand,  the  smoothness  of  a  stack  quotient  doesn't  depend  on  the 
freeness  of  the  action  of  the  group. 

Appendix  A:  Grothendieck  topologies,  sheaves  and  algebraic  spaces 

The  standard  reference  for  Grothendieck  topologies  is  SGA  (Seminaire  de  Geometric 
Algebrique).  For  an  introduction  see  [T]  or  [MM].  For  algebraic  spaces,  see  [K]  or 
[Arl]. 

An  open  cover  in  a  topological  space  U  can  be  seen  as  family  of  morphisms  in  the 
category  of  topological  spaces  ft  :  Ui  —  >  U,  with  the  property  that  ft  is  an  open  inclusion 


26  Tomds  L  Gomez 

and  the  union  of  their  images  is  £7,  Le  we  are  choosing  a  class  of  morphisms  (open 
inclusions)  in  the  category  of  topological  spaces.  A  Grothendieck  topology  on  an  arbi- 
trary category  is  basically  a  choice  of  a  class  of  morphisms,  that  play  the  role  of  'open 
sets'.  A  morphism  /  :  V  -»  U  in  this  class  is  to  be  thought  of  as  an  'open  set'  in  the  object 
U.  The  concept  of  intersection  of  open  sets  is  replaced  by  the  fiber  product:  the 
'intersection'  of  f\  :  U\  —>  U  and  /2  :  Ui  — >  U  is  fa  :  U\  Xu  Ui  — >  U. 

A  category  with  a  Grothendieck  topology  is  called  a  site.  We  will  consider  two 
topologies  on  (Sch/5). 

Jppf  topology.  Let  U  be  a  scheme.  Then  a  cover  of  U  is  a  finite  collection  of  morphisms 
{ft  •"  Ut  — >  U}iel  such  that  each  ff  is  a  finitely  presented  flat  morphism  (for  Noetherian 
schemes,  this  is  equivalent  to  flat  and  finite  type),  and  U  is  the  (set  theoretic)  union  of  the 
images  of  /-.  In  other  words,  U  [/;  — >  U  is  'fidelement  plat  de  presentation  finie* . 

Etale  topology.  Same  definition,  but  substituting  flat  by  etale. 

DEFINITION  4.1  (Presheaf  of  sets) 

A  presheaf  of  sets  on  (Sch/5)  is  a  contravariant  functor  F  from  (Sch/5)  to  (Sets). 

As  usual,  we  will  use  the  following  notation:  if  X  6  F(U)  and  //  :  Ui •  — >  U  is  a 
morphism,  then  X|f  is  the  element  of  F(Ui)  given  by  F(fi)(X),  and  we  will  call  X|f 
the  'restriction  of  X  to  £//',  even  if  ft  is  not  an  inclusion.  If  X;  G  F(Ui),  then  X/|/;. 
is  the  element  of  F(L^)  given  by  F(/-7-,-)(Xf)  where  /-;-f  :  t/£  xv  Uj  -+  Ut  is  the  pullback 
ofJJ- 

DEFINITION  4.2  (Sheaf  of  sets) 

Choose  a  topology  on  (Sch/5).  We  say  that  F  is  a  sheaf  (or  an  5-space)  with  respect  to 
that  topology  if  for  every  cover  {ff :  Ui  — »  U}i€l  in  the  topology  the  following  two 
axioms  are  satisfied: 

1.  Mono.  Let  X  and  Y  be  two  elements  of  F(U).  If  X|f  =  Y\t  for  all  /,  then  X=Y. 

2.  Glueing.  LetX;  be  an  object  of  F(J7,-)  for  each  i  such  that  X^7  =  Xj|f  -,  then  there  exists 
X  G  F(J7)  such  that  X\t  =  Xf  for  each  z. 

We  define  morphisms  of  S-spaces  as  morphisms  of  sheaves  (i.e.  natural  transforma- 
tions of  functors).  Note  that  a  scheme  M  can  be  viewed  as  an  5-space  via  its  functor  of 
points  Horns  (— ,M),  and  a  morphism  between  two  such  S-spaces  is  equivalent  to  a 
scheme  morphism  between  the  schemes  (by  the  Yoneda  embedding  lemma),  then  the 
category  of  5-schemes  is  a  full  subcategory  of  the  category  of  5-spaces. 

Equivalence  relation  and  quotient  space.  An  equivalence  relation  in  the  category  of  S- 
spaces  consists  of  two  5-spaces  R  and  U  and  a  monomorphism  of  5-spaces 

6:R-+UxsU 

such  that  for  all  5-scheme  B,  the  map  S(B)  :  R(B]  -*  U(B)  x  U(B)  is  the  graph  of  an 
equivalence  relation  between  sets.  A  quotient  5-space  for  such  an  equivalence  relation  is 
by  definition  the  sheaf  cokernel  of  the  diagram 


Algebraic  stacks  27 

DEFINITION  4.3  (Algebraic  space)  ([La],  0). 

An  S-space  F  is  called  an  algebraic  space  if  it  is  the  quotient  S-space  for  an  equivalence 
relation  such  that  R  and  U  are  S-schemes,  pi  06,  pi  o  6  are  etale  (morphisms  of  S- 
schemes),  and  S  is  a  quasi-compact  morphism  (of  5-schemes). 

Roughly  speaking,  an  algebraic  space  is  a  quotient  of  a  scheme  by  an  etale  equivalence 
relation.  The  following  is  an  equivalent  definition. 

DEFINITION  4.4  ([K],  Definition  1.1) 

An  S-space  F  is  called  an  algebraic  space  if  there  exists  a  scheme  U  (atlas)  and  a 
morphism  of  ^-spaces  u  :  U  — »  F  such  that 

1.  The  morphism  u  is  etale.  For  any  5-scheme  V  and  morphism  V  — >  F,  the  (sheaf)  fiber 
product  U  Xp  V  is  representable  by  a  scheme,  and  the  map  U  x?  V  — »  V  is  an  etale 
morphism  of  schemes. 

2.  Quasi-separatedness.  The  morphism  U  x?  U  —*  U  x$  U  is  quasi-compact. 

We  recover  the  first  definition  by  taking  R  =  U  XF  U.  Then  roughly  speaking,  we  can 
also  think  of  an  algebraic  space  as  'something'  that  looks  locally  in  the  etale  topology 
like  an  affine  scheme,  in  the  same  sense  that  a  scheme  is  something  that  looks  locally  in 
the  Zariski  topology  like  an  affine  scheme. 

Algebraic  spaces  are  used,  for  instance,  to  give  algebraic  structure  to  certain  complex 
manifolds  (for  instance  Moishezon  manifolds)  that  are  not  schemes,  but  can  be  realized  as 
algebraic  spaces.  All  smooth  algebraic  spaces  of  dimension  1  and  2  are  actually  schemes. 
An  example  of  a  smooth  algebraic  space  of  dimension  3  that  is  not  a  scheme  can  be  found 
in  [H]. 

But  etale  topology  is  useful  even  if  we  are  only  interested  in  schemes.  The  idea  is  that 
the  etale  topology  is  finer  than  the  Zariski  topology,  and  in  many  situations  it  is  'fine 
enough'  to  do  the  analog  of  the  manipulations  that  can  be  done  with  the  analytic  topology 
of  complex  manifolds.  As  an  example,  consider  the  affine  complex  line  Spec(C[jt]),  and 
take  a  (closed)  point  XQ  different  from  0.  Assume  that  we  want  to  define  the  function  ^/x, 
in  a  neighborhood  of  JCQ.  In  the  analytic  topology  we  only  need  to  take  a  neighborhood 
small  enough  so  that  it  does  not  contain  a  loop  that  goes  around  the  origin,  then  we 
choose  one  of  the  branches  (a  sign)  of  the  square  root.  In  the  Zariski  topology  this  cannot 
be  done,  because  all  open  sets  are  too  large  (have  loops  going  around  the  origin,  so  the 
sign  of  the  square  root  will  change,  and  ^/x  will  be  multivaluated).  But  take  the  2:1  etale 
map  V  =  Spec (C^y,*,*"1] /(y  —  jc2))  — >  Spec(C[jt]).  The  function  v/*  cai*  certainly  be 
defined  on  V,  it  is  just  equal  to  the  function  y,  so  it  is  in  this  sense  that  we  say  that 
the  etale  topology  is  finer:  V  is  a  'small  enough  open  subset'  because  the  square  root  can 
be  defined  on  it. 


Appendix  B:  2-categories 

In  this  section  we  recall  the  notions  of  2-category  and  2-ftmctor.  A  2-category  C  consists 
of  the  following  data  [Hak]: 

(i)  A  class  of  objects  ob  C. 

(ii)  For  each  pair  X,  Y  G  ob  C,  a  category  Hom(X,  7). 


28  Tomds  L  Gomez 

(iii)  Horizontal  composition  of  1-morphisms  and  2-morphisms.  For  each  triple  X,  7, 
Z  €  obC,  a  functor 


,  7)  x  Hom(7,Z)  ->  Hom(X,Z) 
with  the  following  conditions 

(O  Identity  1-morphism.  For  each  object  X  6  obC,  there  exists  an  object  id*  €  Horn 
X,X   such  that 


where  idHom(x,K)  is  the  identity  functor  on  the  category  Hom(X,  Y). 
(iij)  Associativity  of  horizontal  compositions.  For  each  quadruple  X,  7,  Z,  T  E  obC, 

^x,z,r  °  (MX,F,Z  x  idHom(z,r))  =  ^x,r,r  °  (idHom(x,y)  x  Mr,z,r)- 

The  example  to  keep  in  mind  is  the  2-category  Cat  of  categories.  The  objects  of  Cat 
are  categories,  and  for  each  pair  X,  Y  of  categories,  Hom(X,  7)  is  the  category  of  functors 
between  X  and  Y. 

Note  that  the  main  difference  between  a  1  -category  (a  usual  category)  and  a  2-category 
is  that  Hom(X,  7),  instead  of  being  a  set,  is  a  category. 

Given  a  2-category,  an  object  /  of  the  category  Hom(X,  7)  is  called  a  1-morphism  of 
C,  and  is  represented  with  a  diagram 


and  a  morphism  a  of  the  category  Hom(X,  7)  is  called  a  2-morphisms  of  C,  and  is 
represented  as 


Now  we  will  rewrite  the  axioms  of  a  2-category  using  diagrams. 

1.  Composition  of  1-morphisms.  Given  a  diagram 

X     f     Y      9      Z  X9°fZ 

• ^  »  __±_^  ,     there  exist     •  ^-^  • 

(this  is  (iii)  applied  to  objects)  and  this  composition  is  associative:  (h  o  g)  o  /  = 
A  °  (#  °  /)  (this  is  (ii')  applied  to  objects). 

2.  Identity  for  l-morphisms.  For  each  object  X  there  is  a  1-morphism  id*  such  that 
/  o  idr  =  idx  o  /  =  /  (this  is  (i')). 

3.  Vertical  composition  of  2-morphisms.  Given  a  diagram 


h 

and  this  composition  is  associative  (7  o  /3)  o  a  =  7  o  (/3  o  a). 


Algebraic  stacks 
4.  Horizontal  composition  of  2-morphisms.  Given  a  diagram 


29 


(7) 


(this  is  (iii)  applied  to  morphisms)  and  it  is  associative  (7  *  /?)  *  a  =  7  *  (/?  *  a)  (this 
is  (ii')  applied  to  morphisms). 

5.  Identity  for  2-morphisms.  For  every  1-morphism  /  there  is  a  2-morphism  id/  such  that 
a  o  idg  =  id/  o  a  =  a  (this  and  item  are  (ii)).  We  have  id^  *  id/  =  idgo/  (this  means 
that  //x,y,z  respects  the  identity). 

6.  Compatibility  between  horizontal  and  vertical  composition  of  2-morphisms.  Given  a 
diagram 


9" 


then  (ft  o  (3)  *  (a'  o  a)  =  (/3'  *  a')  o  (/?  *  a)  (this  is  (iii)  applied  to  morphisms). 

Two  objects  X  and  Y  of  a  2-category  are  called  equivalent  if  there  exist  two  1-morphisms 
f  :X  — »  7,  g  :  Y  — >  X  and  two  2-isomorphisms  (invertible  2-morphism)  a  :  g  o  /  — »  id# 
and  0  :fo  g  -+  idY. 
A  commutative  diagram  of  1-morphisms  in  a  2-category  is  a  diagram 

y 


such  that  a:go/— >/iisa  2-isomorphisms. 

Remark  5.1  Note  that  we  do  not  require  g  o  /  =  h  to  say  that  the  diagram  is  commu- 
tative, but  just  require  that  there  is  a  2-isomorphisms  between  them.  This  is  the  reason 
why  2-categories  are  used  to  describe  stacks. 

On  the  other  hand,  a  diagram  of  2-morphisms  will  be  called  commutative  only  if  the 
compositions  are  actually  equal.  Now  we  will  define  the  concept  of  covariant  2-functor  (a 
contravariant  2-functor  is  defined  in  a  similar  way). 

A  covariant  2-functor  F  between  two  2-categories  C  and  Cr  is  a  law  that  for  each  object 
X  in  C  gives  an  object  F(X)  in  C'.  For  each  1-morphism  / :  X  — >•  Y  in  C  gives  a 
1-morphism  F(f)  :  F(X)  -»  F(Y)  in  C',  and  for  each  2-morphism  a  :/  =*>  g  in  C  gives  a 


30  Tomds  L  Gomez, 

2-morphism  F(a)  :  JF(/)  =>  F(g)  in  C',  such  that 

1.  Respects  identity  1-morphism.  F(idx)  =  idp(x)« 

2.  Respects  identity  2-morphism.  F(idy)  =  id/?(/). 

3.  Respects  composition  of  1-morphism  up  to  a  2-isomorphism.  For  every  diagram 

X  f   Y  g  Z 

e  __i_n  -!+• 

there  exists  a  2-isomorphism  egj-  :  F(g)  o  F(/)  -+  F(g  o  /) 
F(Y) 


F(x]-  W)  '•*•(*) 

(a)  e/,idx  =eidK/  =  idF(/)- 

(b)  6  w  associative.  The  following  diagram  is  commutative 

F(f) 


F(h)oF(gof) 

4.  Respects  vertical  composition  of  2-morphisms.  For  every  pair  of  2-morphisms  a  : 
f~>g,/3:g->h,we  have  F(/3  o  a)  =  F(/3)  o  F(a). 

5.  Respects  horizontal  composition  of  2-morphisms.  For  every  pair  of  2-morphisms 
a  :/  —  >//,  /3  :  g  -+  g1  as  in  (7)  the  following  diagram  commutes 

F(9)oF(}) 


By  a  slight  abuse  of  language,  condition  5  is  usually  written  as  F(/3)  *  F(a)  =  F(/3  *  a). 
Note  that  strictly  speaking  this  equality  doesn't  make  sense,  because  the  sources  (and  the 
targets)  do  not  coincide,  but  if  we  chose  once  and  for  all  the  2-isomorphisms  e  of  con- 
dition 3,  then  there  is  a  unique  way  of  making  sense  of  this  equality. 

Remark  5.2.  Since  2-functors  only  respect  composition  of  1-functors  up  to  a  2-isomor- 
phism (condition  3),  sometimes  they  are  called  pseudofunctors  or  lax  functors. 

Remark  5.3.  In  the  applications  to  stacks,  the  isomorphism  eg/  of  item  3  is  canonically 
defined,  and  by  abuse  of  language  we  will  say  that  F(g)  oF(f)  =  F(g  o  /),  instead  of 
saying  that  they  are  isomorphic. 

Given  a  1-category  C  (a  usual  category),  we  can  define  a  2-category:  we  just  have  to 
make  the  set  Hom(X,  Y)  into  a  category,  and  we  do  this  just  by  defining  the  unit 
morphisms  for  each  element. 

On  the  other  hand,  given  a  2-category  C  there  are  two  ways  of  defining  a  1-category. 
We  have  to  make  each  category  Hom(X,  Y)  into  a  set.  The  naive  way  is  just  to  take  the  set 
of  objects  of  Hom(X,  7),  and  then  we  obtain  what  is  called  the  underlying  category  of  C 


Algebraic  stacks  3 1 

(see  [Hak]).  This  has  the  problem  that  a  2-functor  F  :  C  — >  C'  is  not  in  general  a  functor 
of  the  underlying  categories  (because  in  item  3  we  only  require  the  composition  of  l- 
morphisms  to  be  respected  up  to  2-isomorphism). 

The  best  way  of  constructing  a  1 -category  from  a  2-category  is  to  define  the  set  of 
morphisms  between  the  objects  X  and  Y  as  the  set  of  isomorphism  classes  of  objects  of 
Hom(Jf,  7):  two  objects  /  and  g  of  Hom(X,  y)  are  isomorphic  if  there  exists  a  2- 
isomorphism  a  :f  =>  g  between  them.  We  call  the  category  obtained  in  this  way  the 
1 -category  associated  to  C.  Note  that  a  2-functor  between  2-categories  then  becomes  a 
functor  between  the  associated  1 -categories. 

Acknowledgments 

This  article  is  based  on  a  series  of  lectures  that  I  gave  in  February  1999  in  the  Geometric 
Langlands  Seminar  of  the  Tata  Institute  of  Fundamental  Research.  First  of  all,  I  would 
like  to  thank  N  Nitsure  for  proposing  me  to  give  these  lectures.  Most  of  my  understanding 
on  stacks  comes  from  conversations  with  N  Nitsure  and  C  Sorger. 

I  would  also  like  to  thank  T  R  Ramadas  for  encouraging  me  to  write  these  notes,  and 
the  participants  in  the  seminar  in  TIFR  for  their  active  participation,  interest,  questions 
and  comments.  In  ICTP,  Trieste,  I  gave  two  informal  talks  in  August  1999  on  this  subject, 
and  the  comments  of  the  participants,  specially  L  Brambila-Paz  and  Y  I  Holla,  helped  to 
remove  mistakes  and  improve  the  original  notes.  Thanks  also  to  CheeWhye  Chin  for  a 
very  careful  reading  of  a  preliminary  version  of  this  article. 

This  work  was  supported  by  a  postdoctoral  fellowship  of  Ministerio  de  Educacion  y 
Cultura,  Spain. 

References 

[Arl]  Artin  M,  Algebraic  Spaces,  Yale  Math.  Monographs  3  (Yale  University  Press),  (1971) 
[Ar2]  Artin  M,  Versal  deformations  and  algebraic  stacks,  Invent.  Math.  27  (1974)  165-189 
[DM]  Deligne  P  and  Mumford  D,  The  irreducibility  of  the  space  of  curves  of  given  genus,  Publ. 

Math.  1HES  36  (1969)  75-110 

[E]  Edidin  D,  Notes  on  the  construction  of  the  moduli  space  of  curves,  preprint  (1999) 
[Gi]  Giraud  J,  Cohomologie  non  abelienne,  Die  Grundlehren  der  Mathematischen  Wissenschaf- 

ten,  Band  179  (Springer  Verlag)  (1971) 
[Hak]  Hakim  M,  Topos  anneles  et  schemas  relatifs,  Ergebnisse  der  Math,  und  ihrer  Grenzgebiete 

64  (Springer  Verlag)  (1972) 

[H]  Hartshorne  R,  Algebraic  geometry,  Grad.  Texts  in  Math.  52  (Springer  Verlag)  (1977) 
[HL]  Huybrechts  D  and  Lehn  M,  The  geometry  of  moduli  spaces  of  sheaves,  Aspects  of 

Mathematics  E31  (Vieweg,  Braunschweig/Wiesbaden)  (1997) 
[K]  Knutson  D,  Algebraic  spaces,  LNM  203  (Springer  Verlag)  (1971) 
[La]  Laumon  G,  Champs  algebriques,  Prepublications  88-33,  (U.  Paris-Sud)  (1988) 
[LaM]  Laumon  G  and  Moret-Bailly  L,  Champs  algebriques,  Ergegnisse  der  Math,  und  ihrer 

Grenzgebiete.  3.  Folge,  39  (Springer  Verlag)  (2000) 
[MM]  Mac  Lane  S  and  Moerdijk  I,  Sheaves  in  Geometry  and  Logic,  Universitext,  Springer- Verlag, 

1992 
[S]  Simpson  C,  Moduli  of  representations  of  the  fundamental  group  of  a  smooth  projective 

variety  I,  Publ  Math.  I.H.E.S.  79  (1994)  47-129 

[T]  Tamme  G,  Introduction  to  Etale  Cohomology,  Universitext  (Springer-Verlag)  (1994) 
[Vi]  Vistoli  A,  Intersection  theory  on  algebraic  stacks  and  their  moduli  spaces,  Invent  Math.  97 
(1989)  613-670 


Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  Ill,  No.  1,  February  2001,  pp.  33-47. 
©  Printed  in  India 


Variational  formulae  for  Fuchsian  groups  over  families  of  algebraic 
curves 

DAKSHINI  BHATTACHARYYA 

Last  address:  The  Institute  of  Mathematical  Sciences,  CIT  Campus,  Taramani, 
Chennai  600113,  India 

MS  received  24  February  1998;  revised  4  September  2000 

<2S 

Abstract.  We  study  the  problem  of  understanding  the  uniformizing  Fuchsian  groups 
for  a  family  of  plane  algebraic  curves  by  determining  explicit  first  variational  formulae 
for  the  generators. 

Keywords.    Riemann  surfaces;  Fuchsian  groups;  Ahlfors-Bers  variational  formulae. 

1.  Introduction 

In  this  paper  we  make  a  contribution  to  the  problem  of  understanding  the  uniformizing 
Fuchsian  groups  for  a  family  of  plane  algebraic  curves  by  determining  explicit  first 
variational  formulae  for  the  generators  of  the  Fuchsian  groups,  say  G>,  associated  to  a  t- 
parameter  family  of  compact  Riemann  surfaces  Xt,  where  the  Xt  are  the  Riemann 
surfaces  for  the  complex  algebraic  curves  arising  from  a  ^-parameter  family  of  irreducible 
polynomials.  The  main  idea  of  our  work  is  to  utilize  explicit  quasiconformal  mappings 
between  algebraic  curves,  calculate  the  Beltrami  coefficients,  and  hence  utilize  the 
Ahlfors-Bers  variational  formulae  when  applied  to  quasiconformal  conjugates  of  Fuchsian 
groups. 

We  start  with  a  compact  Riemann  surface  XQ,  corresponding  to  the  plane  algebraic 
curve  P(x,y)  =  X^Zl^i'XV  =  0,  having  genus  say  g  >  1.  Let  us  assume  also  that 
XQ  =  U/GQ  where  GO  (i.e.  the  holomorphic  deck-transformation  group)  is  known.  Then 
we  consider  the  parametrized  family  of  compact  Riemann  surfaces  Xt  corresponding  to 
the  polynomial  equation  Pt(x,  y)  =  0  where  Pt(x,  y)  =  ]T}  ]C  aO'  W^V'  sucn  tnat  aij(f}  are 
holomorphic  functions  of  t  (t  in  a  small  disk  around  the  origin)  with  additional  restriction 
that  dij(G)  =  aij.  For  such  Xt  we  determine  first  variational  formula  for  7,  €  Gt  where 
Xt  =  U/Gt  (Gt  is  the  uniformizing  Fuchsian  group  corresponding  to  Xt) 


where  7  is  an  element  of  GO  (and  7,  7*  are  as  in  eq.  (16)). 

Remark.  Although  we  have  dealt  with  compact  Riemann  surfaces  and  the  torsion-free 
parabolic-free  Fuchsian  uniformizing  group  in  the  introduction  above,  the  theory  of 
Teichmuller  spaces  works  exactly  the  same  for  Riemann  surfaces  of  finite  coriformal  type 


Dr.  Dakshini  Bhattacharyya  tragically  passed  away  in  March  2000.  The  referee  had  indicated 
certain  minor  changes  in  the  paper  as  submitted  for  which  the  editor  could  not  obtain  the  author's 
approval  due  to  her  demise.  These  changes  have  been  incorporated  in  the  final  version. 

33 


34  Dakshini  Bhattacharyya 

-  namely  we  can  allow  distinguished  points  or  punctures  on  the  compact  Riemann 
surfaces  and  correspondingly  allow  elliptic  or  parabolic  elements  in  the  Fuchsian 
groups  under  scrutiny.  Those  results  are  exactly  parallel  and  nothing  new  needs  to  be 
said. 

2.  Invariance  of  sheet  monodromy  over  families  of  curves 

Monodromy  Invariance  Lemma.  To  solve  our  problems  we  have  to  find  a  correspondence 
between  the  ramification  (branch)  points  of  Pt(x,y)  =  0  lying  on  the  x-  sphere  for 
different  values  of  t.  Also  we  will  need  to  make  a  correspondence  between  the  algebraic 
functions  yt(x)  =  y(x,  t)  satisfying  Pr(jt,  y(x,  t))  =  0  for  different  values  of  f,  so  that  the 
monodromy  remains  invariant  at  the  corresponding  branch  points.  That  will  guarantee 
that  the  topological  structure  of  the  branched  covering  is  kept  invariant  as  t  changes. 

In  order  to  do  this  we  assume  certain  restrictions  on  Pt(x,y): 

Assume  degP(jt,y,f)  =  D  for  all  t.  Assume  also  that  there  exists  r,  5-  such  that 
r  +  s  =  D  where  0  <  r  <  m,  0  <  ,s  <  N  and  an(0)  ^  0  i.e  degree  Po(x,y)  =  D. 

Assume 

(1)  Po(x,y)  is  irreducible  in  the  polynomial  ring  C[x,y]. 

(2)  If  degree  Pt(x,y)  =  D,  then  degree  PO(JC,  y)  =  D;  that  is  if  we  substitute  t  =  0  in 
Pt(x,y)  degree  of  the  polynomial  remains  the  same. 

(3)  Suppose  Pt  is  of  degree  TV  in  the  y  variable  for  all  small  t: 


where 


- 

Let  D(t)  denote  the  discriminant  of  PN(x,t).  Then  assume  that  D(0)  ^  0  and 

ak(0)  ^  0. 

Let  D(x,  t)  be  the  discriminant  of  Pt(x,  y)  =  0.  Then  D(JC,  t)  =  PN(x,  r)Q(jc,  t)  where 


We  assume  that  Qo(0)  ^  0  and  D(0)  ^  0,  where  b(t)  =  discriminant  of  Q(jc,f). 
(5)  The  resultant  of  Q(jc,  t)  and  PN(x,  t)  does  not  vanish  at  t  =  0. 
Assume 


is  an  irreducible  polynomial  such  that  x  =  0  and  x  =  oo  are  ordinary  points,  and  the 
set  of  ramification  points  on  the  jc-plane  are  say  located  at: 


Then  it  is  not  hard  to  demonstrate  that: 

(i)  For  all  t  sufficiently  close  to  0,  the  polynomial  Pt(x,y)  is  irreducible  and  0,  oo  are 
ordinary  points. 

(ii)  The  ramification  points  on  the  ^-sphere  for  Pr(jc,  y)  are  holomorphically  dependent 
on  t  and  are  given  by  k  holomorphic  functions:   {&(*)>••  •>&(')}   such 
0(0)  =  (j  for  0  <  j  <  k  and  0  (r)  +  Cy  W  fo^  i  ^  j  and  all  t  small  enough. 


Variational  formulae  for  Fuchsian  groups  35 

(iii)  Assume  N  is  the  degree  of  Pt  in  the  y  variable  (this  follows  from  the  stability 
conditions  mentioned  above.)   Then  there   exists   holomorphic   function   germs 

{y\  0,  t  ),...,  yN(x,  t)}  around  (*,  /)  =  (0,  0)  G  C2  such  that 


for  all  (x,  t)  sufficiently  close  to  (0,0)  and  such  that  N  roots  of  the  y  equation 
P(x,y,  t)  =  0  are  given  by  yj(x,  t). 

(iv)  Analytic  continuation  of  yi(x,  t)  for  every  fixed  t,  |  t  \<  e  in  the  ^-sphere  along 
the  same  route  (avoiding  the  branch  points)  produces  the  same  permutation  of 
{y\  (x,  f),  .  .  .  ,  3>Ar(jc,  t)}  -  i.e.,  the  monodromy  permutations  are  independent  of  t. 

Idea  of  the  proof  for  (iv):  Follow  the  construction,  as  in  Siegel  [S],  for  each  £/(0)  we 
consider  a  circle  C/  with  center  at  £/(0)  such  that  any  two  of  them  does  not  intersect  and 
we  join  the  origin  to  0(0)  by  a  simple  curve  /,-  so  that  if  we  cut  CP1  along  these  curves  it 
remains  simply  connected.  Since  £/'s  are  holomorphic  function  of  t  we  can  find  a 
neighborhood  of  t  =  0  say,  N  =  {t  :\  t  \<  e}  such  that  CiC^O,  .  .  .  ,  CtW  lies  inside 
C\  ,  .  .  .  ,  C*  respectively  and  each  0  W  is  an  °Pen  connected  subset  lying  in  the  interior  of 
Q  1  <  i  <  n.  Now  for  each  point  XQ  on  d,  1  <  z  <  n  we  can  find  mutually  disjoint 
neighborhood  WI(XQ),  .  .  .  ,  WN(xQ)  of  0;(jc0,  0),  1  <  i  <  N  (where  P(JCO,  </>/Oo,  0),  0)  =  0 
and  0,-(;c,  0)  is  an  analytic  function  of  x  I  <  i  <N)  and  an  open  disc  U(XQ)  of  XQ  and  an 
open  disc  V(XQ)  of  t  =  0  such  that  \fx  G  t/(jc0),  Vr  G  V(XQ),  </>i(x,t)  G  W(x0)  and  the 
function  germs  are  analytic  on  U(XQ)  and  U(XQ)  Pi  &(N)  =  (p  for  all  z.  Again  since  the 
points  on  Q  1  <  z  <  n  form  a  compact  set  D  =  Uf=1Cj,  the  open  cover  {U(x)  :  x  G  D} 
has  a  finite  subcover  where  D  C  UjLj  £/(*,-).  Set  V  =  n"=1V(jc,-)  flTV.  Note  that 
</>/(jc,0)  =yX*o,0)  for  somey,  1  <j<N.  Let  us  consider  the  monodromy  permutation 
around  £1  (0).  For  simplicity  let  y\  (jc,  0)  —  »•  V2(jc,  0)  —  »  373  (jc,  0)  —>  yi  (x,  0).  We  shall  prove 
that  for  each  r  G  V  yi(x,  r)  -»•  y2(^,  0  ->  ^s(^,  0  ->  yi(^  0- 

Let  C/(XQ)  is  a  neighborhood  of  JCQ  such  that  U(XQ)  =  f/i(zo)  U  f/2(^o)-  Then 


V^  G  U2(xo),     Vt  G  V,     y3(*,  0  G 

as    )>3  (^,  0)  —  >y  i  (jc,  0)  in  the  neighborhood  of  x  = 


and 


By  construction  we  can  find  finite  number  of  points  jco,...,**  on  C\  and  their 
neighborhood  £/(xb),  .  .  .  ,  t/(jcjt)  and  disjoint  open  set  W\  (jc,-),  .  .  .  ,  WN(XI)  for  each  fixed  i, 

0  <  i  <  *  around  ^(jc/,0),   l<j<N  such  that  Vjc  G  l/(^),  f  G  V,  #(*,*)  €  W/(jcf) 

1  <  7  <  ^V-  Since  yi  (jc,  0)  analytically  continues  to  j2(jc,  0),  Wi  (jt*)  (i.e  the  neighborhood 
of  yi(jct,0))  intersects  W2(jc0)  (which  is  the  neighborhood 


36  Dakshini  Bhattacharyya 

Choose 

x  G  Ufa)  n  U2(XQ) 


=>yi(x,t)  G  W2(x0)  for  f  small  (by  continuity  of  y\  in  t) 

as  only  02(x,  0  G  W2(*0)  V*  G  V 

=^2(x,  0  =  3>i  (*,  0  for  t  small 

=^2(jc,  f)  =  y\(x,  t)     V?  G  V     (as  y\  and  </>2  are  analytic  function  of  t) 

=>yi  (i,  0  G  W2(;to)     Vf  G  V,  VJc  G  £/2(*0)  n  Ufa) 

=*yi  (*,  0  G  W2(*0)     Vf  G  V,  ;c  G  U2(xo) 

(as  for  f  fixed  y{  (jc,  r)  =  (&>(*,  r)    Vjc  G  f/2(j0)  n  Ufa) 

=^yi  (jc,  r)  =  cj)2(x,  t)     \/x  G  U2(xo)    by  analyticity  in  ^). 


So  if  we  continue  y\(x,t)  along  l\   we  get  02(:c,r).  Again  only  J2(^,0  G 
V^:  G  U\(XQ).  Let  us  fix  t  G  V.  If  we  continue  y\(x,t)  across  /i  the  function  we  get  say 
y(x,  t)  which  is  a  solution  of  P(JC,  y,  r)  =  0  (for  fixed  t)  and  hence  belong  to  either 
or  W2(jc0)  or 
Since 


and 

W2(x0)  n  Wi  (x0)  =  <p,    W2(*0)  n 

So 


=»y(^,  0  =  y2(^,  0    v*  G  Ui  (XQ) 

as  only  y2(jc,  ?)  G  W2(^0)     Vx  G  t/i  (JCQ)    V?  G  V. 

Since  t  G  V  is  arbitrary  yifo*)  continues  to  3^^,^)  and  thus  monodromy  remains 
invariant.  D 

3.  Construction  of  quasiconformal  marking  maps 

3.1  Construction  of  a  piecewise-affine  mapping  cj)t:  CP1  -+  CP1  which  carries  ramifi- 
cation points  of  P()(x,y)  to  the  ramification  points  ofPt(x,y) 

Recall  that  the  ramification  points  on  the  Riemann  sphere  for  the  covering  surface  Xt, 
(i.e.,  the  critical  value  set  for  the  branched  covering  map  xt  on  Xt  ),  are  assumed  to  be 
located  at  precisely  K  points  (for  each  t): 

(Ci(0,  •••,&«)• 

Let  g  denote  the  genus  of  each  of  the  Riemann  surfaces  Xt. 

The  aim  now  is  to  consider  XQ  as  the  base  point  for  the  Teichmiiller  space  T(XQ)  =  Tg9 
and  consequently  realise  each  Xt  as  a  point  of  the  Teichmiiller  space  by  constructing  an 
explicit  quasiconformal  (q.c)  marking  homeomorphism  from  XQ  onto  Xt: 


We  shall  have  <J)Q  as  the  identity  mapping.  For  these  see  Nag  [N]. 


Variational  formulae  for  Fuchsian  groups  37 

Thus  the  equivalence  class  of  the  triple  [Xo,  $t,Xt]  is  a  point  of  the  Teichmiiller  space 
T(Xo).  In  fact  we  shall  construct  a  holomorphic  'classifying  map'  (as  the  coefficients  of 
Pt  vary  holomorphically  with  t): 


mapping  the  t  disc  {\t\  <  e}  into  Tg. 
Using  the  Bers  projection 

0  :  Bel(X0)  ->  T(X0) 
we  will  have  a  lifting  of  the  'classifying  map'  rj  to  a  map 


The  marking  homeomorphism  between  the  compact  Riemann  surfaces  XQ  and  Xt  will 
be  obtained  by  lifting  a  mapping  <t>t  between  the  Riemann  spheres  that  carries  corres- 
ponding ramification  points  to  ramification  points.  Construction  of  </>t  :  P1  —  *  P1  is 
detailed  below. 

Recall  that  oo  was  set  up  as  an  ordinary  point  for  the  meromorphic  function  x  on  each 
Xt.  Hence  all  the  ramification  points,  0(0  1  <  i  <  k  lie  in  the  finite  jc-plane.  Restrict  the 
parameter  tin  a  relatively  compact  sub-disc  around  t  =  0:  t  e  Ae  =  {t  :\  t  \<  e}.  (To  save 
on  notation  we  still  call  the  radius  of  the  sub-disc  as  6.) 

Since  the  functions  0  are  analytic  in  t,  we  can  find  a  rectangle  R  containing  in  its 
interior  all  of  the  points  S  =  {0(0  :!<*<£,*€  Ae}.  Outside  R  we  will  define  cj>t  to 
be  the  identity  mapping. 

To  define  </>t  inside  R  we  take  the  first  (domain)  copy  of  CP1  and  triangulate  R  as 
follows:  we  divide  R  into  non-degenerate  triangular  regions  such  that  each  of  the  points 
0(0)  are  used  as  vertices.  Thus  the  triangulation  utilizes  a  set  of  vertices  containing  all 
the  K  points  0(0),  as  weU  as  some  extra  points  £s  for  some  index  set  s  =  K+ 
1,  .  .  .  ,  K  +  L.  (The  four  vertices  of  the  rectangle  jR  are  certainly  included  amongst  these 
last  L  vertices.  Also  note  that  each  triangle  utilized  is,  by  requirement,  non-degenerate  - 
namely  the  vertices  are  always  three  non-collinear  points.) 

Now  consider  another  copy  of  CP1  (which  will  serve  as  the  range  of  the  map  0,)  and 
divide  the  region  inside  the  rectangle  R  in  this  second  copy  into  triangular  regions  in  the 
natural  'corresponding'  fashion,  as  detailed  next:  namely  the  vertices  of  the  triangles  of 
this  second  copy  ofR  consist  of  the  new  ramification  points  0(0's  m  place  of  me  0(0), 
1  <  i  <  K,  -  together  with  the  same  extra  set  of  points  0  (f°r  index  set  s  —  K+ 
1,  .  .  .  ,  £  -h  L)  that  were  used  before.  Note:  these  last  L  vertices  are  left  undisturbed.  Of 
course,  the  edges  of  the  two  triangulations  correspond  exactly  since  the  vertices  have  the 
above  correspondence.  That  is,  if  (0(0),0(0)50t(0))  form  vertices  of  a  triangle  in  the 
first  copy  then  (0(0?  0(0  ?  Ot(0)  form  vertices  of  the  corresponding  triangle  in  the  second 
copy;  similarly,  if  (0(0)jC/»C0)  are  vertices  of  a  triangle  in  the  first  copy  then  (0(0> 
C/»  C$)  will  t>e  the  vertices  of  the  corresponding  in  the  second  copy,  etc. 

Remark.  Since  the  initial  triangulation  is  non-degenerate,  namely  the  vertices  of  any 
triangle  that  was  utilized  were  non-collinear,  then,  by  continuity  of  the  functions  (/(f), 
that  non-degeneracy  of  the  corresponding  triangulation  (on  the  range  copy)  remains  valid 
for  all  small  values  of  /  near  t  =  0. 

Affine  mapping  of  one  triangle  onto  another:  If  (£1,22,23)  are  any  three  non-collinear 
points  in  the  plane,  then  recall  that  their  closed  convex  hull,  (smallest  closed  convex  set  in 


38  Dakshini  Bhattacharyya 

the  plane  containing  these  points),  is  precisely  the  triangle  T  (includes  the  interior  and  the 
edges)  with  the  given  points  as  vertices.  From  elementary  linear  geometry  one  knows  that 
every  point  of  T  has  a  unique  representation  as  a  convex  combination  of  the  vertex 
vectors;  namely,  each  point  of  T  is  representable  as  \Z[  4-  ^11  4-  ^zs,  where  A,  fj,  and  i/  are 
real  numbers  in  the  closed  unit  interval  [0, 1]  such  that  A  4  f^  4  ^  =  1. 

Clearly  then,  given  any  other  set  of  three  non-collinear  vertices  (wi,W2,ws)  for  a 
second  triangle  71',  there  is  a  natural  affine  mapping  of  the  first  triangle  onto  the 
second  which  simply  sends  the  point  Xz\  4-  1*12  4-  z'Zs  of  T  to  the  point  Xw\  4-  ^2  4- 1^3 
of  T'. 

DEFINITION  OF  & 

We  therefore  define  the  desired  homeomorphism  (j)t  inside  the  rectangle  R  by  taking  the 
triangles  of  the  first  triangulation,  by  the  above  affine  mappings,  onto  the  corresponding 
triangles  of  the  second  triangulation.  Notice  that  if  two  triangles  share  a  common  edge, 
then  the  affine  mappings  defined  on  the  two  abutting  triangles  will  coincide  in  their 
definition  along  the  common  edge.  That  is  crucial.  Consequently  we  clearly  get  a  well 
defined  homeomorphism  (j>t  of  the  rectangle  R  on  itself,  and  outside  R  we  simply  extend 
0,  by  the  identity  map  to  the  whole  Riemann  sphere. 

It  is  clear  that  <pt  is  a  C°°-diffeoniorphism  when  restricted  to  the  interiors  of  the 
triangles  used  in  triangulating  R,  and  also,  of  course,  on  the  exterior  of  R. 

Lemma.  <j)r  is  quasiconformal  for  each  t  in  the  t  disc.  The  Beltrami  coefficient  of '  4>t>  is  a 
complex  constant  (of  modulus  less  than  unity)  when  restricted  to  the  interior  of  each 
triangle  in  the  initial  triangulation  of  the  rectangle  R.  Of  course,  the  Beltrami  coefficient 
is  identically  zero  in  the  exterior  of  R. 


3.2  Lifting  of<t>t  :  CP1— ^CP1  to  0,  :  XQ  — »X, 

Consider  the  following  diagram  of  Riemann  surfaces  with  the  vertical  arrows  being,  as 
we  know,  holomorphic  branched  coverings: 

XQ ^  Xt 


np1      <P*        ^  cp1 

PROPOSITION 

There  exists  a  quasiconformal,  orientation  preserving  homeomorphism: 


lifting  the  map  <j)t  :  CP1  -»  CP1  and  making  the  above  diagram  commute.  (Note  that 
is  the  identity.) 


Variational  formulae  for  Fuchsian  groups 


39 


*roof.  In  fact,  in  order  to  deal  with  unbranched  covering  spaces,  we  define  the  following 
>unctured  Riemann  surfaces: 


ind 


X'0  ^x'l{CPl  -all  critical  values  of  x} 


X[  =  Jt^CP1  -  all  critical  values  of  xt}. 


lestricted  to  X'0  and  Xfr  the  vertical  mappings  are  now  smooth  (=unbranched)  covering 
>rojections.  Observe  that  the  <j)t  was  designed  so  as  to  map  the  critical  values  of  x  onto 
hose  of  xt.  Now  we  can  apply  the  standard  lifting  criterion  for  maps  from  the  theory  of 
covering  spaces  to  demonstrate  that  <f>t  lifts.  Consequently,  at  the  level  of  fundamental 
groups  we  need  to  look  at  the  image  of  the  action  on  TTI  of  (<f>t  o  x)  as  compared  with  that 
)f  xt.  (See,  for  instance,  Theorem  5.1,  p.  128,  of  Massey  [M]  for  the  statement  of  the 
isual  lifting  criterion.) 

Since  the  monodromy  permutation  at  any  critical  point  say  Cm(0)  is  the  same  as  that 
iround  the  perturbed  critical  point  CmM>  anc^  since  <MCm(0))  =  CmW»  we  see  *at: 

TTi  (<t>t  O  X)wi  (XfQ,  W0)  =  TTi  (*,)7Ti  (Xj,  /30) , 

where  WQ  €  Xf0  and  JC(WG)  =  Zo  and  $>  £  -Xj  such  that  xt(/3o)  =  (f>t(zo))- 

Clearly  then  the  lifting  criterion  is  satisfied,  and  hence  the  homeomorphism  (f>t  lifts  to  a 
lomeomorphism  <j>t,  as  desired.  Certainly  the  lift  is  quasiconformal  since  the  vertical 
nappings  are  holomorphic.  This  completes  the  proof  of  the  proposition.  In  this  connec- 
ion  recall  the  following  result, 

Cheorem.  If  U  and  V  are  open  subsets  of  compact  surfaces  X  and  Y  respectively  with 
mite  complements,  then  any  homeomorphism  from  of  U  onto  V  extends  uniquely  to  one 
rfX  onto  Y.  D 

Finally  then,  for  our  applications  to  the  variation  of  Fuchsian  groups  we  may  lift  all  the 
vay  to  the  universal  covering  upper  half-planes  and  obtain  the  quasiconformal  homeo- 
norphism  $r(z)  =  $(z,  t)  from  U  to  U,  obtained  by  lifting  the  mapping  to  4>t  :  XQ  — >Xt. 

Thus  we  have  determined  $r(z)  so  that  the  following  diagram  commutes: 


u 


=  *OM) 


u 


.  Xt  =  U/Gt 


x  €  CP1. 


40  Dakshini  Bhattacharyya 

4.  Variational  formulae  for  the  Fuchsian  groups  of  varying  curve 

4.1  The  fundamental  variational  term 

Let  p,t(z)  denote  a  one-parameter  family  of  Beltrami  coefficients  on  the  upper  half-plane 
depending  real  or  complex  analytically  on  the  (real  or  complex)  parameter  t  near  t  =  0. 
Suppose  also  that  ^o(z)  =  0.  We  come  now  to  the  main  formula  that  we  shall  apply.  If 
JJLQ  =  0,  and  if  for  small  t  the  Beltrami  coefficient  is  given  by 

^(z)  =  tfi(z)  +  o(r),  where  0  e  L°°(U),  (2) 

then  one  has  an  important  integral  formula  expressing  the  solutions  of  the  family  of 
Beltrami  equations,  as  a  perturbation  of  the  identity  homeomorphism 


H-0(0,ze  U. 
Indeed,  the  crucial  first  variation  term,  w\  =  w,  for  real  t  is  given  by 


This  perturbation  formula  (see  Ahlfors  [A],  or  section  1.2.13,  1.2.14,  as  well  as  page 
175,  eq.  (1.21)  of  Nag  [N]),  will  be  fundamental  for  us.  We  shall  apply  it  to  the  family  of 
quasiconformal  mappings  $t  (§3)  standing  for  the  family  w^r. 

Since  in  our  set  up  ris  a  complex  parameter  we  may  as  well  deduce  the  form  of  the 
variational  terms  for  general  t  complex  -  which  follows  by  simply  applying  the  real  t 
formula  above  appropriately.  We  show  this: 

If  t  is  complex,  write  in  polar  form:  t  =  \t\eia  then  put  r  =  e~iat  =  t\.  Then  it  is 
straight  forward  to  see  that 


where  a  =  arg  (t).  But  re~2te  is  the  conjugate  of  t.  Therefore,  this  last  formula  says  that 
for  complex  t  we  have  the  final  important  formulae: 

Wto(z)=z  +  tw\(z)+tw\(z)+o(t\     zeU  (3) 

where 


Equation  (4)  will  be  manipulated  to  produce  the  chief  formulae  of  §4. 

Let  F  =  GO  C  PSL(2,  R)  denote  the  uniformizmg  Fuchsian  group  acting  as  deck 
transformations  for  the  covering  TT.  Then  there  is  a  biholomorphic  equivalence: 

x0  =  I//GO.  (5) 

It  follows  from  the  standard  Ahlfors-Bers  deformation  theory  of  Fuchsian  groups  (see 
Nag  [N])  that  the  quasiconformal  homeomorphism  3>r  is  compatible  with  the  Fuchsian 
group  GO,  in  the  sense  that  gt  =  $r  o  g  o  §t~l  is  again  a  Mobius  transformation  in 


Variational  formulae  for  Fuchsian  groups  41 

5SL(2,  R)  for  every  g  G  GO,  and  the  new  Fuchsian  group  (which  evidently  remains 
.bstractly  isomorphic  to  GO)  is  the  Fuchsian  group: 

Gr  =  $roG0o$r-1-  (6) 

Tils  is  the  group  of  deck  transformations  for  the  covering  TT,,  so  that  Xt  is  biholo- 
norphically  equivalent  to  U/Gt.  We  shall  write 

gt  =  $togo<f>-1  eG,  (7) 

or  any  fixed  g  €  GO  =  F. 

In  this  notation,  the  central  problem  of  our  work  is  to  determine  explicit  and  applicable 
ormulae  for  the  variation  of  gt  -  or,  equivalently,  to  compute  the  ^-derivative:  gt  at  t  =  0. 
^s  g  varies  over  any  generating  set  of  elements  for  the  group  GO,  we  shall  then  obtain,  up 
o  first  order  approximation,  a  corresponding  set  of  generating  elements  for  the  deformed 
groups  Gr. 

r.2  The  Beltrami  coefficient  /x,  of  $r 

Rotational  set  up.  Let  us,  for  notational  convenience,  denote  as  jc*  the  meromorphic 
unction  on  U  given  by  x  o  ?r  ,  (this  is,  of  course,  a  holomorphic  branched  covering  of  the 
liemann  sphere  by  the  upper  half  plane).  Clearly,  Jt*  is  automorphic  with  respect  to  the 
nichsian  group  F,  since  x*  descends  onto  the  surface  XQ  as  the  meromorphic  function  x 
hereon.  In  particular,  let  us  note  the  well-known  fact  that  this  function,  jc*,  can  be 
ixpressed  in  terms  of  the  standard  Poincare  theta-series  on  U  with  respect  to  the  group  F. 

Now  recall  from  the  previous  section  that  the  mapping  <pt  was,  by  our  very  definition,  a 
>iecewise  affine  quasiconformal  mapping.  So  the  Beltrami  coefficient  of  c/>t  was  a 
:omplex  constant  on  each  triangle  of  the  triangulation  of  the  domain  rectangle  R.  (The 
ieltrami  coefficient  need  only  be  specified  almost  everywhere  -  therefore  we  will  ignore 
t  on  the  edges  and  vertices  of  the  triangulation.) 

Moreover  we  know  that  the  vertices  of  the  triangulation  (in  the  image  plane)  depend 
lolomorphically  on  t  -  since  the  ramification  points  (/(/)  were  holomorphic  functions  of 
.  Here  is  the  main  proposition  we  require. 


rhe  Beltrami  coefficient  of$t  is 

M,(z)  =  ri>(z)  +  o(t),z  e  17,  v(z) 
vhere 


w 


CP1.  (8) 


lere  the  Beltrami  coefficient  for  the  piecewise-  affine  mappings  <j)t  on  the  Riemann  w- 
iphere  has  been  expanded  up  to  first  order  in  t  as  below: 


Further  note  that  i/(w)  is  a  constant  on  each  triangle  of  the  first  (domain)  triangulation 
)f  R,  and  it  is  zero  for  all  w  outside  R. 


42  Dakshini  Bhattacharyya 

Note.  The  F  invariant  Beltrami  coefficient  v  above,  represents  the  tangent  vector  to  the 
one  parameter  family  of  Beltrami  coefficients  fj,t  which  arise  from  the  one  parameter 
family  of  quasiconformal  mappings  $f. 

Proof.  From  the  above  commutative  diagram  for  the  liftings  we  have 

(xt  o?rr)  o  $f  =  </>t  o  (XOTT).  •  (10) 


Taking  the  d  and  d  derivatives  in  (10),  and  remembering  that  all  the  vertical  maps  are 
holomorphic  coverings  (possibly  branched  as  we  know),  we  obtain  the  Beltrami 
coefficient  of  $t  on  U: 


Clearly  then  the  statements  in  the  Proposition  follow  because  the  0r(w)  are  a  family  of 
piecewise  affine  quasiconformal  homeomorphisms  on  the  w-sphere  which  vary 
holomorphically  in  t.  Thus,  remembering  that  0o  is  the  identity,  we  see  that  the  Beltrami 
coefficients  of  the  family  0r  indeed  must  have  an  expression  as  in  (9)  with  z/(w)  being 
piecewise-constant.  D 

Remark.  Note  that  the  Beltrami  coefficient  of  <j>t  is  a  holomorphic  function  of  the 
parameter  t  in  the  neighborhood  of  t  =  0.  The  map 


takes  values  in  the  complex  Banach  space  /^(CP1),  -  and  the  holomorphy  is  as  a  map 
into  this  Banach  space. 

Beltrami  coefficients  automorphic  with  respect  to  T.  We  must  remember  from  the 
general  theory  (see  §1.3.3  of  Nag  [N])  one  further  fundamental  fact.  Since  the 
quasiconformal  maps  3>r  are  compatible  with  F  their  Beltrami  coefficients  are  (—1,1) 
forms  on  U  with  respect  to  F.  (We  called  them  F-invariant  Beltrami  coefficients.) 

Indeed,  if  jj,  is  the  complex  dilatation  of  a  quasiconformal  mapping  that  conjugates  F 
into  any  group  of  Mobius  transformations,  then 

(/*<>«)(?/*')  =  /*,*•«-,     f°r  aU     £€F.  (12) 

We  denote  the  Banach  space  of  complex  valued  L°°  functions  on  U  that  satisfy  equation 
(12)  for  every  g  e  F,  by  the  notation:  L°°(Z/,  F).  See  p.  49  of  [N].  Thus,  /x,  belongs  to  the 
open  unit  ball  of  this  Banach  space  for  all  small  t  ,  and  also  therefore  0  belongs  to  this 
Banach  space  of  automorphic  objects. 

4.3  The  variational  formula  for  $, 

We  come  to  the  chief  application  of  the  perturbation  formula  (eq.  (4))  in  our  specific 
context  of  varying  algebraic  curves. 

Let  F  denote  a  closed  fundamental  domain,  with  boundary  of  two-dimensional 
measure  zero,  for  the  action  of  F  on  C7;  (for  instance,  we  may  choose  F  as  any  standard 
Dirichlet  fundamental  polygon  for  the  Fuchsian  group  F).  Thus  TT  maps  F  onto  XQ,  and  TT 
is  one-to-one  when  restricted  to  the  interior  of  F. 


Variational  formulae  for  Fuchsian  groups  43 

Recall  that  jc  was  itself  a  meromorphic  function  of  degree  N  on  the  compact  Riemann 
surface  Xo,  (see  §2,  3).  Consequently,  when  restricted  to  the  interior  of  F  the  mapping  x* 
is  a  AMo-1  branched  holomorphic  covering  map  onto  the  Riemann  sphere  -  missing  only 
a  set  of  areal  measure  zero.  Since  this  is  a  finite  covering  space  situation  (aside  from  a 
measure  zero  set  of  branch  points  which  we  may  discard  to  start  with),  we  may  choose  a 
decomposition  of  F  into  N  regions: 

(13) 


Here  the  Dj  are  mutually  disjoint  domains  (except  for  boundary  contact,  as  usual  in 
choice  of  fundamental  regions),  partitioning  F,  with  the  basic  property  that  each  Dj  maps, 
via  x*,  in  a  one-to-one  fashion  onto  the  entire  Riemann  sphere  (missing  atmost  a  measure 
zero  subset).  (Recall  that  the  compact  Riemann  surface  XQ  was  described  as  an  N-  sheeted 
branched  cover  of  the  sphere  -  by  the  degree  N  meromorphic  function  x.) 

A  kernel  function  associated  to  T.  We  introduce  as  an  useful  matter  of  notation,  the 
following  function  of  two  variables:  z  €  U,  r  6  C  (not  lying  on  the  F  orbit  of  z): 

-  ("> 


We  are  now  in  a  position  to  state  a  main  result. 

Theorem.  On  variation  of  $t-  The  lifted  quasiconformal  maps  &t  on  U  satisfy  the 
following  first  order  expansion  for  small  t, 

$,(z)=z  +  twl(z)+-twtl(z)  +  o(t),z£U,  (15) 

where 

~l 


N       r    r 

k^\J  JCPI 

Here  we  have  denoted  by  jc*^  the  restriction  of  the  projection  x*  =  x  o  TT  (which  is  a 
meromorphic  and  T-automorphic  function  on  U),  to  the  region  Dk  C  F,  k  =  1 , . . . ,  N. 
Here  v  denotes  the  function  on  the  w-sphere  appearing  in  formula  (9)  of  the  Proposition 
in  sub- section  4.2  above.  (Recall  that  v  is  simply  a  constant  assigned  on  each  triangle  in 
the  triangulation  ofR,  with  i/  being  identically  zero  outside  R.) 

Note  furthermore,  that  since  x*  is  a  meromorphic  function  on  U,  we  may  replace  in  the 
above  formula  the  derivative  of  its  inverse  by  the  reciprocal  of  its  own  derivative,  as 
shown  below: 

dx~l  ,   %       _   /djt*  L.  ,  % 


f\       \wr J  I      H7    v  y'       •r"~^'*v^y?       +>  >-  *^K- 

These  derivatives  can  therefore  be  calculated  from  the  expression  for  jc*  which  will  be 
available  in  terms  of  the  standard  Poincare  theta  series  on  U  with  respect  to  7.  (Therefore 
we  see  that  1/7  €  GO  then  the  variational  formula  for  7,  =  3>r  o  7  o  $~j  €  Gt  is 

MO,  (16) 


44  Dakshini  Bhattacharyya 

"where 

7  =  wi  07-7'^!, 
7*  =  w*  o  7  —  7'wJ. 

F0r  r/us,  see  Nag  [N].) 

During  the  course  of  the  proof  we  shall  show  that  all  integrals  and  summations 
appearing  in  sight  are  absolutely  convergent.  For  facts  regarding  Poincare  theta  series  and 
their  utilization  in  expressing  meromorphic  functions  on  U/T,  see  [Kra,  Kr]. 

Proof.  We  shall  have  to  manipulate  the  variational  formula  (4)  which  said: 

Wl  (z)  =  —  -  /   /    [£>(w}R(w,  z)  +  i>(w)/e(w,  z)]dw  A  dw 
2J7TJ  y  17 


By  general  theory  quoted  above,  the  integrals  involved  in  (4)  are  necessarily  absolutely 
convergent. 

To  obtain  the  final  result  for  wi  and  w*,  there  are  several  chief  ideas  which  we  first 
explain  in  words: 

(i)  Write  each  of  the  two  integrals  over  U  as  a  sum  of  integrals  over  all  the  tiles  in  the 
F-tessellation  of  U  -  obtained  by  decomposing  U  as  the  union  of  the  fundamental 
domain  F  and  its  translates:  i.e.,  U  =  (Jg^(g(F)). 

(ii)  Utilizing  the  F-automorphic  nature  of  the  Beltrami  coefficient  0  (see  eq.  (12) 
above),  and  making  a  change  of  variables  by  w  =  g(z),  we  can  transform  the  integral 
over  g(F)  to  an  integral  again  over  F  itself. 

(iii)  Consequently,  the  original  expression  for  wi  becomes  simply  an  integration  over  F 
of  a  certain  expression  on  F,  after  interchanging  summation  and  integration.  (The 
validity  of  the  interchange  is  guaranteed  by  the  absolute  convergence  of  the  result, 
together  with  the  dominated  convergence  theorem.  The  main  details  of  this  critical 
interchange  of  sum  and  integral  are  spelled  out  in  the  remarks  attached  at  the  end  of 
the  proof.) 

(iv)  Finally  we  decompose  F  itself  into  the  N  pieces  £>i,  .  .  .  ,  DN  (as  explained  with  eq. 
(13)  above)  -  and  hence  we  may  eliminate  z>  by  replacing  it  with  occurrences  of  v 
itself,  and  thus  express  the  final  result  as  integrations  over  the  Riemann  sphere  CP1  , 
as  desired. 

The  first  three  of  the  above  steps  are  carried  out  e.g.  in  [A].  Let  us  now  get  down  to  the 
main  business  of  showing  the  exact  nature  of  how  these  transformations  come  about  in 
the  expression  for  w\.  First  of  all  note: 


t  f         i>M 

yy,,(,-i)o,-r) 

V^    f  f  ^(w) 

=  /  ^  /  /      ~~7 — ZTv \  ^w  ^  ^w'     F  =  fundamental 


region  of  T  in  (7. 


Variational  formulae  for  Fuchsian  groups  45 

Perform  a  change  of  variables  on  g(F)  by  w  =  u  +  iv  =  g(z) 

rdzAdz 


For  convergence  arguments  we  note  that  since 

71 ifoty 


<  CXD. 


,  I  w  ||  w  —  l  II  w  —  r 
This  demonstrates  that  the  series 


(17) 

is  absolutely  convergent.  Note  that,  for  convenience,  we  have  written  tpg  here  for  the 
following  frequently  recurring  expression: 


We  shall  show  by  a  measure-theoretic  lemma  in  the  remarks  appended  to  the  bottom  of 
this  proof,  that  we  are  allowed  to  change  summation  and  integration  in  the  summation 
(17).  We  shall  utilize  crucially  this  interchange  immediately  in  what  follows.  Returning 
therefore  to  the  actual  expression  for  the  variational  term  M>I,  we  now  obtain: 


l)    rr 

TZ  y  yF 


Similarly 


46 


Dakshini  Ehattacharyya 


That  completes  the  manipulation  of  the  formula  to  a  point  that  already  has  points 
interest;  we  have  carried  out  steps  (i),  (ii),  (iii)  -  and  now  we  are  integrating  over  F  (i. 
over  XQ),  rather  than  over  U. 

The  final  steps  are  for  carrying  out  the  program  outlined  in  point  number  (iv)  abo1 
This  goes  as  detailed  below: 

Let 


(x  o  7r)(A)  = 


and  denote  x  o  TT|D.  =  **,,- 


for  each  i  =  1 , . . . ,  N.  Setting  (x  o  TT)  (C)  =  w,  C  €  U  and  w  e  CP1 ,  and  using  the  relati 
(eq.  (8))  between  v  and  i/9  we  will  have 

_/•_      1  \ 
wi(z)  =: 


2?ri 


N       r   r 

S//c, 


El 


|dw/dC|2 


2m 


Mxr 
...     , 


tlL 


Similarly 


dw  A  dw 


dw  A  dw. 


dw  Adw 


That  at  last  is  exactly  the  expression  desired  and  claimed  in  the  Theorem  and  we 
through. 

The  interchange  of  summation  and  integration  above  in  the  series  (17),  follows  f 
some  straightforward  facts  of  the  theory  of  measure  and  integration.  For  instance, 
purposes  are  adequately  served  by  the  following  result  (see  Rudin  [R]): 

Lemma.  Suppose  {/„}  is  a  sequence  of  complex  measurable  junctions  defined  all 
everywhere  on  a  complete  measure  space  (X,  IJL)  such  that 


|/n  I  d/x<  co. 


Then  the  series  f(x)  =  Y^Tfn(x)  converges  absolutely  for  almost  all  x,  and  f  €  L1 
moreover,  the  summation  and  integration  can  be  interchanged,  namely: 


=  £    ffndp. 
i     JX 


Variational  formulae  for  Fuchsian  groups 

Acknowledgment 

This  work  is  part  of  the  author's  doctoral  thesis  written  under  the  guidance  of  Professor 
Subhashis  Nag  at  the  Institute  of  Mathematical  Sciences,  Chennai. 

References 

[A]  Ahlfors  Lars  V,  Lectures  on  quasiconformal  mappings  (New  York:  Van  Nostrand) 

[B]  Bers  L,  Uniformization,  moduli  and  Kleinian  groups,  Bull  London  Math.  Soc.  4  (1972) 
300 

[C]  Che  valley  C,  Introduction  to  the  Theory  of  Algebraic  Functions  of  One  Variable,  Am. 
Soc.  (Rhode  Island:  Providence)  (1951) 

[FK]  Farkas  H  and  Kra  I,  Riemann  surfaces  (New  York  Inc:  Springer- Verlag)  (1980) 
[G]  Gunning  R  C,  Lectures  on  Riemann  surfaces.  Mathematical  notes  (New  Jersey:  Princeton 

University  Press,  Princeton)  (1966) 
[JS1]  Jones  G  A  and  Singerman  D,  Complex  Functions;  An  algebraic  and  geometric  viewpoint 

(Cambridge:  Cambridge  University  Press)  (1987) 
[JS2]  Jones  G  A  and  Singerman  D,  Belyi  Functions,  Hypermaps  and  Galois  Groups,  Bull.  London 

Math.  Soc.  28  (1996)  561-590 

[K]  Knopp  K,  Theory  of  functions  (New  York:  Dover  Publications)  (1945)  1947  vol.  1,  2 
[Kr]  Kra  I,  Automorphic  forms  and  Keinaian  groups  (W.A.  Benjamin  Inc)  (1972) 
[L]  Lang  S,  Undergraduate  algebra  (New  York:  Springer- Verlag)  (1987) 
[Leh]  Lehner  J,  Discontinuous  Groups  and  Automorphic  Functions,  Am.  Math.  Soc.  (Rhode  Islanci: 

Providence)  (1964) 
[LV]  Lehto  O  and  Virtanen  K,  Quasiconformal  mappings  in  the  plane,  2nd  ed.  (Berlin  and  Nfew 

York:  Springer-Verlag)  (1973) 

[M]  Massey  W  S,  A  basic  course  in  algebraic  topology  (New  York  Inc:  Springer-Verlag)  ( 1 9<3>  1 ) 
[N]  Nag  S,  The  complex  analytic  theory  of  Teichmuller  spaces  (New  York:  John  Wiley  and  Sons) 

(1988) 

[R]  Rudin  W,  Real  and  complex  analysis  (McGraw-Hill  Book  Co.)  (1986) 
[S]  Siegel  C  L,  Topics  in  complex  function  theory  (1969)  vol.  I;  Elliptic  functions 

uniformization  theory  (1971)  vol.  II;  Automorphic  functions  and  abelian  integrals 
[Spa]  Spanier  Edwin  H,  Algebraic  topology  (New  York:  McGraw-Hill)  (1966) 
[Spr]  Springer  G,  Introduction  to  Riemann  surfaces  (Massachusetts:  Addison- Wesley, 

(1957) 
[SV]  Shabat  G  B  and  Voevodsky  V  A,  Drawing  curves  over  numberflelds,  in:  GrothendiooJc 

Festchrift  ffl  (ed.)  P  Cartier  et  al,  Progress  in  Math.  88  (Birkhauser:  Basel)  (1990)  \99 
[W]  Wolfart  J,  Mirror  invariant  triangulations  of  Riemann  surfaces,   triangle  groups 
Grothendieck  dessins:  Variations  on  a  thema  of  Belyi',  preprint  (Frankfurt)  (1992) 


roc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  Ill,  No.  1,  February  2001,  pp.  49-63. 
5)  Printed  in  India 


Amits  of  commutative  triangular  systems  on  locally  compact  groups 


RIDDHI  SHAH 

School  of  Mathematics,  Tata  Institute  of  Fundamental  Research,  Homi  Bhabha  Road, 
Mumbai  400  005,  India 

MS  received  22  December  1999;  revised  28  July  2000 

Abstract.  On  a  locally  compact  group  G,  if  i/ *«  -+  ^,  (kn  —>  oo),  for  some  probabi- 
lity measures  vn  and  JJL  on  G,  then  a  sufficient  condition  is  obtained  for  the  set 
A  =  {z/™|w  <  /:„}  to  be  relatively  compact;  this  in  turn  implies  the  embeddability  of  a 
shift  of  IJL.  The  condition  turns  out  to  be  also  necessary  when  G  is  totally  disconnected. 
In  particular,  it  is  shown  that  if  G  is  a  discrete  linear  group  over  R  then  a  shift  of  the 
limit  p.  is  embeddable.  It  is  also  shown  that  any  infmitesimally  divisible  measure  on  a 
connected  nilpotent  real  algebraic  group  is  embeddable. 

Keywords.  Embeddable  measures;  triangular  systems  of  measures;  infmitesimally 
divisible  measures;  totally  disconnected  groups;  real  algebraic  groups. 


.  Introduction 

Commutative  triangular  systems  of  probability  measures  on  locally  compact  groups  have 
>een  studied  extensively  and  recently  the  embedding  of  the  limit  \JL  (or  a  translate  xp,, 
;  G  G)  have  been  shown  on  a  large  class  of  groups  under  certain  conditions  like  infinite- 
imality  of  triangular  system  and/or  'fullness'  of  the  limit  p,  (see  [S4]  for  the  latest  results 
md  the  literature  cited  therein  for  earlier  results).  Generalizing  the  techniques  developed 
Q  [S3,S4],  we  extend  our  earlier  result  to  some  particular  triangular  systems  on  algebraic 
groups.  We  also  discuss  special  triangular  systems  of  identical  measures,  i.e.  limit 
heorems.  In  particular  if  zA  —*  p.  on  G  then  we  give  a  sufficient  condition  for  the  set 
L  =  {v™  |  m  <  kn}  to  be  relatively  compact;  this  in  turn  would  imply  the  embeddability 
>f  a  shift  of  the  limit  //.  The  condition  turns  out  to  be  also  necessary  if  G  is  totally 
lisconnected.  We  hereby  generalize  our  earlier  results  on  limit  theorems  on  Lie  groups  to 
general  locally  compact  groups.  We  also  show  the  embedding  of  a  shift  of  the  limit  \JL  if  G 
s  a  discrete  linear  group  over  R. 

Let  G  be  a  locally  compact  (Hausdorff)  group  and  let  Ml(G)  be  the  topological 
;emigroup  of  probability  measures  with  weak  topology  and  convolution  as  the  semigroup 
Deration.  Let  \JL,  v  be  any  measures  in  Ml  (G).  Let  the  convolution  product  of  p,  and  v  be 
lenoted  by  //i/.  For  any  compact  subgroup  H  of  G  let  LJH  denote  the  normalized  Haar 
neasure  of  H.  Let  M1H(G)  =  uHMl  (G)o;//,  then  Af#(G)  is  a  closed  subsemigroup  of 
\i{  (G)  with  identity  ujj.  For  any  x  G  G,  let  8X  denote  the  Dirac  measure  at  x  and  let 
c/4  =  8xfr  (similarly,  fjjc  =  ^8X).  Let  /M  =  {x  G  G  \  xp,  =  IJLX}  and  let  I(p]  =  {x  G  G  | 
c/z  =  px  =  /4,  then  /^  (resp.  /(/x))  is  a  closed  (resp.  compact)  subgroup  of  G.  Let  J^  = 
[A  £  M1  (G)  j  AJU  =  //A  =  p}.  Clearly,  JM  is  a  compact  semigroup  and  for  any  A  £  Ml  (G), 
\  G  Jp.  if  and  only  if  supp  A  C  /(M).  Let  G(/x)  be  the  smallest  closed  subgroup  of  G 
:ontaining  supp//.  Let  N(fj)  (resp.  Z(/x))  be  the  normalizer  (resp.  centralizer)  of  G(/x)  in 

49 


50  Riddhi  Shah 

G.  Let  p,  denote  the  adjoint  of  p,,  defined  by  p,(B)  =  p>(B~l),  for  all  Borel  subsets  B 
IJL  is  said  to  be  symmetric  if  p,  =  /L  Let  G°  denote  the  connected  component  of  the  idc 
in  G.  For  a  set  A  C  M1  (G)  and  a  normal  subgroup  C  C  G,  we  denote  A/C  =  7r(A),  v 
TT  :  G  — »  G/C  is  the  natural  projection. 

A  measure  p  G  M1  (G)  is  said  to  be  infinitely  divisible  (resp.  weafc/y  infinitely  divu 
if  for  every  n  G  N,  there  exists  p,n  G  M1  (G)  such  that  /zj|  =  IL  (resp.  /^jcw  =  /x  for 
jtrt  G  G);  and  it  is  said  to  be  embeddable  if  there  exists  a  continuous  one-parai 
convolution  semigroup  {//f},>0  such  that  JJL\  =  /x.  Since  we  aim  to  prove  the  embed 
lity  of  a  given  measure  under  various  conditions,  the  reader  is  referred  to  [M2],  a  si 
article  on  the  embedding  problem  of  infinitely  divisible  measures. 

Let  S  be  a  Hausdorff  semigroup  with  identity  e  and  let  s  G  S.  Let  Ts  denote  the  s 
two  sided  factors  of  5,  that  is,  Ts  =  {r  €  5  |  tr  =  rf  =  s  for  some  r  E  S}.  Elements  5, 
are  said  to  be  associates  if  s  and  t  are  two  sided  factors  of  each  other,  i.e.  s  €  7 
/  E  Ts.  A  subset  A  of  5  is  said  to  be  associatefree  if  51,  f  G  A  are  associates  then  s  = 
element  fe  in  S  is  said  to  be  an  idempotent  if  /z2  =  h.  An  element  5  is  said  to  be  bald 
if  e  is  the  only  idempotent  contained  in  Ts.  For  a  subset  A  of  5,  a  decomposition  o 
s  =  s\--sn,  for  some  n  E  N,  where  si  G  A  and  SjSj  =  sys/  for  all  z,  7,  is  called  s 
decomposition  ofs.  An  element  5  (in  S)  is  said  to  be  infinitesimally  divisible  if  s  has 
decomposition  for  every  neighbourhood  U  of  e  in  5.  A  set  A  =  {s/y-  G  5  |  i  G  N,  1  ; 
W,-,  Hi  — >  oo  as  z  — »  00}  is  said  to  be  a  triangular  system  in  5;  we  will  sometimes 
A  =  (5y)"eNj=i-  A  is  said  to  be  commutative  if  for  every  fixed  z,  5,7  commute  with 
other,  it  is  said  to  be  infinitesimal  if  as  z  — >•  oo,  sy  — »•  e  uniformly  in  7.  We  say  tl 
converges  to  p  if  s-,-1  •  •  •  5^  =  Si  ->  /x. 

In  §  2,  we  prove  a  limit  theorem  for  general  locally  compact  groups,  (see  Theoreir 
In  §  3,  we  show  that  if  iA  -+  ^,  (kn  ->  oo),  on  a  discrete  linear  group  over  R,  then 
embeddable  for  some  x  G  G  (see  Theorem  3.1).  In  §4,  we  show  that  any  infinitesi 
divisible  probability  measure  JJL  on  a  connected  nilpotent  real  algebraic  group  is  er 
dable,  (more  generally  see  Theorem  4.1). 


2.  Limit  theorems  on  locally  compact  groups 

Theorem  2.1.  Let  G  be  a  locally  compact  group  and  let  TT  :  G  — >  G/G°  be  the  m 
projection.  Let  {z/B}  be  a  relatively  compact  sequence  in  Ml(G)  such  that  for  any 
point  v  of  it,  G(TT(Z/))  is  a  compact  group  in  G/G°  and  zA  -*  jj,for  some  p,  G  M1  (C 
for  some  unbounded  sequence  {kn}  C  N.  Suppose  that  for  some  connected  nil\ 
normal  subgroup  N  of  G,  the  closed  subgroup  generated  by  supp  p,  and  N  contair 
Then  the  set  A  =  {i/^  |  m  <  kn}  is  relatively  compact  and  there  exists  x  G  7M  such  t} 
is  embeddable. 

Remarks.  (1)  The  above  theorem  generalizes  Theorem  1.7(1)  of  [S4].  (2)  If  G  is  t 
disconnected  then  G°  =  {e}  and  hence  the  above  theorem  implies  that  if  iA  -*  /x 
{vn}  is  relatively  compact  and  for  any  limit  point  v  of  it,  G(z/)  is  compact  thei 
relatively  compact.  Conversely,  if  A  is  relatively  compact  then  so  are  {z/n}  and  {z/*" 
for  any  limit  point  z/  of  {z/B},  G(v)  is  compact  as  {i/n}cA.  Thus,  for"! 
disconnected  groups  we  get  a  necessary  and  sufficient  condition  for  the  set  A  as  ab 
be  relatively  compact. 
We  first  prove  a  more  general  theorem  for  totally  disconnected  locally  compact  g 


Triangular  systems  of  measures  5  1 

Theorem  2.2.  Let  G  be  a  totally  disconnected  locally  compact  group  and  let  {vn}  C 
Ml(G)  be  such  that  vn—*v  'where  G(y)  is  compact  and  v^i/n  —  >  J^far  some  sequence 
{i/  }  in  Ml  (G)  such  that  ^nz^  =  v'nvnfor  all  n.  Then  given  any  neighbourhood  U  ofe  and 
an  e  >  0  there  exists  an  I  such  that  for  all  large  n,  v™(G(v)l(n)U)  >  (1  -  e/,  for  all 
m  <kn.  In  particular  A  =  {is™  \  m  <  kn}  and  {i/n}  are  relatively  compact. 

Proof.  As  G(v)  is  compact  and  vm  e  !TM,  for  ail  ra,  suppz^  C  xl(fj)  =  I(/~i)x,  for  all 
jc  G  supp  z/  (cf.  [S4],  Theorem  2.4).  Therefore  G(z/)/Qu)  is  a  compact  group. 

Let  V  be  an  open  compact  subgroup  of  G  such  that  V  is  normalized  by  G(z/)/(/i),  and 
VCU.  Since  vn  -»  v,  z/n(G(i/)/(/x)V)  >  1  -  e,  for  all  large  n.  Let  V1  =  {A  |  \(G(v) 
I(fji)V)  >  (1  -  6}l/2}  and  let  U1  =  {A  |  A(G(z/)/(/x)V)  >  1  -  5}  for  some  positive  5  <  e. 
Then  VV  C  £/'.  Let  J  =  {A  €  M!(G)  |  supp  A  C  G(z/)/(/x)V}.  Clearly,  J  is  a  compact 
semigroup  and  JVf  =  V7.  Let  X  e  U'  \V.  If  possible,  suppose  that  A"  €  7^  for  all  n,  then 
by  Theorem  2.4  of  [S4],  supp  A  C  xl(^)  =  I(iJ,)x9  for  all  x  E  supp  A.  Since  A  G  £/', 
supp  A  C  G(i/)I(fj)V9  i.  e.  A  E  J  C  V",  a  contradiction.  Hence  for  A  e  T^  n  E/'  \  V,  there 
exists  n  =  n(A),  such  that  An  ^  TM.  By  Lemma  2.1  of  [S4],  T^  n  U'  \  V"  is  compact.  As  in 
the  proof  of  Lemma  2.5  in  [S4],  one  can  find  /,  such  that  for  any  A  G  7^  n  U1  \  V,  // 
cannot  be  expressed  as  ^  =  AZA',  for  any  A'  which  commutes  with  A. 

Since  vn  -*  v,vn£  V,  for  all  large  n.  Let  such  a  large  n  be  fixed.  Then  there  exists 
an  >  1,  such  that  v™  e  V,  for  all  m  <  an  and  van»  &  V.  Therefore,  i/  J1  €  ^  \  ^  C 
I/'  \  V.  Let  bn^kn-  lan  if  ton  <  Jfcn,  otherwise  bn  =  0.  If  ^n  =  0,  then  i/£  e  (t/')',  for 
all  m  <  jfcn.  Therefore,  for  all  large  n,  z/£(G(z/)/(/z)V)  >  (1  -  5)',  and  hence  i/£(G(i/) 
J(/x)J7)  >  (1  -  e)/  for  all  m  <  ^,  as  V  C  U  and  8  <  e. 

We  now  show  that  bn  =  0,  for  all  large  n.  If  bn  ^  0  for  infinitely  many  «,  then 
iAi^-»^.  Since  {v*"}cUr\Vf,  by  Lemma  2.1  of  [S4],  {i/J«}  is  relatively 
compact  and  it  has  a  limit  point  (say)  A,  such  that  //  =  A'  A7,  for  some  A',  which  is  a  limit 
point  of  {i/J"z^},  i.e.  A  e  T^  n  £/'  \  V  and  AY  =  A'  A.  This  is  a  contradiction  to  the 
choice  of  /  as  above. 

Now  it  is  enough  to  show  that  A  is  relatively  compact  as  this  would  also  imply  that 
{i/n}  is  relatively  compact.  Let  ^n  =  vkjv'n.  Then  nn—^^  and  for  each  n,  v™  G  T^n  for  all 
m  <  kn.  Let  F  =  G(z/)/(/x)V  for  V  as  above.  Then  F  is  compact.  Let  A'  =  {A  G  M^G)! 
A(F)  >  (1  —  c)  /2}.  Then  from  above,  A  C  Af  .  Since  /xn  —  >  /x,  for  every  5  >  0  such 
that  <5  <  (1  -  e)l/29  there  exists  a  compact  set  Kg  such  that  fJn(Ks)  >  l-S  (cf.  [H], 
Properties  L2.20(2)).  Therefore,  for  every  n,m  as  above,  there  exists  jcn?m  such  that 
v™(Ksxnjn)  >  1  —  6.  Now  since  A  C  A',  the  above  implies  that  xn,m  E  ^"^  and  hence 
v™(Krs)  >  1  -  6,  where  ^  =  K^K^1F  which  is  a  compact  set.  In  particular  A  is  relatively 
compact  (cf.  [H],  1.2.20).  This  completes  the  proof. 

We  now  prove  several  results  which  will  be  needed  to  prove  Theorem  2.1. 


Lemma  2.3.  Let  G  be  a  locally  compact  first  countable  group  and  let  {/^n}>  {^«}  and 
{z/n}  be  sequences  in  Ml(G)  such  that  \nvn  —  vn\n  =  fJ>n  —  *  M  for  some  ^  G  M^G). 
Then  there  exists  a  sequence  {xn}  such  thatxn  G  N(^n)  for  each  n  and  {\nxn}  ond  {xn\n} 
are  relatively  compact  and  all  its  limit  points  are  supported  on  supp  ^. 
The  proof  is  quite  similar  to  Proposition  1.2  in  [DM]  and  Theorem  2.2  in  ch.  Ill  of  [P]. 

Proof.  For  any  integer  r  >  0  there  exists  a  compact  set  Kr  C  supp  p,  such  that 
fJL(Kr)  >  1  —  4~(r+1).  Without  loss  of  generality,  we  may  assume  that  Kr  C  Kr+\  for  all 
r.  Let  {Ur}  be  a  neighbourhood  basis  of  e  in  G  such  that  each  Ur  is  relatively  compact, 


52  Riddhi  Shah 

Ur+i  C  Ur  for  all  r  and  C\rUr  =  {e}.  Since  ^n  ->  //,  there  exists  nr  G  N,  such  tl 
Vn(KrUr)  >  1  ~  2~r  and  ^(UrKr)  >  I  -  2~r,  for  all  7i  >  nr.  Let  Ern  =  {x  £  G  \ 
(KrUrx~l)  >  I  -  2-'}  and  let  Fn  =  n{rk<n}££. 

A  simple  calculation  as  in  Theorem  2.2  of  ch.  Ill  in  [P]  shows  that  for  n  > 
vn(G  \  Ern]  >  2-(r+2>  and  hence  i/n(G  \  Fn)  >  1/4.  Similarly,  we  define  Brn  =  {x  G  « 
An(>r  *  UrKr)  >  1  -  2~r}  for  any  r  and  Cn  =  n{r\nr<n}Brn.  Then  z/n(G  \  CB)  >  1/4. 

Therefore  i/n(Fn  n  CB)  >  1/2.  For  each  n,  we  pick  jcn  G  Fn  n  Cn  n  supp  vn  as  it 
nonempty,  xn  e  supp  i/n  c  N([j,n).  Then  for  any  r  >  0,  Xnxn(KrUr)  >  1  -  2~r  and  *, 
(UrKr}  >  1  —  2~r  for  all  n>nr  and  hence  by  tightness  criterion,  {Xnxn}  and  {xnAn} 
tight.  Also,  since  Kr  C  supp  /x  for  all  r,  Anjtn((supp  //)  J7r)  >  1  -  2~r,  for  all  n  >  nr.  Sii 
C]rUr  =  {<?},  it  easily  follows  that  for  any  limit  point  A  of  {An^},  supp  A  C  supi 
Similarly,  the  limit  points  of  {xn\n}  are  also  supported  on  supp^t. 

Lemma  2.4.  Let  G  be  a  locally  compact  group  and  let  fj,n  ->  JJL  in  Ml  (G).  Let  B  b 
subgroup  which  centralizes  an  open  subgroup  H  containing  supp  //.  Then  the  follow 
hold: 

1.  For  any  sequence  {xn}  in  B,  {x~lfj,nxn}  is  relatively  compact  and  it  converges  to 

2.  Let  {j,n  =  Anz/n  =  z/«An,  for  all  n.  If  for  sequences  {xn}  and  {yn}  in  B,  {xn\nyn. 
relatively  compact  then  its  limit  points  belong  to  T^;  in  particular  if  \nan  -»  A 
some  {an}  C  B,  then  A  G  T^  and  the  limit  points  of{xn\nyn}  are  of  the  form  zA  = 
for  some  z^  G  Z(/x). 

Proof  Let  U  be  any  open  set  contained  in  H  and  let  K  C  supp  /x  be  any  compact  set  s 
that  p(K)  >  0.  Then  given  0  <  e  <  //(£),  there  exists  AT  such  that  pn(KU)  >  ^(K]  -  t 
all  n  >  N.  Since  xn  centralizes  KU,  x~lnnxn(KU)  ==  fj,n(KU)  >  p(K)  -  e.  Since  thi 
true  for  aU  K  and  U  as  above,  {x~l  /^.xn}  is  relatively  compact  and  it  converges  to  p.  Le 
be  a  limit  point  of  a  relatively  compact  sequence  {AJ,  =  xn\nyn},  where  ;cn,yn  €  5.  S: 
{^/^n^1}  converges  to  /x,  {y~l^n^n  l}  is  relatively  compact  and  there  exists  a  limit  p 
i/  of  it  such  that  AV  =  /x.  Also,  i/A'  is  a  limit  point  of  {v~  VnVn},  which  converges  t 
Therefore,  z/A7  =  p  and  hence  A'  €  rM.  Now  suppose  Anan  ->  A,  {an}  C  ^,  then  f 
above  A  €  TM.  Therefore,  A  =  x/3,  for  some  x  G  W(p)  and  ft  supported  on  G(/x)  C  //.  1 
j:"1  Artan  ->  j9.  Let  IsT'  be  any  compact  subset  in  H  such  that  /3(K')  >  0.  Then  for  any  c 
subset  U  contained  in  H,  Xnan(xK'U)  =  znX'n(xK'U),  where  zn  =  xy-lanx-lx~l  €  Z 
as  B  c  Z(/x)  and  x  €  N(p)  which  normalizes  Z(/x).  Since  this  is  true  for  aU  n  am 
compact  subsets  K'  of  supp  /J  it  implies  that  {zw}  is  relatively  compact  in  Z(/A).  Therd 
A'  =  zA,  for  Ax  as  above,  where  z  is  a  limit  point  of  {z~l  }.  Now  since  A  G  TM  and  z  e  Z 
zA  =  zr/3  =  xz'jS  =  jc^  =  Az7,  where  z7  =  x~lvc  G 


PROPOSITION  2.5 

Let  G  be  a  locally  compact  group  and  let  C  be  a  closed  normal  (real)  vector  subgroi 
G.  Suppose  that  {/xn}  c  Ml(G)  be  a  sequence  such  that  p,n  ->  p,  the  closed  subg 
(say)  H,  generated  by  the  centralizer  Z(C)  of  C  and  supp/x,  is  open  in  G.  Suppose 
there  exists  a  sequence  {xn}  in  C  such  that  {x^fj^Xn}  is  relatively  compact.  : 
{xn}/(Z(tj}  H  C)  is  relatively  compact.  In  particular  7M  n  C  =  Z(p)  n  C 

Proof.  Suppose  C  C  Z(/x)  then  there  is  nothing  to  prove.  Now  let  V  =  Z(/x)  n  C,  \* 
is  a  proper  closed  subgroup  of  C.  Since  C  is  normal  in  G;  for  any  x  G  G,  ix:C  - 


Triangular  systems  of  measures  53 

^(c)  =  xcx~l  for  all  c  E  C,  is  a  continuous  homomorphism  of  C  and  hence  it  is  a  linear 
operator  in  M(d,  R),  where  d  is  such  that  C  is  isomorphic  to  R^.  Now  V  =  Z(//)D 
C  =  fljtesupp^  ker(/jc)  and  hence  V  is  a  (possibly  trivial)  vector  subspace  and  C  =  V  x  W, 
a  direct  product.  Now  for  each  «,  ;tn  =  zn  -f  yn,  where  zn  €  V  and  yn  €  W.  Let  \in  = 
n>  for  each  n.  Since  V  centralizes  G(//)  and  hence  H  which  is  open,  by  Lemma  2.4, 


Now  it  is  enough  to  show  that  {vn}  is  relatively  compact.  If  possible,  suppose  it  has  a 
subsequence,  denote  it  by  {yn}  again,  which  is  divergent,  i.e.  it  has  no  convergent 
subsequence.  We  know  that  {y^{p!nyn  =  x~lfjinxn}  is  relatively  compact.  Passing  to  a 
subsequence  if  necessary,  we  get  that  v«/||yn||  —  »  y  in  W,  where  ||  ||  denotes  the  usual 
norm  in  the  vector  space  C.  Since  \jln  —  »  y,,  arguing  as  in  Proposition  9  in  [Ml],  we  get 
that  G(IJL)  C  Z(y),  the  centralizer  of  y  in  G,  a  contradiction  as  y  ^Z(p)  n  C  =  V,  for 
y  G  W  and  \\y\\  =  1.  Therefore,  {yn}  is  relatively  compact.  If  x  e  /^  then  jc/jjc"1  =  /z 
therefore,  (7^  n  C)/(Z(/x)  n  C)  is  a  compact  group,  but  since  C  and  Z(/JL)  fl  C  are  both 
vector  groups  so  is  C/(Z(/x)nC)  and  hence  has  no  nontrivial  compact  subgroups. 
Therefore,  /^  n  C  =  Z(^)  H  C. 

PROPOSITION  2.6 


C  be  as  above.  Let  {vn}  be  a  relatively  compact  sequence  in  Ml(G]  such  that 
vk£  —  »  fj,  and  the  closed  subgroup  (say)  H,  generated  by  the  centralizer  Z(C)  of  C  and 
supp  /A,  is  open  in  G.  Let  A  =  \y™  \  m  <  kn}.  IfA/C  is  relatively  compact  then  so  is  A. 

Proof.  Let  A/C  be  relatively  compact.  If  possible,  suppose  that  A  is  not  relatively 
compact.  That  is,  there  exists  a  subsequence  of  {i/n},  denote  it  by  same  notation,  such 
that  {i/n  }  is  divergent,  where  l(n)  <  kn  for  all  n.  Passing  to  a  subsequence  if  necessary, 
we  get  that  vn  —  »  i/  (say).  Let  TT  :  G  —  »  G/G°  be  the  natural  projection.  Since  {n(v}n  \ 
n  e  N}  C  ?r(A)  which  is  compact,  G(TT(I/))  is  compact.  Also,  since  {vn  |  n  G  N}  C  7^, 
by  Theorem  2.4  of  [S4],  suppi/  Cxl(fji)  =/(/x)jc,  for  any  x  E  suppi/.  Since  A/C  is 
relatively  compact,  there  exists  a  sequence  {xn,m}  in  C  such  that  {v™xn,m}  is  relatively 
compact  and  {xn^}  is  divergent.  Also  since  zA  —»  ^  (resp.  z/Jn+1  —  >  //i/  =  z//x)  the 
above  implies  {x^v**-*}  (resp.  {^>^"t"1~m})  and  hence  {*~X"*n,m}  (resP- 
{Jcn,iiz/n'l+l-l'i,w})  is  relatively  compact.  Now  by  Proposition  2.5,  (;tn,m}/(Z(^)nC) 
(resp.  {jcrtiW}/(Z(/zz/)n  C))  is  relatively  compact.  As  Z(fj)  HZ(//z/)  =  Z(/z)  n  Z(i/),  the 
above  implies  that  {jcnjm}/(Z(^)  nZ(i/)  n  C)  is  relatively  compact.  Without  loss  of 
generality  we  may  assume  that  {*n,m}  C  C1  =  Z(/x)  n  Z(i/)  n  C,  which  is  a  vector  group 
centralizing  G(i/)  and  //  .  Therefore,  //'  =  Z(C')  contains  /Y  and  hence  it  is  an  open 
subgroup  in  G  containing  supp  p,  and  supp  v.  We  may  also  assume  that  xn^\  =  xn^n  =  e  for 
every  n  as  {vn}  and  {v%}  are  relatively  compact. 

Let  n  G  N  and  let  1  <  m  <  kn.  From  Theorem  2.2,  v™(H!)  >  S  >  0  and  hence 
v™xn,m(H'}  >  S.  Since  {^xn,m}  is  relatively  compact,  there  exists  a  compact  set  L  C  //', 
such  that  z/™xn?m(L)  >  <5/2.  Let  0  <  €  <  min{<5/2,  1/4}.  There  exists  a  compact  set 
K  C  suppjii  such  that  n(K)  >  1  -  e.  Let  (7  C  H'  be  such  that  U  is  open  in  G.  Then  there 
exists  N,  such  that  for  all  n  >  N,  zA  (KU)  >  1  -  e.  Let  n  >  N  and  let  1  <  m  <  kn.  Then 
there  exists  {yn,m}  c  G,  such  that  i/^y^KU)  >  I  -  e.  Since  e  <  (5/2,  KUy~{mr\ 
Ijc~l  ^  0.  That  is,  y'l  e  ^^,  where  K'  =  (KU^L  C  //'  and  hence  i/JX^i)  > 
1  -  e  and  each  jcrt)m  commutes  with  all  the  elements  of  K\  =  KUK'  C  H'.  Now  for 
mj<kn  such  that  m  +  /<£n,  we  get  that  i/^'^^^jc^1)  >  (1  -c)2.  Since 


54  Riddhi  Shah 

>  1  -  €  and  e  <  1/4,  we  get  that  Kix^+l  n  K^K^  ±  0.  Theref. 
K\K]~l  H  C'.  Since  C  is  a  vector  group,  C"  is  strongly  root  compact 
3.1.12  of  [H]  and  hence  by  the  definition  of  strong  root  compactness  (see  3.1.10  of  [I 
there  exists  a  compact  subset  K"  such  that  xn,m  E  K",  for  all  m,  n.  This  is  a  contradict: 
to  the  fact  that  {xn^}  is  divergent.  Therefore  A  is  relatively  compact.  This  completes 
proof. 

Let  A  €  M1  (G).  For  some  a  =  (n  ,  /i  ,  .  .  .  ,  r^/m),  where  m  €  N,  and  r,-,  //  e  N  U  -| 
fixed,  let  a(A)  =  An  A  '  .  .  .  \rm  X  m,  where  A°  =  A  =  6e.  For  any  such  a,  the  map  A  H->  a 
on  M*(G)  is  continuous.  Also,  G(A)  =  (Ja  suppa(A)  (over  all  possible  choices  of  a 
above). 

Proof  of  theorem  2.1.  Without  loss  of  generality  we  may  assume  that  {z/n}  is  converge 
that  is  i/n  —  >  v  (say).  From  the  hypothesis,  G(TT(Z/))  is  compact,  and  hence  by  Theorem  ^ 
?r(A)  is  relatively  compact.  It  is  enough  to  show  that  A  is  relatively  compact  as 
Theorem  3.6  of  [SI],  there  exists  x  such  that  xp,  =  pjc  is  embeddable. 

Step  1.  Let  K  be  the  maximal  compact  normal  subgroup  of  G°,  then  K  is  characteristic 
G°  and  hence  normal  in  G.  Since  A  is  relatively  compact  if  and  only  if  its  image  on  G 
is  relatively  compact,  without  loss  of  generality  we  may  assume  that  G°  has  no  nontrr 
compact  normal  subgroups.  In  particular,  G°  is  a  Lie  group.  Let  L  be  any  open 
protective  subgroup  of  G.  Let  M  be  any  compact  normal  subgroup  of  L  such  that  L/M 
Lie  group,  then  G°M  =  G°  x  M,  a  direct  product,  as  both  G°  and  M  are  normal  in  L  \ 
G°  flM  =  {e}.  Moreover  H  =  G°M  is  an  open  subgroup  in  G.  Since  /(//)  is  comp; 
without  loss  of  generality,  we  may  assume  that  /(/x)  normalizes  H. 

Step  2.  Now  we  prove  the  assertion  by  induction  on  the  dimension  of  the  Lie  group 
Let  dim  G°  =  0.  Then  G  is  totally  disconnected  and  the  assertion  follows  from  abc 
Now  suppose  that  for  any  k  >  1,  the  assertion  holds  for  G  such  that  dim  G°  <  k.  Now 


Step  3.  Suppose  that  there  exists  a  subsequence  of  {vn},  denote  it  by  {vn}  again,  s 
that  {z/Jj}  is  divergent.  By  Theorem  1.2.21  of  [H],  there  exists  a  sequence  {xn}  C  G,  s 
that  {!/£*„}  and  hence  {x~lvkj~ln}  and  {x~l^xn}  are  relatively  compact  and  we  r 
assume  that  {xn}  is  divergent.  Since  TT(A)  is  relatively  compact,  {n(xn)}  is  relath 
compact  in  G/G°  and  hence  we  may  choose  {;cn}  to  be  in  G°. 

Without  loss  of  generality  we  may  assume  that  the  subgroup  N,  as  in  the  hypothesi 
the  nilradical.  Suppose  that  N  is  trivial.  Then  G°  is  a  connected  semisimple  grc 
Suppose  that  the  center  of  G  is  trivial.  Then  G°  is  an  almost  algebraic  subgrouj 
GLn(R).  By  Propositions  4-6  of  [Ml],  there  exists  a  proper  closed  subgroup  Gr  of 
such  that  given  any  relatively  compact  sequence  {zn}  C  G°,  the  limit  points  of  {xnznx 
are  contained  in  G'.  Now  since  G°  C  G((j),  there  exists  an  A:  e  G(/z)  n  (G°  \  G').  Si 
G°\G'  is  open  in_G°,  there  exists  a  set  U  which  is  open  in  G°  such  that  x£ 
U  C  G°  \  G  and  U  is  compact.  Then  for  some  a  =  (n,  /i,  .  .  .  ,  rm,  Zm),  we  have 
a(/i)(t/Af)  =  6  >  0,  as  UM  =  U  x  M  is  open  in  G,  for  a  compact  group  M  as  ab< 
Since  a(iA)  -»  a(/z),  a(iA)((7M)  >  5/2  for  all  large  n.  Now  since  {x~lvknnxn 
relatively  compact,  so  is  {x~la(v%)xn}.  Therefore,  there  exists  a  compact  set  K  such 
(x-la(vkn»)xn)(K)  =  a(vkn»)(xnKx-1}  >  1-6/4  for  all  n.  From  the  above  equa 
UM  n  xnKx~l  ^  0,  for  all  large  n.  Therefore,  there  exists  a  sequence  {an}  C  K,  such 


Triangular  systems  of  measures  55 


for  all  large  n,  xnanxn  1  =  unvn,  where  un  £  U  and  vn  G  M  and  hence  xnanvn  lxn  l  =  un. 
For  each  n,  put  zn  =  ^v"1,  then  since  jcn,  ww  €  G°,  zn  €  G°.  Also  {zw}  C  KM  is  relatively 
compact.  Therefore  the  limit  points  of  {xnznx~l  =  wn}  belong  to  G'.  But  {wn}  C  U  and 
17  C  G°  \  G',  a  contradiction.  Therefore,  A  is  relatively  compact. 

Step  4.  Now  suppose  G°  is  a  semisimple  group  with  nontrivial  center  Z.  Then  Z  is  a 
discrete  group  normal  in  G  and  Z  =  Zn,  for  some  n,  as  we  have  assumed  that  G°  has  no 
nontrivial  compact  subgroups  normal  in  G.  The  action  of  G  on  Zn  extends  to  the  action  of 
G  on  R/1.  Therefore,  we  can  form  a  semidirect  product  GI  =  G  •  R/1.  Let  D  =  {(z,z)  | 
z  G  Zw}.  Then  Z)  is  normal  in  GI.  Now  G  can  be  embedded  as  a  closed  subgroup  in 
G2  =  G\/D  and  G£  =  (G°  x  Rn)/D.  It  is  easy  to  see  that  the  center  C  of  G\  is 
isomorphic  to  Rrt.  Also,  C  is  normal  in  GI  and  G^/C  is  a  semisimple  group  with  trivial 
center.  Let  ^  :  G2  — >  Ga/C  be  the  natural  projection.  It  is  easy  to  see  that  G(^(/x)) 
contains  G^/C,  the  connected  component  in  G2/C,  and  hence  by  the  above  argument, 
^(A)  is  relatively  compact.  Since  H  centralizes  G°  in  G,  # '  =  H  x  R"  =  G°  x  M  x  R" 
is  open  in  GI  and  hence  H' /D  is  an  open  subgroup  in  G2  which  centralizes  C.  Now  the 
assertion  in  this  case  follows  from  Proposition  2.6. 

Step  5.  Now  suppose  the  nilradical  TV  of  G  is  nontrivial.  Let  C  be  the  center  of  N.  Since 
G°  does  not  contain  any  compact  subgroups  normal  in  G,  C  is  a  vector  group,  i.e.  C  is 
isomorphic  to  Rn,  for  some  n.  Since  N  is  normal  in  G,  so  is  C.  Let  -0  :  G  — »  G/C  be  the 
natural  projection.  Then  since  dimG°/C  <  k,  we  have  that  -0(A)  is  relatively  compact. 
Now  since  C  centralizes  TV  x  M,  M  as  above,  and  supp  /z  and  TV  generate  a  subgroup 
containing  G°,  the  assertion  follows  from  Proposition  2.6. 

Remark.  Theorem  2.1  continues  to  hold  if  the  conditions  in  it  are  replaced  by  the 
following:  iA  — >  /x,  the  closed  subgroup  generated  by  supp  /^  and  N  is  whole  of  G  (where 
N  is  as  in  the  hypothesis  of  the  theorem),  {^n}/G°  is  relatively  compact  and  for  any  limit 
point  v  of  it,  G(v)  is  compact  in  G/G°.  For  the  proof,  A/G°  is  relatively  compact  by 
Theorem  2.2  and  the  first  three  steps  of  the  proof  of  the  above  theorem  will  apply  word 
for  word.  Also,  for  a  normal  subgroup  C  in  steps  4  and  5  above,  Z(JJL)  D  C  is  a  central 
vector  group  in  G  by  the  above  condition  and  hence  by  Proposition  2.5,  the  relative 
compactness  of  A/C  implies  that  of  A/(Z(/x)  n  C).  Therefore  A  is  relatively  compact  by 
Lemma  3.2  of  [SI].  The  above  variation  of  Theorem  2.1  generalizes  Theorem  3.1  of  [SI]. 

3.  Limit  theorems  on  discrete  linear  groups  over  R 

Theorem  3.1.  Let  Gbea  discrete  linear  group  over  R.  Let  {vn}bea  sequence  in  M1  (G) 
such  that  z/*n  — »  frfor  some  ^  G  M^G)  and  some  unbounded  sequence  {kn}  in  N.  Then 
there  exists  x  G  /M,  such  that  XJJL  is  embeddable. 

Remark.  So  far,  in  the  limit  theorems  on  discrete  groups,  one  had  either  the  support 
condition  or  the  infmitesimality  condition  imposed  (see  [S4]  and  Theorem  2.2  above). 
The  above  theorem  gives  a  generalization  of  Theorems  1.5,  1.7(1)  of  [S4]  for  this  special 
class  of  discrete  groups.  It  also  generalizes  Theorem  1.2  of  [DM3].  One  cannot  get  an 
embedding  of  //  itself  or  an  element  x  as  above  to  be  infinitely  divisible  as  in 
G  =  GL(1,  Z)  =  {-1, 1},  for  x  =  -1,  8X  =  <^n+1,  for  all  n,  but  6X  is  clearly  not  infinitely 
divisible  and  hence  not  embeddable. 


56  Riddhi  Shah 


To  prove  the  theorem,  we  need  preliminary  results. 


Lemma  3.2.  Let  V  be  a  finite  dimensional  vector  space  over  R.  Let  {rn}  be  a  diverg, 
sequence  in  GL(V)  such  that  for  some  b  >  0,  |det(rw)|  >  b  for  all  n.  Then  there  exist 
proper  subspace  W  of  V  such  that  the  following  holds:  if  {p,n}  CMl(V)  is  such  ti 
jjin  ->•  n  and  {rn(p>n)}  is  relatively  compact,  then  supp^  C  W. 

The  proof  of  the  Lemma  is  exactly  same  as  the  proof  of  Proposition  3.2  in  [S2]  usi 
Proposition  1.4  in  [DM1].  We  will  not  repeat  it  here. 

PROPOSITION  3.3 

Let  G  be  a  discrete  linear  group  over  R  and  let  {fj,n}  be  a  sequence  converging  to  p 
Ml(G).  Let  Xn  <G  Tnn  for  each  n.  Then  there  exist  sequences  {zn}  and  {z'n}  in  Z(JJL)  si 
that  {XnZn}  and  {z'nXn}  are  relatively  compact  and  all  their  limit  points  belong  to  Tt 

Proof.  There  exists  a  sequence  {\rn}  in  M*(G),  such  that  XnX'n  =  \'n\n  =  //„  -*  //, 
Lemma  2.3,  there  exists  a  sequence  {xn}  in  G  such  that  {Xnxn}  and  {xnXn}  are  relativ 
compact  and  all  its  limit  points  are  supported  on  supp  ^L.  Therefore,  by  Theorem  1 .2.21 
[H],  {jc~l  A'J,  {A^~1}  and  hence  {x~{fj,nxn}  and  {xnp,nx~1}  are  all  relatively  compact 
z/  is  a  limit  point  of  {jc~]  A^}  then  there  exists  a  limit  point  A  of  { Xnxn}  such  that  Azx  = 
Since  supp  A  C  supp  ^  =  supp  A  supp  z/,  supp  v  C  G(IJL).  Therefore  all  the  limit  points 
{x~[Xfn}  and  also  of  {x~ljjinxn}  are  supported  on  G(/z). 

Similarly,  the  limit  points  of  {xn^nx^1}  are  also  supported  on  G(yu),  and  {x~l  a(p,n). 
and  {xna(ij,n)x~l }  are  relatively  compact  and  their  limit  points  are  supported  on  G(p), 
any  a  (where  a  and  a(/xn)  are  defined  as  in  §  2).  Also,  for  any  e  >  0,  there  exist 
compact  set  K  such  that  (x~ljj,nxn)(K)  >  1  —  e  f or  all  n.  Now  for  any  limit  point  ^ 
{x~lfj,nxn},  i(KD  G(fj,))  >  1  -  e.  Therefore  it  is  easy  to  see  that  (jc"1  p,nxn)(K')  >  1  - 
for  all  large  n,  where  Kf  —  K  n  G(/x). 

We  know  that  G  C  GL(n,R)  C  Af(n,R).  Let  V^  be  the  vector  space  generated 
G(/x)  in  M(n, R).  There  exists  a  finite  set  {vi, . . .  ,;ym}  C  G(JJL)  such  that  {yi ,...,] 
generates  VM.  Since  G(/x)  =  U^supp  a(/x),  where  a  and  a(/x)  are  as  defined  in  §  2,  th 
exist  aj, . . .  ,am  such  that  j,  G  supp  a/ (/x),  for  each  i.  Therefore,  as  G  is  discrete, 
some  6  >  0,  a/(/x){y/}  >  5  for  all  i.  Since  a/(/^n)  -+  a/(/z),  there  exists  /V  such  1 
<*i(Mn){;y/}  >  V2'  f°r  a^  w  >  ^  for  all  i. 

Now  since  {j~1a/(/^w)jcM}  is  relatively  compact  and  all  its  limit  points  are  supported 
G(/x),  arguing  as  above  we  can  get  a  compact  set  K\  C  G(/x),  such  that  (;c~la/(/^) 
(/iTi)  >  1  -  6/2  for  all  f,  for  all  large  n.  That  is,  ai(^n)(xnK}x~})  >  1-6/2  for  all  i, 
all  large  n.  Therefore,  y,-  €  xnK\x~l,  or  x~{yiXn  €  K\  C  G(p)  C  VM,  for  all  large  n.  Si 
Vp  is  generated  by  {yb  . . .  ,ym},  the  above  implies  that  x"^^  =  VM,  for  all  large  j 

Let  G  be  the  Zariski  closure  of  G  in  GL(rf,R)  and  let  N(V^  (resp.  Z(VM))  be 
normaliser  (resp.  centraliser)  of  VM  in  G.  Then  Z(VAl)  and  N(V^)  are  algebraic  subgro 
of  G  and  Z(VM)  is  normal  in  N(V^).  Now  A^(VM)  acts  on  V;,  linearly  and  the  r 
p  :  N(V^)  -+  GL(VM)  is  a  rational  morphism,  as  in  the  proof  of  Theorem  3.2  in  [DN 
Therefore,  the  image  of  p,  lm(p)  is  closed  in  GL(V^)  and  since  kerp  =  Z(1 
p' :  N(V^)/Z(V^)  — >  Imp  is  a  topological  isomorphism. 

We  know  that  {xn}cN(V^).  Now  if  possible,  suppose  that  {xn}/Z(V^)  is 
relatively  compact.  Going  to  a  subsequence  if  necessary,  without  loss  of  generality, 


Triangular  systems  of  measures  57 

may  assume  that  {jtn}/Z(VM)  is  divergent;  i.e.  it  has  no  convergent  subsequence,  and  for 
some  6  >  0,  either  Idetp't^Z^))!  =  |detp(jcn)|  >  S  or  |detp/(^~lZ(V/I))|  >  6. 

Suppose  \d&pf(xnZ(V^))\  =  |detp(jtn)|  >  6  for  all  n.  By  Lemma  3.2,  there  exists  a 
proper  subspace  W  of  V^  such  that  suppa(/x)  C  W  for  all  a,  as  a(p,n)  —  >  a(/x)  and 
{(p/(jcwZ(V/i)))(ce(/^n))  =^na(^n)jc~1}  is  relatively  compact.  This  implies  that  G(p)  = 
Uasuppo:(/i)  C  W,  a  contradiction  as  G(fjb)  generates  V^  and  W  is  a  proper  subspace. 

Now  suppose  \detp'(x~lZ(V^))\>  8.  Now  using  the  fact  that  for  every  a, 
{(p/(^-IZ(V//)))(a(M«))  -x^{a(fjLn)xn}  is  relatively  compact  and  replacing  {xn}  by 
{x~1}  in  the  above  argument  we  arrive  at  a  contradiction.  Therefore,  {xn}/Z(V^)  is 
relatively  compact. 

Clearly,  N(V^)  n  G  normalizes  Z(V^).  Let  H  =  (Af(VM)  n  G)Z(V^)  and  let  jc  €  H. 

Then  jc(VM  n  G)jT  !  =  VM  n  G.  Let  GM  be  the  closed  subgroup  generated  by  V^  n  G  in  G. 

Then  G(/x)  C  GM  and  *  normalizes  G^.  Therefore  H  is  a  closed  subgroup  (in  G) 

normalizing  G^.  Since  G^  is  discrete,  the  connected  component  H  of  //,  centralizes  GM 

and  hence  ff  C  Z(VM)  C  #  as  VM  is_generated  by  G(^)  and  G(p)  C  GM.  Since  #°  is 

open  in  H,  it  follows  that  H  is  open  in  77.  That,  is  H  =  H  and  /f  is  a  closed  subgroup.  This 

'  implies  that  ((N(VJ  n  G)Z(^))/Z(V^)  is  isomorphic  to  (N(VJ  nG)/(Z(VM)  n  G). 

/  Therefore    {jc,,}/Z(/x)    is   relatively    compact    as   Z(V^)  Pi  G  =  Z(/x).    Therefore 

f  JCn  =  zwtfn  =  ^n4'  ^or  some  relatively  compact  sequence  {an}  in  G  and  some  sequences 

f  {zn}  and  {^}  in  Z(//).  Also,  since  {An;crt}  and  {xnXn}  are  relatively  compact,  so  are 

&.  {A«zn}  and  {^An},  and  all  their  limit  points  belong  to  T^  by  Lemma  2.4. 


Proof  of  Theorem  3.1.  Since  iA  —*  //,  by  Proposition  3.3,  for  any  m,  there  exists  a  se- 

}  quence  {zm,n}  C  Z(/z)  such  that  {z^zm,n}  is  relatively  compact.  Passing  to  a  subsequence 

)r  if  necessary,  without  loss  of  generality,  we  may  assume  that  {^rtzi,n}  is  convergent, 

a  with  the  limit  v.  Then  v  G  7^  by  Lemma  2.4.  Also,  for  any  m,  {^mjW  =  z 

3f  is  relatively  compact  and  its  limit  points  are  of  the  form  zv  =  i/z7,  for  some  z,  zf  € 

€;  (cf.  Lemma  2.4). 

Suppose  for  any  fixed  m,  the  limit  points  of  {v™zm,n}  are  of  the  form  vmzm  for  some 

Dy  zm  €  Z(//).  Then  combining  the  above  two  statements,  we  get  that  the  limit  points  of 

„}  {^C^SiH-i,*}  have  ^  form  vmZmZv  =  i/m+1zm+i,  for  some  zm+\  €  Z(/z).  By  induction, 

sre  for  any  m,  the  limit  points  of  {v™zm,n}  are  of  the  form  z/mzm,  for  some  zm  €  Z(/u). 

"or  Moreover,  by  Lemma  2.4,  i/m  G  7^,  as  it  is  a  limit  point  of  {^^Zm,/!^1  },  for  each  m.  Also 

iat  suppi/cAT(/i). 

Now  by  Proposition  3.3,  {i/n}/Z(^)  is  relatively  compact.  Therefore  G(i/)Z(//)/Z(/z) 

on  is  compact  and  hence  finite  of  order  (say)  s,  as  G  is  discrete.  Let  x  G  supp  i/,  then 

Xn)  Jcff  €  Z(/x).  Let  /?  =  i/5z  =  Zf  *  for  z  =  x~s  G  Z(//).  Then  €  G  supp  (3  and  /3n  G  7^  for  all  n. 

for  Therefore  by  Theorem  2.4  of  [S4],  supp/3  C  /(//)  and,  furthermore,  /3n  ~*  a;//,  where 

llce  //  =  G(/3)  C  /(/x).  Hence  supp  i/  C  xH  n  Hx.  Therefore  xp,  =  i/y^  =  /xz/  =  //jc,  and  hence 

t  z  G  /M,  for  all  jc  G  supp  v. 

the  Now  we  show  that  /x  has  a  shift  which  is  infinitely  divisible.  Let  /  G  N  be  fixed.  Let 

UpS  an  —  [kn/l]  and  bn  =  kn~  lan.  Then  for  any,  m  <  /,  i/jw«i/J»-'lw»  —  ».  ^  and  hence  there 

nap  exist  sequences  {z!m^n}  in  Z(^)  such  that  {v^^mn}  are  relatively  compact.  Arguing  as 

,j2],  above,  we  get  that  the  limit  points  of  {vfiz!^}  are  of  the  form  A{z,  for  some  z  G  Z(/x)  and 

I/  ^  some  limit  point  A/  of  {i^J1^}.  Let  r  G  N  be  fixed.  Since  an  -*  oo,  for  large  n  such  that 

an  >  r,  z/  ^zx,iW  =  vrnZr,n'yn,  where  {7^  =  z~^ann~rz^n}  which  is  relatively  compact  and 

not  hence  A/  =  vr^  for  some  7.  Also  z^z^  ,=  ^S"~r^^i|B.  By  Proposition  3.3,  there  exists 

e  {yn}  in  Z(/z)  such  that  {f^"~ryn}  is  relatively  compact  and  hence  so  is  {j^^^z7^} 


58  RiddhiShah 

and  all  its  limit  points  are  of  the  form  zVr  for  some  z'  G  Z(p)  (cf.  Lemma  2.4).  1 
is,  \i  =  7'z/r,  for  some  y  and  hence  for  /3  =  z/*z  =  z^5  defined  as  above,  A/  =  / 
=  ff'ff  for  some  0,  /3".  Since  this  is  true  for  all  r,  UH  G  TV  That  is,  XIUH  =  u;#A/  = 
for  all  /. 

For  each  n,  let  zn  =  (^,n)~1-  Then  the  sequence  {znvbnn}  is  relatively  compact.  Cles 
Z?n  <  /  for  all  n.  Let  r  <  /  be  such  that  r  =  fonjk  for  infinitely  many  nk.  Then  clearly 
limit  points  of  {znvbnn}  are  contained  in  {gi/r  \  r  <  /,g  G  Z(yu)}  and  hence  if  p/  is 
such  limit  point  then  supp  p/  C  G(i/)Z(/z)  C  7^  and  p/o;#  =  */u;#  (resp.  u;#p/  =  u>/ 
where  xj  6  Z(/x),  where  5  is  the  cardinality  of  G(i>)Z(/x)/Z(/z). 

Combining  the  above  we  get  that  ^  =  Ajp/  =  AJCJ//P/  =  A'JC/(=  x/Aj)   for   s< 

*/  6  supp  p/  C  7/x,  for  each  /.  That  is,  ^  is  weakly  infinitely  divisible.  As  A/  G 

supp  A/  C  yiG(jj)  for  some  v/  6  supp  A/  C  N(^).  Since  for  each  /,  ^  =  A}JC/  and  ;c/  G  ( 

Z(ji4),  we  get  that  yj  G  G(i/)Z(/i)G(/i).  Hence  (y,)*  G  G(/i)Z(ju),  as  G(I/)Z(/Z)/Z(AX) 

finite  group  of  order  5.  Since  TM/Z(jLt),  is  relatively  compact,  arguing  as  in  Theorem  3 

[DM3],  we  get  that  F  =  7M/G(jit)Z(^)  is  finite  and  it  obviously  consists  of  c 

measures.  Also,  the  above  implies  that  the  image  of  A/  on  G  =  N(fj,)/G(p)Z(p)  is 

where  yz  =  v/G(/x)Z(^)  in  G'  and  J/5  =  e,  the  identity  in  G'.  Let  #  =  {7  €  F  \  Y 

for  some  r  £  N}.  Since  F  is  finite,  so  is  B  and  there  exists  an  element  of  maximal  c 

in  B;  let  i  be  the  maximal  order.  Then  7l!  =  6e  for  all  7  E  B.  Since  the  image  of  > 

N(^)/G(/x)Z(/i)  belongs  to  B,  we  have  that  supp  A?  C  G(ju)Z(/z),  for  all  /.  Now  for 

m,  let  j9m  =  Afjm,  where  p,  =  Ajj^j,  for  some  x  €  7M.  Then  /i  =  /3%x  and  supp  ^m  C  < 

Z(/x).  Also,  since  supp  ^m  c  vG(/x)  for  some  v€supp/3m,  v  =  zy'=yz,  for  s 

y  €  G(M),z  G  Z(/x).  Then  ^  =  z~%  =  ^z"1  is  supported  on  G(/x).  Also,  ^  =  /9J 

(/yjVoc  =  (jfli.ry,  where  jc7  =  z"1*  €  7^  n  G(/x)  as  supp/3^  C  G(M).  That  is, 

weakly  infinitely  divisible  on  G(ju).  Moreover,  from  the  above  equation,  we  have 

{#i}/z/*  is  relatively  compact,  where  ZM  =  G(/^)nZ(/z)  is  the  center  of  G(» 

[DM3],  Theorem  2.1).  In  fact,  {/3'mzm}  is  relatively  compact  for  some  sequence  {zn 

ZM.  Let  im  =  /3fmzm.  Then  (im)m  =  (^w)w^  and  hence  M  =  (Ym)mxm  for  some  . 

1^  n  G(/j),  for  all  m.  Now  if  y  is  a  limit  point  of  {^  then  (7' )n  G  TM  for  all  n  and  h< 

as  earlier,  supp  7'  c  x7(/x)  =  7(/z)jc,  for  some  jc  G  7M  n  G(».  Since  (7M  n  G(//))/i 

finite  (cf.  [DM3],  Theorem  2.1),  if  a  is  its  cardinality  then  supp  (i)a  C  zl(p)  =  /(M) 

some  z  G  ZM.  Therefore  limit  points  of  {(7^)*}  are  supported  on  zl(p)  =  7(//)z,  z  < 

Let  7m  =  (yani)a.  Then  //  =  ^am,  where  jcflm-G  7M  n  G(/A).  Let  {7Cm}  be  a  conve: 

subsequence  of  {7^1}  converging  to  7,  Then  from  above,  supp  7  C  z7(/x)  =  7(/x). 

some  z  G  ZM.  Therefore,  for  each  m,  replacing  %m  by  7Cwz~1  (and  using  the  same  : 

tion),  we  get  that  /z  =  7^ym,  ym  G  7^  n  G(p)  and  7Cm  4  7  and  0(7)  C  7(/x),  whi 

compact.  Also  {ym}/ZM  is  finite,  and  hence  passing  to  a  subsequence  again,  we 

assume   that   y«  =  o4  =  4a»    where    ^e7MnG(/i)   and   4€ZM.    Therej 

^4  =  a  V  =  M^'1-  Now  applying  Theorem  2.2,  we  get  that  A  =  {7^  |  n  < 

and  (4}  are  relatively  compact.  Now  if  /?  is  a  limit  point  of  {«£»}  then  a"1]^  =  (3. 

some  z7  G  Zp.  Since  for  all  m,  cm  =  /„!,  where  /m  ->  oo,  any  w  divides  cm  for  all  larj 

Also  since  A  is  relatively  compact,  it  is  easy  to  see  that  ft  has  an  n-th  root  in  A,  na] 

anyjimit  of  the  sequence  {^/n}.  Therefore,  y/x  =  /3  is  infinitely  divisible  in  the  con 

set  A,  where  y  =  (z7)  V1  G  7A  n  G(/^).  Now  as  in  the  proof  of  Theorem  3.1.32  oJ 

y^  is  rationally  embeddable,  i.e.  there  exists  a  homomorphism/  :  Q^|_  ->  M1  (G)  sue! 

/(]0, 1[HQ+)  C  A  is  relatively  compact  and  /(I)  =  /z.  Now  since  G  is  discrete 

compact  connected  subgroup  of  G  has  to  be  {e}.  Therefore,  as  in  the  proof  of  The 

3.5.4  of  [H],/  extends  to  R+  and  hence  yp,  is  embeddable. 


Triangular  systems  of  measures  59 

4.  Infinitesimally  divisible  measures  on  algebraic  groups 

We  first  recall  that  an  element  s,  in  a  Hausdorff  semigroup  S  with  identity  e,  is  said  to  be 
infmitesimally  divisible  if  for  every  neighbourhood  U  of  e  in  S,  s  has  a  ^/-decomposition, 
i.e.  there  exist  5-1,  .  .  .  ,sn  G  U  such  that  s/'s  commute  and  s  =  s\  -  -  •  sn.  The  following 
theorem  generalizes  Theorem  1.2  of  [S3]  in  a  certain  sense. 

Theorem  4.1.  Let  G  be  a  real  algebraic  group  and  let  fji  G  M1  (G)  &e  infmitesimally 
divisible  in  M*(G).  TTz^n  f/zere  exist  a  closed  semigroup  S  C  M1H(G),  with  identity  ujHfor 
some  compact  subgroup  H  of  I  (IJL),  and  an  equivalence  relation  ~,  such  that  IJL  G  5  tm<f  £/ 
p  :  S  —*  S*  —  S/  ~  is  the  natural  map  then  P(IJL)  is  bald  and  infmitesimally  divisible  in  S*, 
and  Tpfa)  is  compact  and  associatefree  in  S*.  Moreover,  if  G  is  connected  and  nilpotent 
then  IJL  is  embeddable. 

Before  proving  the  above  theorem,  we  define  an  equivalence  relation  on  a  certain  kind 
of  subsemigroup  of  M^G),  for  any  locally  compact  (Hausdorff)  group  G.  For  a 
/xGM^G),  let  Sfj,  be  the  closed  subsemigroup  generated  by  TM  in  Ml(G).  Since 
rM  C  Ml(N(p,)),  5M  C  MI(N(IJL)).  In  fact,  for  any  A  E  TM,  supp  A  C  JcG(/Lt),  for  some 
x  G  supp  A  C  N(p,).  Therefore,  it  easily  follows  that  for  any  /3  G  5M,  supp  /3  C  xG(p,)9  for 
any  x  G  supp/3  C  N(p).  We  also  know  that  Z(/x)  CT^cS^  andZQu)^  =  TMZ(/x)  =  3^. 
Let  us  define  an  equivalence  relation  *«'  on  5M  as  follows:  for  any 

/?,  A  G  S^,  /3  «  A  if    /?  =  zA  for  some    z  e  Z(/x). 

For  {/?„},  {An}  C  5M,  suppose  /3n  «  An,  i.e.  /3n  =  zwAw  for  some  zn  G  Z(/z),  for  each  n. 
Now  if  /3W  -^  /3  and  An  —  >  A,  then  we  have  that  {zn}  is  relatively  compact  and  for  any 
limit  point  z  of  it,  z  G  Z(p)  and  0  =  z\.  Therefore,  /3  w  A. 

Now  for  A  G  5^,  for  any  fixed  jc  G  supp  A,  supp  (Ax"1)  C  G(/u).  For  any  z  G  Z(/x), 
z7  =  xzx~l  G  Z(ju)  as  Z(p)  is  normal  in  A^(/x)  and  hence 

\z  =  (\x-l)xz  =  (A^-1)^  =  ^(Ax"1)^  =  z'A. 

Similarly,  one  can  also  show  that  z\  =  Az",  for  some  z"  G  Z(^). 

Now  for  f  G  {1,2},  j9|,  A,-  G  5M,  let/3,-  «  A/,  i.e.  there  exist  zz-  G  Z(/z),  such  that  fa  =  ztXt, 
Then  from  the  above  equation,  fa  fa  =  z\X\Z2\2  =  ZiZ^i^2  f°r  some  4  €  Z(M)-  That  is, 
fa  fa  &  Xi\2.  Let  ij}  :  5M  —  >  »S*  =  5^7  «  be  the  natural  projection.  Then  if)  is  a  continuous 
open  homomorphism  and  it  is  also  easy  to  show  that  5*  is  Hausdorff. 

In  case  of  a  real  algebraic  group  G,  we  define  an  analogous  equivalence  relation  «' 
with  respect  to  Z°(/^),  the  connected  component  of  the  identity  in  Z(/A),  i.e.  for  /?,  A  G  5M, 
/3  «'  A  if  /3  =  zA,  for  some  z  G  Z°(^).  It  is  easy  to  verify  as  above  that  this  is  an 
equivalence  relation  using  the  fact  that  Z°(/z)  is  normal  in 


Proof  of  Theorem  4.1.  Let  G  be  a  real  algebraic  group  and  let  /^  be  infinitesimally 
divisible  in  Ml(G).  Since  G  is  metrizable,  so  is  Ml(G). 

Step  1.  Let  Sp,  wx,  5*  and  ^  :  5M  —  »•  5*  be  as  above.  Clearly,  SM  and  5*  are  second 
countable  and  ^(/x)  is  infinitesimally  divisible  in  5*. 


Since  G  is  algebraic,  by  Theorem  3.2  of  [DM2],  rM/Z°(^)  is  relatively  compact. 
Clearly,  ^(rM)  C  r^j.  Now  for  any  {AB}.C  rM,  there  exists  a  sequence  {zn}  C 


60  Riddhi  Shah 

such  that  {XnZn}  is  relatively  compact  and  hence  {i/>(Xn)  =  i/>(Artzw)}  is  also  relatively 
compact.  Since  7)tZ(^)  =  7^,  {Xnzn}  C  T^  and  the  above  implies  that  ^(T^)  is  compact 


Since  /x  is  infmitesimally  divisible  so  is  ?/>(/x)  in  5*  .  We  can  choose  a  neighbourhood 
basis  {£//}/€N  of  6e  in  M1  (G).  For  any  f,  there  exist  /x/i  ,  -  .  •  ,  Mm,  £  I//  H  TM,  such  that  /^-s 
commute  and  /x  =  /XH  •  •  -/x^..  Therefore  T/J(^)  =  ^(/x/i)  ••  -i/K^m,)  is  a  ^(f//)-decom- 
position  of  ib(/x)  in  ^(7^).  Let  A  =  (Mu)?eNj=i  and  <KA)  =  MttV^Nj-i-  Then  A 
(resp.  ^(A))  is  a  commutative  infinitesimal  triangular  system  in  SM  (resp.  in  5*) 
converging  to  p,  (resp.  ^>(/x)).  In  fact,  IJL  =  HjLifrj  and  ^(/x)  =  nji 


Step  2..  Since  /£  is  open  in  7^,  one  can  choose  U  and  W  to  be  neighbourhoods  of  I^Jp, 
such  that  U  =  (i/  e  M1  (G)  |  i/C/J/^)  V)  >  6},  for  some  6  >  0,  U  D  7^  =  7J^,  for 
some  relatively  compact  neighbourhood  V  of  e  in  G°  and  WW  C  £/.  Now  let  A  €  S^H 
17  \  W  be  such  that  <0(A'Z)  €  7^)  in  5*  for  all  n,  then  /x  =  \nvn  =  z^An,  for  some  i/rt,  i^ 
in  5^  for  all  n.  Then  the  concentration  functions  of  both  A  and  A  do  not  converge  to  zero. 
Since  A  commutes  with  /x,  as  in  the  proof  of  Theorem  2.4  of  [S4],  supp  A  C  *7(/x)  = 
I(n)x9  for  some  x  G  supp  A  C  /^  n  U.  i.e.  A  6  /jjj^,  a  contradiction  as  A  ^  W.  Now  as  in 
the  proof  of  Lemma  2.5  in  [S4],  there  exists  n  such  that  for  any  m  >  «,  ^(/x)  cannot  be 
expressed  as  ^(^)  =  ^(Ai)  •  •  •  V7(Am)^(^),  where  -0(A/)s  commute  with  each  other  and 
also  with  ^&(i/)  for  any  A/  €  S^  n  17  \  W,  for  all  jf. 

Since  /M  C  7^,  T/;(/M)  is  compact.  Let  ^  =  ^(^M)-  T116111  ^  is  a  compact  semigroup 
and  i/>(U  n  SM)  and  ^(  W  n  S^)  are  neighbourhoods  of  K  in  5*  . 

Since  ip(^)  is  a  limit  of  a  triangular  system  as  above,  as  in  Lemma  2.6  of  [S4],  given 
any  neighbourhood  U'  of  K  in  5*  ,  one  can  choose  small  neighbourhoods  U  and  W  as 
above  such  that  ip(U  n  S^)  C  t/7  and  show  that  there  exists  a  ^/'-decomposition  of  ^(/x) 
in  ^(T^,),  namely,  ^(^)  =  ^(/X])  •  -  "0(/xn),  where  each  ^(/x,-)  €  £/'  is  a  limit  of  a  sub- 
system of 


3.  Let  {U'n}  be  a  neighbourhood  basis  of  K  in  S*  such  that  U'n+l  C  I/^  for  all  n  and 
=  ^.  Now  let  ^(M)  =7i  ••*7/i  he  a  £/{  -decomposition  of  ^(/x)  in  ^&(TM) 
obtained  as  above.  Given  any  t/^-decomposition  of  ^(/x)  as  ^(/x)  =  i/  1  -  •  •  i/r,  i//  =  n/6ytf 
^(^(i)/)'  where  U/J//  =  {!,...,  %(/)}  we  get  ^+r  decomposition  of  each  i//  in  such  a 
way  that  i//  =  i/n  •  •  -  i//n/,  i///K  €  C/J+I,  where  i//mi^w  =  i/wi//m,  for  all  /,m,p,  ^,  and  all 
the  i//w  are  limits  of  a  subsystem  of  (^(/^(*+i)(iy))»  where  {(*  +  l)(i)}  is  a  subsequence  of 
{*(/)}.  Clearly  ?/>(/x)  =  n/,mz//m  is  a  U'k+l-  decomposition  for  ^(/x). 

For  each  fc  €  N,  let  Aft  be  the  subsemigroup  of  5*  generated  by  ^-decomposition 
obtained  in  above  manner.  Then  each  Mk  is  abelian,  /x  €  M*  and  M*  cM*+i-  Let 
M  =  \JkMk  and  let  A:7  =  K  n  M  =  ^(I^J^HM.  Then  Af  (resp.  ^)  is  a  closed  (resp. 
compact)  abelian  semigroup.  Also,  given  any  neighbourhood  U'  ofK'  in  M,  there  exists  a 
neighbourhood  £7"  of  K  in  5*  ,  such  that  U"  n  M  C  (/.  Hence  /x  has  a  ^/'-decomposition 
in  M  for  every  neighbourhood  U1  of  #'. 


4.  We  now  show  that  T^  is  compact  in  M  .  Let  17,  \¥  and  V  be  as  in  Step  2.  Let 
i/  G  SM  be  such  that  ^(i/)  €  7^)  in  M  .  Now  /x  =  i/i/  =  i/V  for  some  i/,  i/7  €  SM. 
Arguing  as  in  Step  2,  there  exists  n  (which  does  not  depend  on  the  choice  Qfip(iS)  € 
such  that  for  any  m  >  n,  t/?(i/)  cannot  be  expressed  as  $(v)  =  -0(Ai)  •  •  •  ^»(Am)^(/3)  in 
for  AJ  €  Sp  H  17  \  W,  for  all  7,  and  V>(A/)s  commute  and  they  also  commute  with  ^(/3). 
Here,  ^(v)  is  a  limit  of  a  commutative  A"'-infinitesimal  triangular  system  in  M,  i.e. 


Triangular  systems  of  measures  61 


^(i/)  =lim/-»(jon?i1V'(^o')  for  some  j^O'  €  «V  Again  arguing  as_in  Step  2,  ^(*/)  = 
^(z/i)  -  -  •  7/>(z/n)  for  ^(z/0  G  r^)  n  ^(Z7  \  W).  That  is,j&(i/)  €  (^(17  \  W)n.  Since  n  does 
not  depend  on  the  choice  of  ^(ii)  in  7^),  7^)  C  (^(^  \  W))"-  Hence  it  is  easy  to  show 
as  in  the  proof  of  Lemma  2.1  of  [S4]  that  7*^)  is  relatively  compact. 


Step  5.  Let  J  =  ^(J^)  n  M.  Then  7  is  a  compact  semigroup  and  there  exists  a  maximal 
idempotent  h\  in  7.  Then  7'  =  Jh\  is  a  group.  Let  //  =  {x  G  /(^)  |  ^(x)h\  G  7'}.  It  is  easy 
to  check  that  H  is  a  compact  group.  Let  h  =  UH  and  let  /z*  =  ^(u;//).  Then  7/z*  =  J'h*  = 
/**  and  #"  =  K'h*  =  (^)  fiM)/i*,  which  is  a  compact  group.  Let  M*  =  Aft*.  M*  is  a 
closed  abelian  semigroup  with  identity  h*  and  ^"  C  M*  C  M.  Now  if  17  is  a  neigh- 
bourhood of  K"  in  M*  then  there  exists  a  neighbourhood  £/'  of  ^'  in  M  such  that 
[/'/z*  C  t/,  and  hence  if  V>(M)  =  AI  •  •  •  An  is  a  ^/'-decomposition  of  ^(/x)  in  M  ,  then  since 
p,=z  Hh  =  nhn,  il>(fjt)  =  AI  •  •  •  Xnip(hn)  and  hence  t/^/z)  =  AI/Z*  •  •  •  A,./z*  is  a  £/~decom- 
position  of  -0(/i)  in  M*.  Now  we  define  an  equivalence  relation  ~'  on  M*  as  follows:  For 

A,  z/  G  M*,  A  ~'  i/  if    A  =  kv  for  some    &  G  K"  . 

Let  S*  —M^  I  ~'  and  let  0  :  Af*  —  »•  S*  be  the  natural  projection  and  let  p  =  (/>  o  ^.  Then 
5*  is  a  Hausdorff  abelian  semigroup  with  identity  <t>(h*\  p~l(S*)=S  is  a  closed 
semigroup  in  M1H(G),  the  relation  ~  is  defined  by  p  on  5,  each  p(\)  in  Tp(^  is  inifinite- 
simally  divisible  in  5*  and  by  step  4,  J^)  is  compact.  Now  if  a,  b  G  Tp^  are  associates 
then  «  =  a'b  and  fc  =  b'a.  Let  /?,/?'  €  5i  be  such  that  p(/3)  =  b  and  'p(/3')  =  bf  and 
p(7)  =  a',  then  since  b  =  fcVfc,  ^(/3)  =  fa/;(/3')'0(7)'0(/3)  for  some  ^  €  X"  and  hence 
^  ^V»(/3)  ^or  ^  w-  AS  in  step  2,  supp  /3X  C  Jt/(/3)  =  /(/3)jc,  for  some  x  £  7M,  and  since 
=  fc'  is  infinitesimally  divisible,  it  is  easy  to  show  that  x  G  /^-  Therefore  b'  is 
identity  in  5*  and  b  =  0,  i.e.  7^)  is  associatefree. 

Now  if  /3  G  5M  be  such  that  p(/3)  G  T^,^)  is  an  idempotent  then  tjj(/3)n  G  r^,(M)  for  all  n 
and  hence  as  in  step  2,  supp/3  C  xl(^)  =  I(^L)X  for  some  x  G  7^.  Since  p(/3)  is  also  in- 
finitesimally divisible  in  S*  one  can  easily  show  that  x  G  /^  and  /3  =  jco;///  =  UH'*  for  some 
//'  C  /(^)  and  hence  ^(/3)  G  K"  and  hence  p(/3)  is  identity  in  5*.  Therefore  P(IJL)  is  bald. 


Step  6.  Now  let  G  be  connected  and  nilpotent  and  let  Z  be  the  center  of  G.  Then  G/Z  is 
simply  connected  and  hence  so  are  N(Z(fj))/Z  and  N(Z(^))/Z(JLL),  where  7V(Z(/x))  is  the 
normaliser  of  Z(/x),  and  both  of  them  are  connected.  Therefore,  7^  =  Z(/i)  as  7/z/Z(/x)  is 
compact.  Hence,  in  the  above  equation  K"  —  /i*  and  ~'  is  a  trivial  relation,  i.e.  5*  =  M* 
and  also  p  =  i£>. 

Now  we  show  that  for  s  G  7^)  \  ^(A)  in  5*,  there  exists  a  continuous  5-norm/5  on  Ts 
(in  5*)  such  that  fs(s)  >  0,  (an  ^-norm  on  7^  (in  S*)  is  a  map  /5  :  Ts  —  *  R+  which  is 
continous  at  the  identity  and  it  is  a  partial  homoporphism,  i.e.fs(s\S2)  =/?Csi)  +^(^2)  if 
$i,  52,  5i^2  G  Ty).  This  would  imply  the  embedding  of  t/>(ju)  in  a  continuous  real  one- 
parameter  semigroup  {7r},€R  in  5*  (cf.  [S3],  Theorem  2.3  or  [S4],  Theorem  4.1)  and  in 
particular,  /x  =  \nnxn  xn  G  Z(/I)  =  Z°(/i). 

Let  A  G  iS1  be  such  that  ^(A)  =  s.  If  A  is  not  a  translate  of  an  idempotent  then  as  in  the 
proof  of  Theorem  5.1  in  [S3],  there  exists  a  continuous  A-norm  f\  on  S  such  that 
/A  (A)  >  0,  (it  is  easy  to  see  that  one  does  not  need  the  underlying  semigroup  to  be  abelian 
in  that  proof).  Moreover,  if  ^(v\)  =  ^(1/2)  then  v\  =  z/2*  for  some  x  G  Z(/x).  Then 
v\v\  =  v2V2  and/A(^i)  =/\(^2)  (see  the  proof  of  Theorem  5.1  in  [S3]).  Therefore,  we 
can  define  a  .s-norm/j  on  Ts  in  S*  such  that  fs(ijj(v))  =/A(/^).  Now  if  A  is  indeed  a 
translate  of  an  idempotent,  i.e.  A  =  XLUK  =  UKX  for  some  compact  group  K  C  7(/z)  C  Z, 


62  Riddhi  Shah 

then  clearly  ;c  G  /M  =  Z(p.)  and  hence  s  =  ^(A)  is  an  idempotent.  Now  since  ^(/x)  is  bald 
s  =  tp(h)y  a  contradiction. 

The  embeddability  of  ^(fj)  in  particular  implies  that  T/>(^)  =  ^(Xn)ny  and  hence 
^  =  A%,  jcn  G  Z(/Lt)  for  all  n.  Therefore,  supp  Xnn  C  GO)Z(/x).  Here,  supp  Xn  C  ynG(p.) 
for  some  yn  €  suppAn  C  N(p,).  Therefore,  3^  G  G(/z)Z(jLi)  C  G(//)Z(/x),  where  G(»  is 
the  Zariski  closure  of  G(/x).  Since  N(fjL)/G(p,)Z(iJ,)  is  simply  connected,  yn  G  G(/x)Z(/x) 
for  all  n.  That  is,  for  each  n,  supp  Art  C  G(/^)Z(/x)  and  hence  An  =  /3nzn  for  some 
zw  G  Z(p)  and  supp  /?„  C  G(^)  and  /i  =  /3£4,  where  4  =  zjjxn  G  Z(/x).  Now  we  have  that 
z'n  G  C  =  G(/z)  H  Z(ju),  which  is  the  center  of  G(^).  Therefore,  CZ  C  Z(p)  is  an  abelian 
algebraic  subgroup  containing  the  center  Z  of  G.  Therefore  CZ  is  connected,  and  hence  it 
is  divisible.  In  particular,  each  z'n  is  infinitely  divisible  in  CZ,  and  hence  //,  is  infinitely 
divisible  on  G  which  is  a  connected  nilpotent  Lie  group,  therefore  fj,  is  embeddable 
(cf.  [BM]). 

Remark.  As  remarked  in  [S4],  Theorem  4.1  also  holds  for  p,  G  M1H(G]  which  is 
infinitesimally  divisible  in  Af^(G). 

We  now  state  the  following  theorem  for  maximally  almost  periodic  groups  without  a 
proof.  A  locally  compact  group  G  is  said  to  be  maximally  almost  periodic  if  its 
irreducible  finite  dimensional  unitary  representations  separate  points  of  G. 

Theorem  4.2.  Let  G  be  a  maximally  almost  periodic  first  countable  group.  Let  A  be  a 
commutative  infinitesimal  triangular  system  of  probability  measures  converging  to  \i>  in 
M1  (G).  Then  there  exists  an  x  G  G°  such  that  xp,  =  //jc  is  embeddable. 

If  G  is  as  above  then  there  exists  a  normal  vector  subgroup  V,  such  that  G°/V  is 
compact  and  V  centralises  an  open  subgroup  of  finite  index  in  G  (cf.  [RW],  Theorems 
1,  2].  The  above  theorem  can  be  proven  using  the  above  fact,  Proposition  2.5,  Lemma 
2.4,  Proposition  3.3  and  Theorem  4.2  of  [S4]  and  the  techniques  developed  above. 

Acknowledgement 

The  author  would  like  to  thank  the  referee  for  useful  suggestions. 

References 

[BM]  Burrell  Q  L,  Infinitely  divisible  distributions  on  connected  nilpotent  Lie  groups  II.  /. 

London  Math.  Soc.  II  9  (1974)  193-196 
[DM1]  Dani  S  G  and  McCrudden  M,  Factors,  roots  and  embeddability  of  measures  on  Lie  groups. 

Math.  Z.  190  (1988)  369-385 
[DM2]  Dani  S  G  and  McCrudden  M,  Embeddability  of  infinitely  divisible  distributions  on  linear 

Lie  groups.  Invent.  Math.  110  (1992)  237-261 
[DM3]  Dani  S  G  and  McCrudden  M,  Infinitely  divisible  probabilities  on  discrete  linear  groups.  J. 

Theor.  Prob.  9  (1996)  215-229 
[H]  Heyer  H,  Probability  measures  on  locally  compact  groups  (Berlin-Heidelberg:  Springer- 

Verlag)  (1977) 
[Ml]  McCrudden  M,  Factors  and  roots  of  large  measures  on  connected  Lie  groups,  Math.  Z.  177 

(1981)  315-322 

[M2]  McCrudden  M,  An  introduction  to  the  embedding  problem  for  probabilities  on  locally 
compact  groups,  in:  Positivity  in  Lie  Theory:  Open  Problems.  De  Gruyter  Expositions 
in  Mathematics  26,  (Eds)  J  Hilgert,  J  D  Lawson,  K-H  Neeb  and  E  B  Vinberg  (Berlin-New 
York:  Walter  de  Gruyter)  (1998)  pp.  147-164 


Triangular  systems  of  measures  63 

[P]  Parthasarathy  K  R,  Probability  measures  on  metric  spaces  (New  York-London:  Academic 

Press)  (1967) 
[RW]  Robertson  L  and  Wilcox  T  W,  Splitting  in  MAP  groups,  Proc.  Am.  Math.  Soc.  33  (1972) 

613-618 
[SI]  Shah  Riddhi,  Semistable  measures  and  limit  theorems  on  real  and/?-adic  groups.  Mh.  Math. 

115  (1993)  191-213 
[S2]  Shah  Riddhi,  Convergence-of-types  theorems  on  ^-adic  algebraic  groups.  Proceedings  of 

Oberwolfach  conference  on  Probability  measures  on  groups  and  related  structures  XI  (ed.) 

H  Heyer  (1995)  357-363 
[S3]  Shah  Riddhi,  Limits  of  commutative  triangular  systems  on  real  and  p-adic  groups.  Math. 

Proc.  Camb.  Philos.  Soc.  120  (1996)  181-192 
[S4]  Shah  Riddhi,  The  central  limit  problem  on  locally  compact  group,  Israel  J.  Math.  110 

(1999)  189-218 


Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  Ill,  No.  1,  February  2001,  pp.  65-94. 
©  Printed  in  India 


Topological  * -algebras  with  C* -enveloping  algebras  II 


S  J  BHATT 

Department  of  Mathematics,  Sardar  Patel  University,  Vallabh  Vidyanagar  388  120, 

India 

E-mail:  sjb@spu.ernetin 

MS  received  17  October  1997;  revised  14  March  2000 

Abstract.  Universal  C* -algebras  C*  (A)  exist  for  certain  topological  * -algebras  called 
algebras  with  a  C* -enveloping  algebra.  A  Frechet  * -algebra  A  has  a  C* -enveloping 
algebra  if  and  only  if  every  operator  representation  of  A  maps  A  into  bounded 
operators.  This  is  proved  by  showing  that  every  unbounded  operator  representation  TT, 
continuous  in  the  uniform  topology,  of  a  topological  * -algebra  A,  which  is  an  inverse 
limit  of  Banach  * -algebras,  is  a  direct  sum  of  bounded  operator  representations, 
thereby  factoring  through  the  enveloping  pro-C* -algebra  E(A)  of  A.  Given  a  C*- 
dynamical  system  (G,A,a),  any  topological  *-algebra  B  containing  CC(G,A)  as  a 
dense  *-subalgebra  and  contained  in  the  crossed  product  C*-algebra  C*(G,A,a) 
satisfies  E(B)  =  C*(G,A,a).  If  G  =  IR,  if  B  is  an  a-invariant  dense  Frechet  *- 
subalgebra  of  A  such  that  E(B)  =  A,  and  if  the  action  a  on  B  is  m-tempered,  smooth 
and  by  continuous  * -automorphisms:  then  the  smooth  Schwartz  crossed  product 
5(R,B,a)  satisfies  E(S(U,B,a}}  =  C*(lR,A,a).  When  G  is  a  Lie  group,  the  C°°- 
elements  C°°(A),  the  analytic  elements  CW(A)  as  well  as  the  entire  analytic  elements 
CeuJ(A)  carry  natural  topologies  making  them  algebras  with  a  C* -enveloping  algebra. 
Given  a  non-unital  C* -algebra  A,  an  inductive  system  of  ideals  Ia  is  constructed 
satisfying  A  =  C*-ind  lim/a;  and  the  locally  convex  inductive  limit  ind  lim/a  is  an  m- 
convex  algebra  with  the  C* -enveloping  algebra  A  and  containing  the  Pedersen  ideal  KA 
of  A.  Given  generators  G  with  weakly  Banach  admissible  relations  R,  we  construct 
universal  topological  *-algebra  A(G,  R)  and  show  that  it  has  a  C*-enveloping  algebra  if 
and  only  if  (G,/?)  is  C* -admissible. 

Keywords.  Frechet  *-algebra;  topological  *-algebra;  C* -enveloping  algebra; 
unbounded  operator  representation;  O*  -algebra;  smooth  Frechet  algebra  crossed 
product;  Pedersen  ideal  of  a  C* -algebra;  groupoid  C* -algebra;  universal  algebra  on 
generators  with  relations. 


1.  Statements  of  the  results 

In  [5],  a  functor  E  has  been  considered  that  associates  C*-algebras  E(A)  with  certain  topo- 
logical *-algebras  A,  called  algebras  with  a  C*-enveloping  algebra.  By  a  classic  construc- 
tion due  to  Gelfand  and  Naimark,  a  Banach  *-algebra  A  admits  a  C* -enveloping  algebra 
C*(A)  =  £(A)  ([14],  2.7,  p.  47).  By  ([15],  Theorem  2.1),  a  complete  locally  m-convex 
* -algebra  has  a  C* -enveloping  algebra  if  and  only  if  it  admits  a  greatest  continuous  C*- 
seminorm.  The  following  extrinsic  characterization  of  such  algebras  has  been  motivated 
by  the  simple  observation  that  any  *-homomorphism  from  a  Banach  *-algebra  into  the 
* -algebra  of  linear  operators  on  an  inner  product  space  maps  the  algebra  into  bounded 
operators. 

65 


66  S  J  Bhatt 

Theorem  1.1.  Let  A  be  a  Frechet  *  -algebra.  Then  A  is  an  algebra  with  a  C*  -enveloping 
algebra  if  and  only  if  every  *  -representation  of  A  is  a  bounded  operator  representation. 

The  above  theorem  is  false  without  the  assumption  that  A  is  metrizable  (see 
Remark  4.4).  By  a  ""-representation  (TT,  £>(TT),#)  of  a  *-algebra  A  [37]  is  meant  a  homo- 
morphism  TT  from  A  into  linear  operators  (not  necessarily  bounded)  all  defined  on  a 
common  dense  invariant  subspace  £>(TT)  of  a  Hilbert  space  H  such  that  for  all  x  in  A, 
7r(jc*)  C  TT(*)*.  In  the  general  theory  of  *  -algebras,  following  Palmer  [24],  A  is  called  a 
BG*  -algebra  if  every  *-homomorphism  from  A  into  linear  operators  on  a  pre-Hilbert 
space  maps  A  into  bounded  operators.  The  absence  of  a  complete  algebra  norm  on  a  non- 
Banach  *-algebra  A  indicates  that  A  may  contain  elements  that  fail  to  be  bounded  in  any 
natural  sense.  Hence  an  appropriate  framework  for  the  representation  theory  of  A  is  that 
of  unbounded  operator  representations.  However,  this  natural  point  of  view  was 
developed  rather  late,  following  [30,20].  Prior  to  (and  later,  in  spite  of)  this,  bounded 
operator  representations  of  A  have  been  investigated  in  detail,  especially  when  A  is  a 
locally  w-convex  *-algebra,  i.e.,  A  =  proj  limAQ,  the  inverse  limit  (also  called  the 
projective  limit)  of  Banach  "-algebras  [9,  15],  (see  [16]  for  a  summary  of  bounded 
operator  representations  of  A).  In  fact,  such  an  A,  when  *-semisimple,  admits  sufficiently 
many  continuous  irreducible  bounded  operator  representations  [9].  Then  the  enveloping 
pro-C*~algebra  (projective  limit  of  C*  -algebras)  E(A)  of  A,  discussed  in  [10],  [19]  and 
[15],  turns  out  to  be  E(A)  =  proj  lim  E(Aa),  E(Aa)  =  C*(Aa)  being  the  enveloping  C*- 
algebra  of  the  Banach  *-algebra  Aa  ([15],  Theorem  4.3).  Thus  A  has  a  C"  -enveloping 
algebra  if  E(A)  is  a  C*-algebra.  By  the  construction,  E(A)  is  universal  for  norm- 
continuous  bounded  operator  representations  of  A.  Theorem  1.2,  to  be  used  to  prove 
Theorem  1.1,  shows  desirably  that  E(A)  is  also  universal  for  representations  into 
unbounded  operators.  The  uniform  topology  ([37],  p.  77,  78)  on  an  unbounded  operator 
algebra  is  defined  at  the  end  of  this  section. 

Theorem  1.2.  Let  A  be  complete  locally  m-convex  *  -algebra.  Let  (TT,  £>(?]-),  #)  be  a 
closed  "-representation  of  A  continuous  in  the  uniform  topology  on  ?r(A).  Then  there 
exists  a  unique  *  -representation  (a,D(o),Hff)  of  E(A]  such  that  the  following  hold. 


(1)  Ha  =  H 

(2)  As  a  representation  of  E(A),  a  is  closed  and  continuous  in  the  uniform  topology  on 
a(E(A}). 

(3)  a  is  an  'extension'  of  TT  to  E(A)  in  the  sense  that  for  all  x  in  A,  (croj)(x)  =  TT(JC), 
j  :  A  ->  E(A)  being  the  natural  map,  jf(jc)  =  ;c  •+  srad  (A),  srad  (A)  denoting  the  star 
radical  of  A. 

(4)  On  the  unbounded  operator  algebra  ?r(A),  the  uniform  topology  r£(A)  is  a  (not 
necessarily  complete)  pro-C*  -topology  which  coincides  with  the  relative  uniform 
topology  r£(£(A))  from  a(E(A)). 

COROLLARY  1.3 

Let  TT  be  a  closed  irreducible  *  -representation  of  a  complete  locally  m-convex  *  -algebra  A 
continuous  in  the  uniform  topology  on  ?r(A).  (In  particular,  let  A  be  Frechet  and  IT  be 
irreducible).  Then  TT  maps  A  into  bounded  operators. 

AO*-algebras  (abstract  O*-algebras)  [36,  37]  provide  the  unbounded  operator  algebra 
analogues  of  C*-algebras.   Starting  with  a  topological  (not  necessarily  m-convex) 


C*  -enveloping  algebras  67 

"-algebra  A,  one  can  construct  an  enveloping  A0*-algebra  O(A)  universal  for 
* -representations  continuous  in  the  uniform  topology,  and  declare  A  to  have  a  C*- 
enveloping  algebra  if  the  uniform  topology  on  O(A)  is  normable.  On  the  other  hand,  by 
modifying  the  construction  in  [15],  the  pro- C* -algebra  E(A)  can  also  be  considered  as  the 
universal  object  for  norm-continuous  bounded  operator  * -representations  of  more  general 
locally  convex,  non-m-convex,  *-algebras  A.  In  general,  the  completion  of  O(A)  differs 
from  E(A).  For  a  barrelled  A,  O(A)  is  normable  implies  that  E(A)  is  a  C*-algebra,  but  the 
converse  does  not  hold.  In  the  present  context,  the  following  shows  that  both  the 
approaches  are  consistent  in  the  metrizable  case. 

Theorem  1.4.  Let  A  be  a  Frechet  ""-algebra.  Then  the  pro-C* -algebra  E(A)  is  the  com- 
pletion of  the  AO* -algebra  O(A}.  Thus  O(A)  is  normable  if  and  only  if  A  is  an  algebra 
with  a  C* -enveloping  algebra. 

There  are  several  situations  in  C* -algebra  theory  in  which  topological  * -algebras  arise 
naturally  [27].  Enveloping  C* -algebras  provide  a  standard  method  of  constructing  C*- 
algebras;  and  frequently,  lurking  behind  such  a  construction  is  a  topological  * -algebra  B 
such  that  E(B)  =  A.  Let  a  be  a  strongly  continuous  action  of  a  locally  compact  group  G 
by  * -automorphisms  of  a  C* -algebra  A.  The  crossed  product  C* -algebra  C*  (G,  A,  a)  is  the 
enveloping  C*-algebra  of  the  Z^-crossed  product  Banach  *-algebra  Ll(G,A,ct).  If  B 
is  a  topological  *-algebra  such  that  CC(G,A)  C  B  C  C*(G,A,a)  and  CC(G,A)  is  dense 
in  B,  then  E(B)  =  C*(G,  A,  a).  Let  G  be  a  Lie  group.  Then  the  *-subalgebra  C°°(A)  of 
C°°-elements  of  A  is  a  Frechet  *-algebra  with  an  appropriate  topology  such  that 
£(C°°(A))  =  A.  The  '-algebras  CW(A)  and  Ceu(A)  consisting  of  analytic  elements  and 
entire  elements  of  A  are  shown  to  carry  natural  topologies  making  them  algebras  with 
C*-enveloping  algebras.  We  also  consider  the  smooth  crossed  product  [29,34].  For 
simplicity,  we  take  G  =  R,  and  prove  the  following. 

Theorem  1.5.  Let  a  be  a  strongly  continuous  action  of  R  by  * -automorphisms  of  a 
C* -algebra  A.  Suppose  that  B  is  a  dense  Frechet  *-subalgebra  of  A  satisfying  the  following. 

(a)  A  has  a  bounded  approximate  identity  contained  in  B  and  which  is  a  bounded 

approximate  identity  for  B. 
(b)E(B)=A. 
(c)  B  is  a-invariant;  and  the  action  a  of  R  on  B  is  smooth,  m-tempered  and  by 

continuous  *  -automorphisms  of  B. 

Then  the  smooth  Schwartz  crossed  product  5(R,B,  a)  is  a  Frechet  ""-algebra  with  a 
C* -enveloping  algebra,  and  E(S(R,  B,  a))  =  C*  (R,  A,  a).  Further,  if  the  action  ofRonB 
is  isometric  (see  §  5),  then  the  L1  -crossed  product  L1  (R,  B,  a)  is  also  a  Frechet  * -algebra 
with  a  C* -enveloping  algebra,  and  E(Ll(R,B,a))  =  C*(R,A,a). 

It  follows  that  E(S(R,  C°°(A),  a)  =  C*(R,  A,  a).  In  particular,  if  a  is  a  smooth  action 
of  R  on  a  C°° -manifold  M,  then  £(S(R,  C°°(M),  a)  =  C*(R,  C(Af),  a),  the  covariance 
C*-algebra  of  the  R-space  M. 

For  a  locally  compact  Hausdorff  space  X,  let  1C  be  the  directed  set  consisting  of  all 
compact  subsets  of  X.  For  K  €  /C,  let  CK(X)  =  {/  €  CC(X)  :  supp/  C  K}9  CC(X)  denoting 
the  compactly  supported  continuous  functions  on  X.  It  is  well  known  that  {C#(X)  : 
K  e  JC}  forms  an  inductive  system;  and  Co(X)  =  C*-ind  limC^(X)  (C*-inductive  limit), 
CC(X)  =  ind  HmC^(X)  (locally  convex  inductive  limit).  Further,  CC(X)  with  the  locally 


68  S  J  Bhatt 

convex  inductive  limit  topology  is  a  complete  locally  w-convex  Q-algebra  and 
E(CC(X))  =  Cb(X).  The  following  provides  a  non-commutative  analogue  of  this.  We 
refer  to  the  last  paragrapgh  in  this  section  for  the  relevant  definitions  pertaining  to 
topological  algebras. 

Theorem  1.6.  Let  A  be  a  non-unital  C*  -algebra.  Let  KA  denote  its  Pedersen  ideal  For 
a  G  K%,  let  Ia  denote  the  closed  two  sided  ideal  of  A  generated  by  aa*.  Let  Ky  = 
\]{Ia  :aeK%}.  Then  the  following  hold. 

(1)  {Ia  :  a  G  K%}  forms   an   inductive   system,   A  =  C*  -  ind  lim{/fl  :  a  G  KA},   and 
KnAc  =  ind  lim{/a  :  a  E  K+}. 

(2)  Ky  with  the  locally  convex  inductive  limit  topology  t  is  a  locally  m-convex  Q-algebra 
satisfying  E(KnAc]  =  E(KA)  =  A. 

(3)  If  A  has  a  countable  bounded  approximate  identity,  then  (K™,  t)  is  an  LFQ-algebra. 

In  general  KA  ^  Ky,  though  KA  C  Ky.  Now  KA  has  been  interpreted  as  a  non- 
commutative  analogue  of  CC(X).  Then  Ky  may  be  interpreted  as  continuous  functions  on 
a  non-commutative  space  vanishing  at  infinity  in  'commutative  directions'  and  having 
compact  supports  in  'non-commutative  directions'.  This  interpretation  is  suggested  by  the 
remarks  preceeding  ([28],  Theorem  8). 

The  universal  C*-algebra  C*(G,R)  on  a  C*-admissible  set  of  generators  G  with 
relations  R  provides  another  method  of  constructing  C*-algebras.  Motivated  by  some 
problems  in  C*  -algebras,  Phillips  introduced  more  general  weakly  C*  -admissible 
generators  with  relations  (G,R)  leading  to  the  construction  of  the  universal  pro-C*- 
algebra  C*(G,7?)  on  (G,R}  [21].  In  §8,  we  construct  a  universal  topological  "-algebra 
A(G,jR)  on  (G,/?)  with  weakly  Banach  admissible  relations  R,  and  prove  the  following. 

Theorem  1.7.  Let  (G,R)  be  weakly  Banach  admissible. 


(1) 

(2)  A(G,R)  has  a  C*  -enveloping  algebra  if  and  only  if(G,R)  is  C*  -admissible. 

The  paper  is  organized  as  follows.  Proofs  of  Theorems  1.1,  1.2  and  1.4  are  presented  in 
§  3.  The  preliminary  lemmas  and  constructions  in  the  locally  convex,  non-m-convex  set 
up  more  general  than  in  [5],  are  discussed  in  §  2.  Section  4  contains  a  couple  of  remarks 
including  some  corrections  in  [5].  The  smooth  crossed  product  is  discussed  in  §5 
culminating  in  the  proof  of  Theorem  1.5.  Section  6  contains  the  proof  of  Theorem  1.6. 
This  is  followed  by  a  brief  discussion  on  the  C*-algebra  of  a  groupoid  in  §  7.  Universal 
C*-algebras  on  generators  with  relations  are  discussed  in  §  8.  In  what  follows,  we  briefly 
recall  the  relevant  ideas  in  unbounded  operator  representations. 

For  the  basic  theory  of  unbounded  operator  *  -representations  (7r,P(7r),#)  of  a 
*-algebra  A,  we  refer  to  [37,  30].  Let  A1  denote  the  unitization  of  A.  The  graph  topology 
f7r  =  ^(A»)  on  2>M  is  defined  by  seminorms  £->  ||f||  +  |br(x)f||,  where  x  €  A.  The 
closure  TT  of  TT  is  the  *  -representation  (TT,  /)(??),#),  where  D(TT)  =  P|{D(TT(JC))  :  x  G  A1}, 
D(ir(x))  being  the  domain  of  the  closure  TT(JC)  of  TT(JC)  ;  and  TT(JC)  =  n(x)\D^  for  all  x  in  A1  . 
Throughout,  ?r  is  assumed  non-degenerate,  i.e.,  the  norm  closure  (ir(A)H)~  =  H  and  the 
^-closure  (^(A^TT))'*  =  P(TT).  If  TT  =  ?f,  then  TT  is  closed.  The  hermitian  adjoint  TT*  of  TT 
is  the  representation  (not  necessarily  a  ^representation)  (7r*,D(7r*),#),  where 
D(**)  =  fK^M*)*  :  x  e  A1},  and  <K*(X]  =  Tr^^.for  all  jc  G  A1.  If  TT  =  TT*,  then 
TT  is  self-adjoint.  Further,  TT  is  standard  if  7r(jt*)*  =  TT(JC)  for  all  x  in  A1.  If  each  TT(JC)  is  a 


C* -enveloping  algebras  69 

bounded  operator,  then  TT  is  bounded.  If  TT  is  a  direct  sum  of  bounded  representations,  then 
TT  is  weakly  unbounded.  An  O* -algebra  is  a  collection  U  of  linear  operators  T  all  defined 
on  a  dense  subspace  D  of  a  Hilbert  space  H  such  that  for  all  T  G  W,  one  has  TZ)  C  £>,  and 
r*D  C  £>;  and  W  is  a  *-algebra  with  the  pointwise  linear  operations,  composition  as 
the  multiplication,  and  T  — >  T+  :=  7*1^  as  the  involution.  Given  a  * -representation 
(7r,D(7r),/f)  of  a  *-algebraA,  the  wm/flrm  topology  [20],  ([37],  p.  77-78)  rD  =  r^))  on  the 
0*-algebra  ?r(A)  is  the  locally  convex  topology  defined  by  the  seminorms  {qK  \  K  is  a 
bounded  subset  of  (D(7r),/7r)},  where 

qK(n(x}}  =  sup{|(7r(*)£,7?)|  :  &»7  in  AT}. 

A  vector  £  in  D(TT)  is  strongly  cyclic  [30]  (called  cyclic  in  [37])  if  Z>(TT)  =  (7r(A)£)~f7r 
the  closure  of  (TT(A)£)  in  (Z)(7r),  f^).  By  a  cyclic  vector,  we  mean  £  in  D(TT)  such  that  the 
norm  closure  (TT(A)£)~  =  //.  For  topological  *-algebras,  we  refer  to  [21].  A  Q-algebra  is 
a  topological  algebra  whose  quasi-regular  elements  form  an  open  set.  An  LFQ-algebra  is 
a  Q-algebra  which  is  an  LF- space  [41].  The  topology  of  a  locally  convex  (respectively 
locally  m-convex)  * -algebras  A  is  determined  by  the  family  K(A)  (respectively  KS(A)),  or 
a  separating  subfamily  P  thereof,  consisting  of  continuous  *-seminorms  (repsectively 
continuous  submultiplicative  *-seminorms)  p.  If  A  has  a  bounded  approximate  identity 
(ei),  then  it  is  assumed  that  p(ei)  <  1  for  all  i  and  all  p.  A  pro-C* -algebra  is  a  complete 
locally  m-convex  * -algebra  whose  topology  is  determined  by  a  family  of  C* -seminorms. 
A  Frechet  * -algebra  (respectively  locally  convex  F* -algebra)  is  a  complete  metrizable 
locally  m-convex  (respectively  locally  convex)  * -algebra.  A  cr-C* -algebra  means  a  Frechet 
pro-C* -algebra.  For  pro-C* -algebras,  we  refer  to  [26,27]. 

2.  Preliminary  constructions  and  lemmas 

Let  A  be  a  * -algebra,  not  necessarily  having  an  identity  element.  Let  /  be  a  positive 
linear  functional  on  A.  Then  /  is  representable  if  there  exists  a  closed  strongly  cyclic 
* -representation  (7r,D(7r),#)  of  A  having  a  strongly  cyclic  vector  £  E  Z)(TT)  such  that 
f(x)  =  {TT(.X)£,  £)  for  all  x  €  A.  If  TT  can  be  chosen  to  be  a  bounded  operator  representation, 
then  /  is  boundedly  representable.  The  first  half  of  the  following  is  an  unbounded 
representation  theoretic  analogue  of  ([39],  Theorem  1),  whereas  the  remaining  half 
improves  a  part  of  ([39],  Theorem  1)  even  in  the  bounded  case.  The  proof  exhibits  the 
unbounded  analogue  of  the  GNS  construction  in  the  case  of  non-unital  algebras.  This 
provides  a  useful  supplement  to  ([37],  §  8.6).  It  is  well-known  that  a  representable 
functional  is  boundedly  representable  if  and  only  if  it  is  admissible  in  the  sense  that  for 
each*  €  A,  there  exists  k  >  0  such  that/^Vjcy)  <  kf(y*y)  for  all  y  E  A.  In  the  following, 
Lemma  2.1(3)  is  very  close  to  ([39],  Theorem  1)  in  which  a  C*-seminorm  p  is  taken. 

Lemma  2.1.  Let  f  be  a  positive  linear  functional  on  a  * -algebra  A.  The  following  are 
equivalent. 

(1)  /  is  representable. 

(2)  There  exists  m  >  0  such  that  \f(x)\2  <  mf(jfx)  for  all  x  E  A. 

Further,  f  is  boundedly  representable  if  and  only  iff  satisfies  (2)  above  and  the  following. 

(3)  There   exists   a  submultiplicative   *-seminorm  p  on  A   and  M  >  0   such   that 

\f(x)\<Mp(x}forallxeA. 

When  A  is  a  Banach  *-algebra,  Lemma  2.1  is  given  in  ([7],  Theorem  37.11,  p.  199).  In 
the  framework  of  unbounded  representation  theory,  it  is  discussed  in  [2].  There  is  a  gap  in 


70  SJ  Bhatt 

the  proof  in  ([7],  Theorem  37.11)  in  that  hermiticity  of/  has  been  implicitly  used. 
Regrettably  it  remained  unnoticed  in  [2].  This  was  rectified  in  [39]  in  the  formalism  of 
bounded  representations.  The  following  proof  provides  an  analogous  correction  in  the 
context  of  unbounded  representations. 

Proof.  Suppose  (1)  holds  with/(jc)  =  (TT(JC)^,  f  )  for  all  x  G  A.  Then  for  all  x  in  A. 


as  9r(A)£  C  Z>(TT)  =  D(x(x))  =  D(TT(JC*))  and  TT(JC*)  C  TT(JC)*.  Thus 


for  all  jc  €  A,  giving  (2). 

Conversely,  assume  (2).  We  adopt  the  GNS  construction.  Let  N/  =  {x  G  A  :  f(jfx)  =  0}. 
By  the  Cauchy-Schwarz  inequality,  Nf  is  a  left  ideal  of  A.  Let  X/  =  A/TV/,  and  A/  :  A  —  >  Xf 
be  A/(JC)  =  x  +  Nf.  Then  (A/(;c),  A/(y)}  =  /(/X)  defines  an  inner  product  on  Xf.  Let  Hf  be 
the  Hilbert  space  obtained  by  completing  Xf.  Let  (p  :  Xf  —  >  C  be  <p(A/(;t))  =  /(*),  a  linear 
functional.  Then  for  all  jc  G  A, 

MA/«)|2  =  |/M|2  <  mf(x**)  =  m(\f(X),Xf(X))  =  m\\Xf(x)\\2. 

Thus  (p  extends  uniquely  to  Hf  as  a  bounded  linear  functional;  and  by  Riesz  theorem, 
there  exists  a  £  G  #/  such  that  for  all  x  G  A,/(JC)  =  <p(Xf(x))  =  (A/(jc),  f  }.  Further,  if  m  is 
the  minimum  possible  constant  in  the  assumed  inequality,  then  ||£||  =  m1/2.  The  idea  of 
using  Riesz  theorem  at  this  stage  is  borrowed  from  [39].  Define  a  *  -representation 
(7ro,Z)(7To),#/)  of  A  by:  D(TTQ}  =  Xf,  and  for  any  x  in  A,  7ro(*)A/(y)  =  A/(*y)  for  all  >?  in 
A.  Let  TT  be  the  closure  of  TTQ.  Then  for  all  x,  y  in  A, 


Assertion  1.  X/  = 

Let  jc  G  A.  For  all  y  e  A, 


showing  that  the  linear  functional  A/(y)  ->•  (7r0(^)A/(y),f)  on  D(TTO)  is  ||  ||  -continuous. 
Hence  ^  G  D(KQ(X)*)  for  all  ^  e  A.  It  follows,  by  the  definition  ofDfa),  that  £  G  £>K). 
Now  (i)  becomes  (A/  (x)  ,  A/  (y)  )  =  (A/  (jc)  ,  TTO  (y*  )  *0  for  all  x  G  A.  Since  Xf  is  dense  in  #/, 
we  obtain  \f(y)  =  7T0(y*)*C  =  -tf>(y)£  for  all  y  in  A.  Thus  Xf  =  7rJ$(A)£. 

Assertion  2.  £  G  £>(TT). 

Since  TTO(X)  =  TTO(X)  **,  we  show  that  £  G  D(7r0  (*)**)  for  all  jc  G  A,  i.e.,  for  all  x,  the 
functional  on  D(TTO  (x)*)  given  by  77  —  >  {TTO  (x)*?;,  g)  is  1  1  1  1  -continuous.  Fix  an  ;<:  G  A.  Now 
C  G  D(fl{;),  hence  g  G  D(TTO(X*)*)  so  that  the  functional  g  on  D(7r0(x*))  =  Xf  defined  by 
^(7?)  =  (^o(^)»?,0  is  ||  1  1  -continuous,  and  extends  continuously  to  #/.  Now  let 
^  G  Z>(TTO(X)*).  Let  (77^)  be  a  sequence  in  X/  such  that  rjk  ->  ^  in  ||  |[.  Then  ^  G  ^(TrJ) 
implies  that  for  any  x  G  A, 


C*  -enveloping  algebras  71 

showing  that  V  —*  {^oMVj?)    is    II    |  (-continuous   on  D(TTO(JC)*).   This   proves   the 
assertion  2. 
Now  by  the  proof  of  assertions  1  and  2  above,  it  follows  that  for  any  x  €  A, 


Clearly  £  is  a  strongly  cyclic  vector  for  TT.  Thus  (2)  implies  (1). 

Now  assume  (2)  and  (3).  Let  Np  =  {x  e  A  :  /?(*)  =  0},  a  *-ideal  in  A.  Let  Ap  be  the 
Banach  *-algebra  obtained  by  completing  A/Np'in  the  norm  \\xp\\  =  p(x)  where 
xp  =  x  H-  Wp.  By  (3),  F(xp)  =  /(*)  gives  a  well-defined  continuous  positive  functional  on 
Ap.  By  standard  Banach  "-algebra  theory,  for  all  x,  y  in  A, 

>  =  /(yV*y)  =  F(y*pX;Xpyp) 


Since  7r(A)£  is  dense  in  H/,  TT  is  a  bounded  operator  representation. 

COROLLARY  2.2 

L£?A  &e  a  *  -algebra. 

(1)  A  positive  functional/  on  A  is  representable  if  and  only  iff  is  extendable  as  a  positive 
functional  on  the  unitization  A1  of  A. 

(2)  A  representable  positive  functional  on  A  satisfies  f(x*)  =/(jc)~  for  all  x  in  A. 

(3)  Let  A  be  a  topologicdl  *  -algebra  having  a  bounded  approximate  identity.  Then  every 
continuous  positive  functional  on  A  is  representable. 

COROLLARY  2.3 

Let  A  be  a  complete  locally  m-convex  *  -algebra  with  a  bounded  approximate  identity  (e^) 
satisfying  p(e^)  <  1  for  all  p  in  a  defining  family  of  seminorms. 

(1)  Letf  be  a  continuous  positive  functional  on  A.  Thenf  is  boundedly  representable  and 
there  exists  p  G  KS(A)  such  that  \f(x)\  <  (lim  sup  /(e7£*  ))/?(*)  for  all  x  E  A. 

(2)  Let  (TT,  D(TT),#)  be  a  *  -representation  of  A.  Then  each  ir(e^)  is  a  bounded  operator 
and  ||7r(e7)||  <  1  for  all  7.  Further,  if  it  is  strongly  continuous  (in  particular,  if  -K  is 
continuous  in  the  unifrom  topology,  which  is  the  case  if  A  is  locally  convex  F*  ([37], 
Theorem  3.6.8,  p.  99)),  then  \\n(ej£  -  £\\  ->  0  for  each  £. 

Proof.  (1)  By  continuity,  there  exist  p  G  KS(A)  and  m  >  0  such  that  \f(x)\  <  mp(x)  for 
all  x  G  A.  Now  Lemma  2.1  applies  by  Corollary  2.2(3).  Let  /  =  lim  sup  /(e7e*  ),  which  is 
finite.  Let  c  —  sup{|/(jc)|  :  p(x)  =  1}.  Choose  a  sequence  (jcn)  in  A  such  that  f(xn)  -^  c 
andp(^:n)  =  1  for  all  n.  Then,  by  the  Cauchy-Schwarz  inequality, 

\f(xn)  |2  =  lim  \f(e,xn}  |2  <  (lim  sup/(^;  ))/(*>„)  <  /c, 

as  p(x^Xn)  <  p(xn)2  =  1.  Hence  c2  <  /c,  i.e.  c  <  /,  and  the  assertion  follows. 

(2)  Let  P=  (pa)  be  a  cofinal  subset  of  KS(A)  determining  the  topology  of  A.  Let 

Ap  =  {x  €  A  :  supapa(x)  <  oo}.  Then  Ap  is  a  *-subalgebra  of  A  containing  each  e 

is  complete,  Ap  is  a  Banach  "-algebra  with  norm  p(x)  =  suppa(jc).  For  any  £  G 


72  S  J  Bhatt 

consider  the  positive  functional  ^(x)  =  (ir(x)^  ^)  on  A.  Then  for  all  .xGA, 
|^M|2  <  ||£||2  u$(x*x).  By  Lemma  2.1,  o;^  is  representable,  hence  extends  as  a  positive 
functional  uj  on  the  unitization  A1  of  A.  In  view  of  the  inclusion  map  (A^)1  —  »  A1,  LJ  is  a 
positive  functional  on  (Ap)1  .  By  ([7],  Corollary  37.9,  p.  198),  cj  is  continuous  in  the  norm 
of  (Ap)1.  It  follows  that  0;$  restricted  to  Ap  is  continuous  in  the  norm  of  Ap  and 


IM^)£||2  =  w^ 

showing  that  ||7r(^7)||  <  1.  Now  suppose  that  TT  is  strongly  continuous.  Let  77  G  D(TT)  and 
e  >  0.  There  exists  x  G  A  and  77'  G  £>(TT)  such  that  HTT^)?/  —  rj\\  <  e/3.  Since  e^x  —  »  *, 
there  exists  70  such  that  for  all  7  >  70, 

I  to  -  *-(*yMI  <  I  to  -  *•(*)*/!  I  +  IkW  -  fl- 


showing  that  ^(e^}r]  -^  77  for  each  77  G  £>(TT).  This  completes  the  proof  of  Corollary  2.3. 

The  enveloping  pro-C*  -algebra  E(A) 

We  construct  the  enveloping  pro-C*  -algebra  E(A)  for  a  locally  convex  *-algebra  A  with 
jointly  continuous  multiplication.  This  extends  the  consideration  in  [10,  15,  19]  in  which 
A  is  additionally  assumed  m-convex.  The  added  generality  will  include  several 
constructions  relevant  in  C*-algebra  theory  (like  the  C*-algebra  of  a  groupoid).  Let 
R(A)  denote  the  set  of  all  continuous  bounded  operator  *  -representations  TT  :  A  —  >  B(H^} 
of  A  into  the  C*-algebras  B(H^}  of  all  bounded  linear  operators  on  Hilbert  spaces  Hn.  Let 
Rf(A)  =  {?r  G  R(A)  :  TT  is  topologicaUy  irreducible}.  For  p  G  K(A),  let 

Rp(A)  =  {TT  G  jR(A)  :  for  some  k  >  0,  ||7r(jc)||  <  kp(x)  for  all  jc}, 
and  Rfp(A)  =  Rp(A)  n  R'(A).  Then      . 


2.4.  Lef  A  i?e  a5  a^ove,  p  G  ^(A).  Then  rp(  )  w  a  continuous  C*-seminorm  on  A 
ft'sfrfhg  rp(x)  <  p(**x)l/2.  Ifp  G  KS(A),  then  rp(x}  =  sup{||<j:)||  :  TT  G  /^(A))  <  p(x) 
for  all  ;c  G  A. 


/  Let  5p(x)  =  p^)12.  Let  A  =  A*  G  A  and  TT  G  ^(A).  Then  \\7r(hn)\\  <  Ap(An)  for 
all  n  G  N.  By  standard  Banach  algebra  arguments,  the  spectral  radius  satisfies 


=  liminf||7r(^)||1/w  =inf  ||TT(^)||I/W  <  Mp(hn)l/n  <  p(h). 
Hence,  for  any  x  G  A, 


so  that  rp(x)  <  sp(x).  We  use  the  joint  continuity  of  multiplication  to  conclude  the 
continuity  of  the  C*-seminorm  x  ->  rp(x).  Now  suppose  p  e  ATf(A).  Then 


C*  -enveloping  algebras  73 

Further,  let  Np  =  {x  G  A  :  p(x)  =  0},  a  closed  *-ideal  in  A.  Let  Ap  be  the  Banach 
*  -algebra  obtained  by  completing  A/NP  in  the  norm  \\x  +  Np\\p=p(x).  Then  RP(A) 
(respectively  Rp(A))  can  be  identified  with  R(AP)  (respectively  R'(AP)).  The  assertion 
follows  from  the  fact  that  for  all  z  G  Ap, 


([14],  2.7,  p.  47).  This  completes  the  proof  of  the  lemma. 
Define  the  star  radical  to  be 

srad  (A)  =  {x  G  A  :  rp(x)  =  0  for  all  p  G  K(A)} 
=  {x  G  A  :  TT(JC)  =  0  for  all  TT  G  R(A)}. 

For  each  p  G  AT  (A),  ^(jc  4-  srad  (A))  =  rp(jc)  defines  a  continuous  C*-seminorm  on  the 
quotient  locally  convex  *-algebra  A/srad  (A)  with  the  quotient  topology.  Let  r  be  the 
Hausdorff  topology  on  A/srad  (A)  defined  by  {qp  :  p  G  K(A)}.  The  enveloping  pro-C*- 
algebra  E(A)  of  A  is  the  completion  of  (A/srad  (A),  r).  When  A  is  metrizable,  E(A)  is 
metrizable.  In  view  of  Corollary  2.2,  when  A  is  m-convex,  this  coincides  with  the 
enveloping  l.m.c.  *  -algebra  defined  in  [10,  19,  15]. 

Lemma  2.5.  Let  A  be  a  locally  convex  *  -algebra  with  jointly  continuous  multiplication. 

(a)  Let  A  be  the  completion  of  A.  Then  E(A)  =  E(A). 
(b) 


Proof.  Since  A  has  jointly  continuous  multiplication,  A  is  a  complete  locally  convex 
*-algebra.  The  map  i  :  A/srad  (A)  —  >  A/srad  (A),  where  i(x  4-  srad  (A))  =  x  4-  srad  (A),  is 
a  well  defined  ^isomorphism  into  E(A).  Note  that  for  any  p  G  K(A\  Rp(A)  =  RP(A)  via 
the  restriction  (in  fact,  also  K(A)  =  K(A))9  hence  srad  (A)  =  A  n  srad  (A).  For  any 
p  G  K(A),  let  p  G  K(A)  be  the  unique  extension  of  p.  Then,  for  any  x  G  A, 

qp(x  +  srad(A))  =  rp(x)  =qp(x  +  srad(A)); 

and  for  any  p  G  K(A\  qp(x  +  srad(A))  =  qp\A(x  4-  srad  (A)).  Thus  /  is  a  homeomorphism 
for  the  respective  pro-  C*  -topologies.  On  the  other  hand,  i  has  dense  range  in  A/srad(A). 
Indeed,  let  z  G  A.  Choose  a  net  (*/)  in  A  such  that  *,•  —  >•  z  in  the  topology  t  of  A.  Then 

qp(xt  -  z  4-  srad(A))  =  rp(xt  -  z)  =  sup{||7r(xz-  -  z)  \  :  TT  G 
<  tp(^.  -  z)  -+  0 


for  all  p  G  £(A).  Thus  £"(A),  which  is  the  completion  of  A/srad(A),  coincides  with  the 
completion  E(A)  of  A/srad(A).  This  completes  the  proof  of  (a).  We  omit  the  proof  of  (b). 

A  representation  (TT,  D(TT),//)  of  A  is  countably  dominated  if  there  exists  a  countable 
subset  B  of  A  such  that  for  any  x  G  A,  there  exists  b  G  B  and  a  scalar  k  >  0  such  that 
IK^H  <  *||7r(i)£||  for  all  £  €  £>(TT)  ([22],  p.  419). 


Lemma  2.6.  (a)  Let  A  be  a  locally  convex  *  -algebra.  Letj  :  A  —  »  E(A)9j(x)  =  x  4-  srad(A). 


(1)  T/V  :  A  —  »•  B(H)  is  a  continuous  bounded  operator  *  -representation,  then  there  exists 
a  unique  continuous  *  -representation  a  :  E(A)  —  >  #(//)  swc/z  t/ia^  TT  =  a  o  j.  Further, 
TT  is  irreducible  if  and  only  if  a  is  irreducible. 


74  S  J  Bhatt 

(2)  Let  (7r,2?(7r),//)  be  a  closed  *  -representation  of  A  continuous  in  the  unifor? 
topology.  Let  TT  be  weakly  unbounded.  Then  there  exists  a  closed  weakly  unboune* 

*  -representation  (a,T>(<j),H)  of  E(A)  such  that  TT  =  croj  and  D(cr)  is  dense  in  th 
locally  convex  space  (P(7r),rff). 

(3)  Let  A  be  unital  and  symmetric.  Assume  that  A  is  separable  or  nuclear  (as  a  locall 
convex  space).   Let   (TT,  D(TT),//)   be  a  separably  acting,    countably   dominate 

*  -representation  of  A  continuous  in  the  uniform  topology.  Then  there  exists  a  close 

*  -representation  (<7,P(a),/?)  of  E(A)  such  that  TT  =  aoj. 

(b)  (1)  There  exists  a  unital,  locally  convex,  non-m-convex,  F*-algebra  A  such  that  , 
admits  a  faithful  family  of  unbounded  operator  *  -representations,  but  admits  n 
non-zero  bounded  operator  *  -representation. 

(2)  There  exists  a  unital  non-locally-convex  F*  -algebra  that  admits  no  non-zer 
*  -representation. 

Proof,  (a)  (1)  follows  by  the  definition  of  E(A). 

(2)  Let  TT  =  ®7Tf,  where  each  TTJ  is  a  norm  continuous  bounded  operator  *-representatio 
TT/  :  A  -»  B(Ht)  on  a  Hilbert  space  ft.  We  take  D(TIV)  =  ft  .  Let  Et  :  H  ->  ft  be  th 
orthogonal  projection.  By  (1),  there  exist  continuous  *-homomorphisms  <r/  :  E(A)  - 
B(Hi)j  &i  oj  =  7T/.  Let  a  =  ®crz-  on  the  Hilbert  direct  sum  ©ft  =  H  having  the  domaii 

V(a)  =  {77  =  E£z-7?  €  H  :  £||<7f(z)£;T?||2  <  oo  for  all  z  €  E(A)} 

C  P(TT)  =  {??  =  ££;??  €  #  :  T}\\Xi(x)Eiri\\2  <  oo  for  all  x  G  4}. 


On  £>(cr),  the  <j-graph  topology  ^(£(A))  is  finer  than  the  relativized  7r-graph  topolog 
txlvfoy  Being  closed  and  weakly  unbounded,  both  a  and  TT  are  standard  representation 
Hence,  for  all  h  =  h*  in  A,  the  operators  &(j(h))  having  domain  I>(cr)  and  TT(/Z)  wil 
domain  P(TT)  are  essentially  self-adjoint.  Since  self-adjoint  operators  are  maximal] 
symmetric,  D(o)  is  dense  in  P(TT(/I))  for  the  graph  topology  defined  by  g  —  »  ||£||- 
||7r(/i)^||.  Thus  T>(d)  is  dense  in  the  locally  convex  space  P(TT)  =  n{Z>(7r(A))  :  h  =  / 
in  A}. 

(3)  By  ([22],  Theorem  3.2  and  remark  on  p.  422)  and  ([37],  Theorem  12.3.5,  p.  343 
there  exists  a  compact  Hausdorff  Z  with  a  positive  measure  /x  such  that 

TT=  /    7rAdM(A),     Z>(TT)=  /    D(7TA)dM(A),     H  =   I 
Jz  Jz  Jz 

and  each  n\  is  irreducible.  Since  A  is  symmetric,  each  TT  and  TTA  are  standard  ([37 
Corollary  9.1.4,  p.  237)  (the  commutativity  assumption  in  this  reference  is  not  require 
as  the  arguments  in  ([2],  Theorem  3.5)  shows);  and  by  [3],  each  KX  is  a  bounded  operat 
representaion,  being  irreducible.  Then  we  can  proceed  as  in  (2). 
(b)  (1)  Take  A  =  Lw[0,  1]  =     fl     LP[°>  1]  (*e  Arens  algebra)  with  pointwise  operation 

!</?<oo 

complex  conjugation,  and  the  topology  of  //-convergence  for  each  p,  1  <  p  <  oo.  Tl 
algebra  A  is  a  unital,  symmetric,  locally  convex  ^-algebra,  admitting  a  faithful  standa 
'-representation  (7r,P(7r),#)  such  that  ?r(A)  is  an  extended  C*-algebra  with  a  commc 
dense  domain  [13].  However,  there  exists  no  non-zero  bounded  operator  representation 
A,  as  A  admits  no  non-zero  multiplicative  linear  functional;  and  hence  no  non-ze 
submultiplicative  *-seminorm.  Thus  srad  (A)  =  A  and  E(A)  =  (0).  (2)  Take  A  =  M  [OJ 
the  algebra  of  all  Lebesgue  measurable  functions  on  [0,1]  with  the  topology 


C*  -enveloping  algebras  75 

convergence  in  measure.  It  admits  no  non-zero  positive  linear  functional,  and  hence  no 
non-zero  *  -representation. 

Remark.  2.7.  We  call  a  *  -representation  (TT,  T>(-K\H)  of  a  *-algebra  A  boundedly 
decomposable  if  it  can  be  disintegrated  as  TT  =  fz  7TAd/z(A)  with  each  TT\  a  bounded 
operator  *  -representation.  One  may  show  that  E(A)  is  universal  for  all  closed  boundedly 
decomposable  *  -representations  of  a  locally  convex  F*  -algebra  A.  We  do  not  know 
whether  in  (2)  and  (3)  of  Corollary  (2.4)  (a),  a  is  continuous  in  the  uniform  topology. 

The  bounded  vectors  [4]  for  a  *  -representation  TT  of  a  *  -algebra  A  are  B(-K)  = 
ri{/?(7r(jc))  :  x  G  A},  where,  for  an  operator  T,  the  bounded  vectors  for  T  are 

B(T)  =  {£  £  V(T)  :  there  exists  a  >  0,  c>  0  such  that 

<acn  for  all  ne  N. 


The  following  is  motivated  by  [35].  It  shows  that  unbounded  representations  of  locally  ra- 
convex  *  -algebras  cannot  be  wildly  unbounded, 


Lemma  2.8.  Let  (TT,  P(TT),  H)  &e  a  closed  *  -representation  of  a  complete  locally  m-convex 
*  -algebra  A  continuous  in  the  uniform  topology  on  7r(A).  TTierc  the  following  hold. 


(1)  £>(TT)  =  #(TT);  awd  TT  is  a  direct  sum  of  norm-continuous  cyclic  bounded  operator 
*  -  representations.  _ 

(2)  TT  is  standard.  For  commuting  normal  elements  x,  y  of  A,  the  normal  operators  TT(X) 
and  TT(V)  have  mutually  commuting  spectral  projections. 

(3)  The  uniform  topology  TV  on  7r(A)  is  a  pro-C*  -topology,  i.e.,  it  is  determined  by  a 
family  of  C*-seminorms. 

(4)  If  A  is  Frechet,  then  TV  is  metrizable  and  TT  is  direct  sum  of  a  countable  number  of 
cyclic  bounded-operator  *  -representations. 

Proof.  Let  £  £  ^(^r).  Let  LO^  on  A  be  the  positive  functional  w$(x)  =  (Tr(jt)f  ,  £}  for  x  €  A. 
By  Lemma  2.1,  0;$  is  representable  and  admissible.  Hence  the  closed  GNS  representation 
(7rWf,  P(7rW€),  //u€)  associated  with  u>£  is  a  cyclic,  norm-continuous  bounded  operator 
^representation  with  2>(7rW|)  =  AT^.  Let  £w  denote  the  cyclic  vector  for  TTW€.  Let 
P(Tr^)  =  (7r(A)^)~r7r  and  //^  =  [7r(A)£]~.  Since  TT  is  closed,  T>(^)  C  D(TT).  The  TT- 
invariant  subspace  P(TT^)  defines  a  closed  subrepresentation  (^^V(TT^)^H^  of  TT  as 
TT^OC)  =  TT^)!^^.  Since  (^(jc)^,^}  =  w^(jc)  =  (^(xJ^O  for  all  *  e  A,  it  follows 
that  TT^  and  TT^  are  unitarily  equivalent.  Thus  TT^  is  a  bounded  operator  representation,  and 
£>(7re)  =  H£  C  B(TT).  This  also  implies  that  ^  is  reducing  in  the  sense  of  ([37],  §  8.3). 
Thus  the  following  is  established. 

Assertion  I.  For  any  f  in  X>(TT),  [7r(A)£]~r7r  =  [TT(A)£]~  C  B(TT). 

It  follows  that  7r(A)P(7r)  C  B(TT),  hence  5(?r)  is  dense  in  (^(TT),  rT)  and  norm  dense  in 
H.  Since  B(TT)  forms  a  set  of  common  analytic  vectors  for  TT(A),  the  conclusion  (2) 
follows,  using  ([40],  Theorem  2).  Also,  a  standard  Zorn's  lemma  argument  gives 
TT  =  07T,-,  with  each  TT,-  a  cyclic,  continuous,  bounded  operator  representation. 

Assertion  II.  For  each  bounded  subset  M  of  (P(?r),  r^),  there  exists  p  6  £j(A)  such  that 
IM*MI  <  I  WIPW  f°r  aU  jc  €  A,  ry  6  M. 


76  S  J  Bhatt 


By  continuity,  given  M  as  above,  there  is  k  >  0  and  p  G  ^(A)  such  that 
ftp  (A:)  for  all  x  G  A.  Hence,  for  each  77  €  M  and  x  G  A,  ||7r(jc)r7|f  <  kp(x*x]  <  kp(xf.  ] 
Corollary  2.3,  ||7r(z)77||2  <  /p(*)2,  where  /  =  limsup  w^**)  <  ||r?||2.  Hence  ||7r(jt)7}|| 
||ry||p(jc)  for  all  x  G  A,  all  77  G  M. 

Now  let  £  G  X>(TT).  By  (II)  above,  there  exists  p  £  KS(A)  such  that  for  all  n  £  M, 


showing  that  £  G  B(TT(*)).  Thus  X>(TT)  =  B(TT)  proving  (1). 

The  proof  of  (3)  is  based  on  arguments  in  ([35],  Theorem  1).  Let  T  be  the  collection 
all  subspaces  (linear  manifolds)  K  of  D(TT)  such  that  K  is  7r-invariant,  and  -K\K  is 
bounded  operator  "-representation.  For  K  G  T,  let  SK  be  the  C*-serninorm 

SK(TT(X))  =  sup{||7r(*)r?||  :  rj  G  AT,  |M|  <  1}. 

Let  n  be  the  topology  on  ?r(A)  defined  by  {SK  :  K  G  JT}.  We  show  that  TX>  =  n-  Clea 
n  <  rp.  Let  M  be  a  bounded  subset  of  (£>(TT),  rff).  Choose  Jk  and  p  as  in  assertion  ( 
above.  By  Corollary  2.3,  \u^(x)\  <  \\£\\p(x)  for  all  x  e  A,  all  ^  €  M.  Thus 

M  C  Vp  :=  {77  G  D(TT)  :  \(n(x}ri,r])\  <  \\rj\\2p(x}  for  all  x  inA}. 

Then  Dp  G  ^;  ||7r(jc)r?||  <  ||??||VW2  for  all  77  e  Xy,  and,  as  TT  is  closed,  ([35],  Lemma 
implies  that  Vp  is  ||  |j-closed.  Let  S  =  {^  G  X>p  :  ||f  ||  <  1}.  As  M  is  also  ||  •  ||  bound 
||r?||  <  r  for  all  77  G  M;  and  M  C  rS.  Then,  for  all  x  G  A,  qM(n(x))  <  r2  :^(TT(JC))  .  Tl 
TX>  <  r.  This  gives  (3).  Finally  (4)  is  consequence  of  the  fact  that  the  topology  o 
metrizable  A  is  determined  by  a  countable  cofmal  subfamily  of  KS(A).  This  completes 
proof  of  Lemma  2.8. 

Now  let  A  be  commutative.  Let  M(A)  be  the  Gelfand  space  consisting  of  all  non-z< 
continuous  multiplicative  linear  functionals  on  A.  Let  M*(A)  =  {(p  G  M(A]  :  (p  =  y 
and  (£>*(*)  =  <p(x*).  For  each  x  G  A,  let  x  :  M*(A)  —  >  C  be  the  map  x(tp)  =  <p(x).  1 
following,  which  incorporates  the  spectral  theorem  for  unbounded  normal  operate 
describes  all  unbounded  ^representations  of  A.  The  proof  can  be  constructed  us 
Lemma  2.8  and  ([9],  Theorem  7.3),  in  which  all  bounded  "-representations  of  A  hi 
been  realized. 

COROLLARY  2.9 

Let  A  be  a  commutative  complete  locally  m-convex  ""-algebra.  Let  (7r,£>(7r),#)  h 
closed  *  -representation  of  A  continuous  in  the  uniform  topology.  Then  there  exis 
positive  regular  Borel  measure  p,  on  M*  (A)  and  a  spectral  measure  E  on  the  Borel  set 
M*(A)  with  values  in  B(H)  such  that  the  following  hold. 

(1)  TT  is  a  unitarily  equivalent  to  the  representation  (a,  T>(a]  ,  H  a)  by  multiplical 
operators  in  H^  =  L2(Ai*(A),  /x)  with  domain 

£>(<j)  =  {/  G  Ha  :  (p  ->  ;c(c/?)/(c£>)is  in  Hff  for  all  x  G  A} 

defined  as  (cr(x)  f)  (tp)  = 

(2)  For  each  x  G  A,  TT(*)  = 

We  say  that  a  locally  convex  *-algebra  A  is  an  algebra  with  a  C*  -enveloping  algebr 
the  pro-C*  -algebra  E(A)  is  a  C*-algebra.  In  view  of  Lemma  2.5,  we  do  not  neec 
assume  A  to  be  complete  or  unital.  In  [5],  A  is  further  assumed  to  be  m-convex.  ' 


C* -enveloping  algebras  77 

following  extends  the  main  results  in  ([5],  §  2)  to  the  present  more  general  set  up,  and  can 
be  proved  as  in  [5].  A  is  called  an  sQ-algebra  if  for  some  k  >  0,  p  E  K(A)9  the  spectral 
radius  r  satisfies  r(x*x)1/2  <  kp(x)  for  all  x  E  A;  A  is  *-sb  if  r(x*x)  <  oo  for  each  x, 
equivalently,  r(h)  <  oo  for  all  h  =  A*.  Thus  g  =>  sQ  =>  *-*&. 

Lemma  2.10.  Lef  A  fee  a  complete  locally  convex  * -algebra  with  jointly  continuous 
multiplication. 

(1)  A  is  an  algebra  with  a  C* -enveloping  algebra  if  and  only  if  A  admits  greatest 
continuous  C*-seminorm. 

(2)  If  A  is  sQ,  then  A  admits  a  greatest  C*-seminorm,  which  is  also  continuous. 

(3)  Let  A  be  an  F* -algebra.  If  A  is  *-sb,  then  A  has  a  C* -enveloping  algebra;  but  the 
converse  does  not  hold  (see  ([5],  Example  2.4)). 

The  enveloping  AO* -algebra  O(A) 

For  a  locally  convex  *-algebra  (A,  f)  (t  denoting  the  topology  of  A),  let  Pc(A,i) 
(respectively  PCa(A,  t])  be  the  set  of  all  continuous  (respectively  continuous  admissible) 
representable  positive  functional  on  A.  For  each/  in  PC(A,  t),  let  (TT/,  £>(TT/),  H/)  denote 
the  strongly  cyclic  GNS  representation  defined  by/  as  in  Lemma  2.1.  Let  /  =  n{ker  TT/  : 
/  E  PCfl(A,  0}  and  J  =  H{ker TT/  :/  E  PC(A,  t)}.  Then  /  and  J  are  closed  *-ideal  of  A, 
J  C  /,  and  I  =  srad  (A)  in  view  of  the  cyclic  decomposability  of  any  TT  E  R(A).  The 
universal  representation  of  (A,  t)  is  TTM  =  ®{TT/  :  /  E  PC(A,  t)}.  This  is  a  slight  variation  of 
([37],  p.  228).  Then  o-u(x  +  J)  =  TTU(X)  define  a  one-one  *-homomorphism  of  A/J  into 
the  maximal  0*-algebra  £+(X>(TTM)).  Let  au(t)  be  the  topology  on  A/J  induced  by  the 
uniform  topology  on  ?rM(A);  viz.  cru(t)  is  determined  by  the  seminorms  {qM  :  M  is  a 
bounded  subset  of  (X?(TTM),  ^J}»  where  ^A/ (-^  +  /)  =  sup{|(7rM(;c)£,  77)!  :  ^,  rj  in  M}.  Then 
(A/J,  au(f))  is  an  AO*-algebra  [36]  in  the  sense  that  it  is  algebraically  and  topologically 
*-isomorphic  to  an  0*-algebra  with  uniform  topology  [37].  We  call  (A/7,  au(i)  the 
enveloping  AO* -algebra  of  A,  denoted  by  O(A). 

Lemma  2.11.  Let  A  be  as  above. 

(1)  Every  * -representation  of  A  which  is  continuous  in  the  uniform  topology  and  which  is 
a  direct  sum  of  strongly  cyclic  representations  factors  through  O(A).  When  A  is  either 
complete  and  m-convex,  or  is  countably  dominated,  every  * -representation  of  A 
continuous  in  the  uniform  topology  factors  through  O(A). 

(2)  Let  A  be  barrelled.  Then  au(t)  is  coarser  than  the  quotient  topology  tq  on  A/J. 

(3)  There  exists  a  continuous  * -homomorphism  from  O(A)  into  the  pro-C* -algebra  E(A). 

(4)  The  following  are  equivalent. 

(i)  cru(t)  is  normable. 
(ii)  cru(t)  is  C* -normable. 
(iii)  There  exists  a  linear  norm  on  A/J  defining  a  topology  finer  than  au(t). 

When  any  of  these  conditions  hold,  and  if  A  is  barrelled,  then  A  has  a  C* -enveloping 
algebra;  but  the  converse  does  not  hold. 

Proof.  (1)  follows  from  the  construction  of  O(A)  and  Lemma  2.8.  (2)  Let  A  be  barrelled. 
Since  /  is  closed.  (A//,  tq)  is  barrelled  ([32],  ch.  II,  §7,  Corollary  1,  p.  61).  Further,  au  is 


78  5  J  Bhatt 

weakly  continuous.  Hence,  au  is  continuous  in  the  uniform  topology  ([20j,  Th 

(3)  Since  J  C  sradA,  the  map 

0  :  A/J  ->  A/sradA  ->  E(A),  r/>(.i*  4-  7)  -  .v  f  srad.4 

is  a  well  defined  *-homomorphism.  Now,  as  E(A)  is  a  pro-  ('*  -algebra,  /:*i.4  \     k 
C*~algebra  for  any  p  €  K(A),  denoted  by  £7,(A),  with  the  norm  |  j-  *  kcr  </,, 
£(A)  =  proj  lim£;,(A),  inverse  limit  of  C*  -algebras  [26|.  Let   /,,  :  /A  At      *  i 
(pp(z)  sz  +  ker^.  For  the  continuity  of  0  :  (O(A),  <TM(/))     *  (A'fAL  ri.  it  is  suf 
show  the  continuity  of  the  *-homomorphism  0;,  -  ^,  o  <;>  :  <V(A  )     >  Kr(A  't.  N^*vt» 

-0:-4  ~-»A/srad(A)  ->  E(A)  ->£;,(A),i;iv)  -  (,v  I  sradf/iil  *  *"•*»  - 

is  a  continuous  bounded  operator  ""-representation;  and  //      c>/(  <  *  /u,  ;w(  .1  )       i     ! 
0/;  is  continuous  for  each  /;  €  AT(A). 

(4)  (i)  if  and  only  if  (ii)  if  and  only  if  (iii)  follows  from  (  [  2()|,  Theorems  *  2.  3   * 
barrelled.  Let    j  be  a  norm  on  A/J  determining  <rM(r).  Since  ttj    -  <7M(;  h  />  .     * 
defines  a  continuous  C*-seminorm  on  A.  Let  /;  be  any  continuous  C'Vseitunoif 
Ap  be  the  completion  of  A/ker/?  in  the  C*-norm  |.v  1  ker  />|      i>(.\).  Then    . 
7Tp(jc)  =  jc-H  kerp  defines  a  continuous  bounded  operator  *-represcntalion.  Hv 
exists  a  continuous  *-homomorphism  crp  such  that  o/}  •.•>/K      n/t.   Since    tK*- 
topology  on  A/;  is  the  |  [^-topology,  and  since  <rM(0  is  determined  by  I  •.  it  t'olU  •  ^ 
some  fc  >  0,  0^(2)1  <  /:|z|  for  all  z  €  A/J.  Thus  ;>(.v)  '    kp  ^  (A);  ami  sv>  />'  x  '  " 

all  j  E  A,  both  being  C*-seminorms.  Thus  p,x,  is  the  greatest  continuous  (  '*  %***  i   : 
A.  By  Lemma  2.8,  E(A)  is  a  C™"*-algebra.  That  the  converse  does  not  hukl  is  til  , 
Arens*  algebra  A  =  L^[0,  1],  wherein  £(A)  =  (/>},  O(A)      A  topt)logica!l>  a-     v 

3.  Proofs  of  theorems  1.1,  1.2  and  1.4 

Proof  of  Theorem  1.2.  First  we  prove  the  following. 

Assertion  L  Given  a  bounded  subset  Af  of  (P(7r),/ff),  there  exists  />  f-  A\b4 
such  that  ^M(TT(JC))  <  ^rp(j)  for  all  x  £  A. 

By  the  continuity  of  TT,  given  M,  there  exists  A-  >  0  and  p  i,  K%{A 
<!M(K(X))  <  kp(x)  for  ail  jc  e  A.  Let  f  €  M.  Then 

^W|  =  |(7r(j)eOI  <  ^(TrW)  <  kp(jc) 

for  all  ^.  Since  o;^  is  representable,  it  is  extendable  to  A1.  The  arguments  in  fit 
Corollary  2.3(1)  applied  to  the  extension  of  cj^  to  A1  give 


for  all  x  in  A.  Thus  ||7rwe(*)f||  <  \\$\\p(x):  and  by  the  definition  of  rr.  n 
l^llrpW  for  all  x  in  A.  Since  M  is  ||  ||-bounded,  there  exists  /  >  C)  such  thai  1 
M,  all  x  in  A, 


It  follows  that  for  all  x  in  A,  and  all  f  ,  77  in  Af  , 

IW^^I^IWk^)1/2^/2^ 

Thus  <?M(TT(*))  <  /2rp(jc)  for  all  jc  in  A. 


C*  -enveloping  algebras  79 

Now,  by  Lemma  2.8,  TT  =  ©TT/,  with  each  TT,  :  A  —  »  #(///)  norm  continuous.  By  Lemma 
2.6,  there  exists  a  closed  representation  (V,  D(V)3#)  </  =  ®<j/  of  £(A),  with  each 
<ji  :  E(A)  —  »  B(Hi)  norm  continuous,  cr/  ojf  =  ?r/  for  all  /.  We  shall  eventually  show 


On  the  other  hand,  consider  the  *  -representation  (cr,  U(a),H)  of  A/srad(A)  having 
domain  T>(d)  =  P(TT),  and  given  by  cr(j(x))  =  TT(JC)  for  all  x  €  A.  By  ([37],  Proposition 
2.2.3,  p.  39),  on  P(TT),  ^  =  ^+(p(w))  which  is  the  graph  topology  on  D(TT)  due  to  the 
maximal  O*-algebra  £+(£>(TT)).  Hence,  on  TT(A),  the  uniform  topology  r^  =  TD 
ITT(A)  =  ri  (sa)0>  which,  by  lemma  2.8,  is  a  pro-  C*-  topology.  By  ([37],  Proposition  3.3.20, 
p.  85),  <j(A/srad  (A))  is  contained  in  a  r^+(v(7r}}  -complete  *-subalgebra  of  £+(P(TT));  and 
a  can  be  extended  as  a  continuous  *-homomorphism  a(E(A),r)  —*  [£+(D(7r)),  rP  ^  ] 
giving  a  closed  *  -representation  cr  of  £(A)  on  //  with  domain  D(cr)  =  D(TT).  Next  we 
prove  the  following. 

Assertion  II.  As  representations  of  E(A),  cr  =  d  . 
This,  we  do,  in  the  following  steps. 

(a)  a  is  an  extension  of  cr7. 

Clearly,  £>(</)  C  P(TT)  =  V(a).  We  show  o-(z)'|p(c/)  =  </(z)  for  all  z  €  B(A).  Fix  z  € 
E(A).  Let  7]  e  C(?r).  Choose  a  net  (jcr)  in  A  such  that  for  all  p  G  ^(A), 
qp(j(xr)  —  z)  —  >  0.  Choose  an  appropriate  p  by  (I)  above.  Then 


<*/>  (*,--*,/) 

=  *4,(jWr)-;(^))^0. 

Hence  7r(A:r)77  is  norm  Cauchy  in  T>(TT);  and  similarly,  7r(jc)7r(xr)r7  is  norm  Cauchy  in  T>(K) 
for  all  jc  G  A.  Thus  7r(xr)rj  is  Cauchy  in  (P(TT),  r^-),  which  is  complete  as  TT  is  closed.  Thus 
there  exists  £  €  D(TT)  such  that  lim(;cr)77  =  £  in  ^.  This  defines  cr(z)  as  cr(z)ry  =  £,  which 
gives  cr 


(b)  cr  is  a  closed  representation  of  E(A). 
Indeed,  as  TT  is  closed. 


hence  V(cr)  =  D(oy)  .  This  also  follows  from  the  fact  that  TT  is  closed:  on  T>(a)  =  T>(K), 
tv  —  ?£+(X>(TT))  =  ^<r((£(A)l))»  as  we^  as  7r(^)  c  ^(^(A))  C  £+(X>(7r)).  This  further  implies 
rp  L(E(A))  =  TS  ;  which,  in  turn  gives  the  following. 

(c)  a1  is  continuous  in  the  uniform  topology  as  a  *  -representation  of  (£"(A),r). 

Now,  by  (c),  Lemma  2.8  implies  that  the  closed  representation  a1  is  standard;  hence 
self-adjoint,  and  so  maximal  hermitian  ([31],  (I),  Lemma  4.2).  Then  (a)  gives  d  =  cr, 
thereby  verifying  (II).  This  completes  the  proof  of  Theorem  1.2. 


80  SJ  Bhatt 

If  TT  is  irreducible,  then  <j  is  irreducible,  hence  is  a  bounded  operator  representation  by 
[3],  ([6],  Theorem  4.7).  This  gives  Corollary  1.3. 

Proof  of  Theorem  LI.  Let  A  be  Frechet.  Then  A  =  proj  limAn,  an  inverse  limit  of  a 
sequence  of  Banach  *  -algebras  An.  Assume  that  each  *  -representation  (and  hence  the 
universal  representation  TTM)  of  A  is  a  bounded  operator  representation.  Since  A  is  Frechet, 
TTW  is  continuous.  Let  a  be  the  representation  of  E(A)  defined  by  Theorem  1.2 
corresponding  to  TTM.  Then  <r  is  also  a  bounded  operator  *  -representation.  Further,  as  A  is 
Frechet,  E(A)  =  proj  limC*(Aw)  is  also  Frechet.  Thus  a  is  continuous  and  there  exists  a 
continuous  C*-seminorm  qQ  on  E(A)  such  that  \\cr(z)\\  <  q0(z)  for  all  z  €  E(A).  Now  the 
bounded  part  of  E(A) 

b(E(A))  =  {z  G  E(A)  :  q(z)  <  oo  for  all  continuous  C*-seminorm  q} 
is  a  C*  -algebra  with  the  norm 

:  q  is  a  continuous  C*-seminorm  on  E(A)} 


Since  a  is  one-one,  the  restriction  c/  =  <r  b@(A))  is  a  *  -isomorphism  of  the  C*-algebra 
b(E(A))  into  B(Hff).  Hence,  for  all  z  e 


It  foUows  that  b(E(A))  =  E(A).  As  E(A)  is  Frechet,  the  continuous  inclusion  map 
)  is  a  homeomorphism.  The  converse  follows  from  Theorem  1.2. 


Proof  of  Theorem  L4.  By  Corollary  2.3,  I  =  J  =  srad  (A)  in  the  notations  of  Lemma  2.9. 
Let  K=A/J,  a  Frechet  *-algebra  in  the  quotient  topology  from  A.  By  Lemma  2.8,  the 
uniform  topology  TX>  on  ?rw(A)  is  a  cr-C*  -topology;  and  the  topology  au(t)  on  #  is 
determined  by  the  (continuous)  C*-seminorms  {SG(-)  '  G  €  J7},  where  ^  is  the  collection 
of  all  subspaces  T>  of  X>(TTM)  such  that  V  is  7ru-invariant  and  iru\v  is  a  bounded  operator 
^-representation;  and  ^G(z)  =  ||7rM|G(jc)||  for  all  i  =  jc  +  J,  jc  €  A.  Thus  crw(r)  <  r  where  r 
is  the  relative  topology  from  E(A)  defined  by  all  C*-seminorms  on  E(A).  To  show  that 
r  <  su(t),  let  zn  =  Jn  +  /  €  £,  zn  -*  0  in  crM(f).  Let  ^  be  any  C*-seminorm  on  A.  There 
exists  7TE#(A)  such  that  q(x)  =  ||7r(jc)||,  and  TT  =  e{7r/|/  €  Fj  for  a  suitable 
F7rcPc(A,/).  Now  fl;  =  ®f^fHf  C  D(7rB),  HT€^,  and  |Wjcn)||  =  SH,^)  -*  0. 
Hence  zn  -*  0  in  r.  Thus  r  =  <7M(f),  and  F(A)  =  (0(A),att(0),  the  completion.  The 
remaining  assertion  follows  from  Lemma  2.11. 

4.  Remarks 

PROPOSITION  4.1 

Let  A  be  a  *-sb  Frechet  *  -algebra.  If  A  is  hermitian,  then  A  is  a  Q-algebra. 

Proof  We  can  assume  that  A  is  unital.  Let  P  be  a  sequence  of  submultiplicative 
*-seminorms  defining  the  topology  of  A.  Let  A  =  proj  UmA^  be  the  Arens-Michael 
decomposition  expressing  A  as-  an  inverse  limit  of  a  sequence  of  Banach  *  -algebras; 
where,  for  q  €  ?,  Aq  is  the  Banach  *-algebra  obtained  by  completing  A/  ker  q  in  the  norm 

||*  +  ker  #||  =  q(x).  Let  7rq  :  A  -»  Aq  be  7rq(x)=  x 


Case  1.  Assume  that  A  is  commutative.  By  herrniticity,  spA(h)  =  {<p(/z)  :  0  G  -M(A)}  C 
R  for  all  A  =  A*  €  A.  Note  that  since  A  is  hermitian.  M(A)  =  M  (A).  Using  ([23], 
Proposition  7.5),  it  follows  that  for  each  q,  M(Aq)  =  M*(Aq)\  hence  by  ([7],  Theorem 
35.3,  p.  188),  each  Aq  is  hermitian.  Now  by  ([17],  Lemma  41.2,  p.  225),  for  each  z  G  Aq, 
the  spectral  radius  satisfies 


I  I  denoting  the  Gelfand-Naimark  pseudonorm  on  Aq.  Then  m^(jc)  =  \nq(x)\q  defines  a 
continuous  C*-seminorm  on  A.  By  Lemma  2.10,  there  exists  a  greatest  continuous  C*- 
seminorm  p^  )  on  A.  By  ([23],  Corollary  5.3),  for  each  x  G  A, 

rA(x)  =  sup{rAf)(7rq(x)}}  <  sup{mq(x)}  <poo(x). 

By  the  continuity  of  p^,  there  exists  a  p  G  ATy(A)  and  fe  >  0  such  that  for  all  x  in  A, 
K*)  <Poo(x)  <  kp(x).  It  follows  from  ([23],  Proposition  13.5)  that  A  is  a  Q-algebra. 

Case  2.  Let  A  be  non-commutative.  Let  M  be  a  maximal  commutative  *-subalgebra  of  A 
containing  the  identity  of  A.  Since  M  is  spectrally  invariant  in  A,  M  is  also  hermitian.  By 
*  -spectral  boundedness  and  herrniticity,  each  positive  functional  on  M  can  be  extended  to 
a  positive  functional  on  A  ([17],  Theorem  9.3,  p.  49).  It  follows  from  ([15],  Corollary  2.8) 
and  the  continuity  of  positive  functionals  on  unital  Frechet  *-algebras,  that  for  all  z  G  M, 
Poo(z)  =/^W  <  rM(z?z)i/2,  p%>  being  the  greatest  C*-seminorm  on  M  and  rM(-) 
denoting  the  spectral  radius  in  M.  Thus  M  is  a  commutative  hermitian  algebra  with  a  C*- 
enveloping  algebra.  By  case  1,  M  is  a  Q-algebra.  Further,  M  being  hermitian,  the  Ptak's 
function  jc  —  >  rM(x*x)  /2  is  a  C*-seminorm  on  M  ([17],  Corollary  8.3,  p.  38;  Theorem 
8.17,  p.  45). 

Now  let  ;c  G  A,  and  take  M  to  be  the  maximal  commutative  *-subalgebra  containing 
x*x.  Let  rK(-)  denote  the  spectral  radius  in  an  algebra  K.  Then  by  Ptak's  inequality  in 
hennitian  Frechet  *-algebras  ([17],  Theorem  8.17,  p.  45) 

rA(x)  <  rA(x*x)[/2  =  rM(x*x)l/2  =  p£(x*x)l/2  =  Poc(x*x)l/2  <  q(x), 

q  being  a  *-algebra  seminorm  on  A  depending  on  p^  only.  It  follows  from  ([23], 
Proposition  13.5)  that  A  is  a 


(4.2)  (i)  It  is  claimed  in  ([5],  Corollary  2.4)  that  a  complete  hermitian  m-convex  *-algebra 
with  a  C*-enveloping  algebra  is  a  Q-algebra.  Regrettably,  there  is  a  gap  in  the  proof.  The 
author  sincerely  thanks  Prof.  M  Fragoulopoulou  for  pointing  out  this.  It  is  implicitely 
used  in  the   'proof  therein  that  the  completion  of  a  hermitian  normed  algebra  is 
hennitian.  By  Gelfand  theory,  this  is  certainly  true  in  the  commutative  case,  but  is  not 
true  in  non-commutative  case  (see  ([17],  p.  18)).  Thus  ([15],  Corollary  2.4)  remains  valid 
in  commutative  case;  and  the  above  proposition  partially  repairs  the  gap  in  the  non- 
commutative  case.  Consequently  ([15],  Lemma  2.15,  Theorem  2.14)  remains  valid  for 
Frechet  algebras.   Is  a  hermitian  Frechet  algebra  with  a  C*  -enveloping   algebra  a 
Q-algebra?  (ii)  The  algebra  C(R)  of  continuous  functions  on  IR  exhibits  that  the  condition 
*-sb  can  not  be  omitted  from  the  above  proposition.  It  also  follows  from  above  that  a  *-sb 
a-C*-algebra  is  a  C*-algebra. 

(4.3)  In  Theorem  1.2,  the  assumption  that  TT  is  closed  can  not  be  omitted.  Let  A  =  C^~  (  R). 
the  Frechet  *-algebra  of  C°°  functions  on  R,  with  pointwise  operations  and  the  topology 


82  SJ  Bhatt 

of  uniform  convergence  on  compact  subsets  of  R  of  functions  as  well  as  their  derivatives. 
Then  E(A)  =  C(R),  the  algebra  of  continuous  functions  on  R  with  the  compact  open 
topology.  On  the  Hilbert  space  /?  =  L2(R),  the  "-representation  TT  of  A  with 
P(TT)  =  C£°(R),  7r(a)f  =  Qf,  cannot  be  extended  to  a  * -representation  of  C(R)  with 
the  same  domain  ([10],  Example  4.7). 

(4.4)  Theorem  1.1  means  that  a  Fechet  *-algebra  has  a  C*-enveloping  algebra  if  and  only 
if  it  is  a  BG*  -algebra  [24].  In  the  non-metrizable  case,  it  follows  from  Theorem  1.2  that  if 
A  is  a  complete  topological  w-convex  "-algebra  with  a  C*-enveloping  algebra,  then  every 
* -representation  of  A  which  is  continuous  in  the  uniform  topology  is  a  bounded  operator 
representation.  However,  the  converse  does  not  hold.  This  is  exhibited  by  the  BG* -algebra 
C[0, 1]  of  continuous  functions   on   [0,1]   with  the  pro-C*-topology  r  of  uniform 
convergence  on  all  countable  compact  subsets  of  [0,1].  Thus  Theorem  1.1  is  false  without 
the  assumption  that  A  is  Frechet.  It  would  be  of  interest  to  find  an  example  of  a 
topological  algebra  with  a  C*-enveloping  algebra  which  is  not  a  J?G*-algebra.  i 

(4.5)  Yood  [42]  has  shown  that  a  *-algebra  A  admits  a  greatest  C*-seminorm  if  and  only  if  ! 
sup  \f(x)\  <  oo  for  each  *,  where  the  sup  is  taken  over  all  admissible  states  5;  and  by 
Lemma  2.10,  this  happens  for  a  Frechet  A  if  and  only  if  A  has  a  C*-enveloping  algebra. 

Yood's  result  is  an  algebraic  version  of  ([5],  Corollary  2.9)  that  states  that  a  complete 
m-convex  algebra  has  a  C* -enveloping  algebra  if  and  only  if  S  is  equicontinuous. 

(4.6)  (i)  Let  TT  be  a  * -representation  of  a  complete  locally  m-convex  * -algebra  A  with  a 
bounded  approximate  identity.  Let  A  have  a  C* -enveloping  algebra.  Is  TT  continuous  in  the 
uniform  topology?  In  particular,  let  TT  be  a  bounded  operator  "-representation.  Is  TT  norm- 
continuous? 

(ii)  Let  A  be  a  pro-C*-algebra  (more  generally,  a  complete  m-convex  *-algebra  with  a 
bounded  approximate  identity).  Let  /  be  a  representable,  not  necessarily  continuous, 
positive  functional  on  A.  Is  the  GNS  representation  TT/  a  bounded  operator  representation? 
Is  every  ""-representation  of  A  weakly  unbounded? 

These  are  motivated  by  the  point  of  view  ([5],  Remark  2.1 1,  p.  207)  that  a  topological  \ 

* -algebras  with  a  C*-enveloping  algebra  provide  a  hermitian  analogue  of  a  commutative 
Q-algebra.  It  is  easy  to  see  that  a  * -representation  TT  of  a  locally  convex  j2-al?<sbra  is  a  4" 

bounded  operator  representation  and  is  norm  continuous.  i 

j . 

5.  Crossed  product  constructions  * 

i 

We  recall  the  crossed  product  of  a  C*-dynamical  system  (G,  A,  a).  Let  a  be  a  strongly  * 

continuous  action  of  a  locally  compact  group  G  by  * -automorphisms  of  a  C*-algebra  A. 
Let  CC(G,A)  be  the  vector  space  of  all  continuous  A-valued  functions  with  compact 
supports.  It  is  a  *-algebra  with  twisted  convolution  ' 


=  /  x(h)ah(y(h-lg))dh 
JG 

and  the  involution  x^(g)  =  A(g)~lag(x(g-l))\  The  Banach  "-algebra  Ll(G,A)  is  the 
completion  of  CC(G,A)  in  the  norm  \\x\\  1  =  /G  ||jc(A)||dA;  and  the  crossed  product  C"- 
algebra  C*(G,A,a)  is  the  completion  of  Ll(G,A)  in  its  Gelfand-Naimark  pseudonorm 
\\x\\  =  wip{\\ir(x)\\:ireR(Ll(G,A))},  which  is,  in  fact,  a  norm.  Thus  it  is  the 
enveloping  C*-algebra  of  the  Banach  "-algebra  Ll(G,A).  The  C*-algebra  C*(G,A,a) 


can  also  be  realized  as  the  enveloping  C*  -algebra  of  non-normed  topological  *  -algebras 
smaller  than  LJ(G,  A). 

Let  3C  be  the  collection  of  all  compact,  symmetric  neighbourhoods  of  the  identity  in  G. 
For  K  e  Jf,  let  CK(G,A)  =  {/  €  CC(G,  A)  :  supp/  C  K},  a  Banach  space  with  the  norm 
H/ll  =  sup{||/(jc)||  :  x  G  K}.  The  inductive  limit  topology  r  on  Q(G,A)  is  the  finest 
locally  convex  topology  on  CC(G,A)  making  each  of  the  embeddings  Q;(G,A)  -> 
CC(G,  A),  for  all  AT  G  JT,  continuous.  Then  CC(G,  A)  is  a  locally  convex,  non-  w-convex, 
topological  *  -algebra  with  jointly  continuous  multiplication  and  continuous  involution. 
From  ([18],  p.  203),  £(CC(G,  A))  =  C*(G,  A,  a).  This  immediately  leads  to  the  following. 

PROPOSITION  5.1 

Let  (G,A,a)  be  a  C*  -dynamical  system.  Let  B  be  any  topological  ""-algebra  containing 
CC(G,A)  as  a  dense  *-subalgebra  and  satisfying  CC(G,A)  C  B  C  C*(G,A,a).   Then 


For  1  <p  <  oo,letA^(G,A)  =  Ll(G,A)  nif(G,A),  aBanach  *  -algebra  with  the  norm 
\X\P  =  I  Wli  +  \\x\\p-  The  above  applies  to  B  =  (~}{AP(G,A)  :  1  <  p  <  ex)},  a  locally  m- 
convex  Q-Frechet  "-algebra  with  the  topology  of  |  [^-convergence  for  each  p. 

Smooth  elements  of  a  Lie  group  action 

Let  A  be  a  unital  C*  -algebra  and  G  be  a  Lie  group  acting  on  A.  Let  A  denote  the 
infinitesimal  generators  of  actions  of  1-parameter  subgroups  of  G  on  A,  viz., 


is  a  continuous  homomorphism  of  R  into  G}. 

Then  A  consists  of  derivations  and  it  is  a  finite  dimensional  vector  space  ([11],  p.  40) 
hawing  basis,  say  <5i  ,  #2,  •  •  •  ,  &d-  Then  Cn  -elements  (1  <  n  <  oo)  and  C°°  -elements  of  A  for 
the  action  a  are  defined  as  follows. 


C"(A)  ={jc€A  :^€Dom(^,^2...<5/n)  for  all  n-tuples  {<5/n  .  .  .  ,  <5,n  }  in  A} 


By  ([11],  Proposition  2.2.1),  each  C"(A)  and  C°°(A)  are  dense  *-subalgebras  of  A;  and 
Cn(A)  is  a  Banach  *-algebra  with  the  norm 

iwin  =  iwi+E  E  iift,«fe...«4(*)ii/«- 

'  k=l  i\,i2,...jk=l 

Then  C°°(A)  =  proj  HmCn(A)  is  a  Frechet  *-algebra  with  the  topology  defined  by  the 
norms  {||  ||n  :n=  1,2,,...}. 

Lemma  5.2.  C°°(A)  has  a  C*  -enveloping  algebra  and  E(C°°(A))  =  A. 

Proof.  It  is  well  known  that  Cn(A)  and  C°°(A)  are  spectrally  invariant  in  A.  Hence 
(Cn(A),  ||  ||)  and  (C°°(A),  ||  ||)  are  Q-algebras  in  the  norm  ||  ||  from  the  C*-algebra  A. 
Since  ||  ||  <  ||  ||;,  (C00(A),r)  is  also  a  Q-algebra.  By  Lemma  2.10,  (C°°(A),r)  is  an 
algebra  with  a  C*  -enveloping  algebra.  Let  TT  :  B  —  >  B(H),  where  B  =  Cn(A)  or  C°°(A),  be 


84  S  J  Bhatt 

a  bounded  operator  "-representation  on  a  Hilbert  space  H.  Then  for  all  x  e  B9 


2 


<  rB(x*x)  <  |W|2. 

Hence  TT  is  ||   |  [-continuous;  and  by  the  density  of  C°°(A)  in  A,  TT  extends  uniquely  to  a 
*  -representation  of  A  on  #.  It  follows  that  JE(C°°(A))  =  C*(Cn(A))  =  A  for  all  n. 

An  element  x  E  A  is  analytic  if  *  €  C°°(A)  and  there  exists  a  scalar  /  >  0  such  that 

E( 

*=0 

whereas  x  is  e/ztfre  if  x  G  C°°(A)  and  for  all  t  >  0,  it  holds  that 


Let  CW(A)  (respectively  C^(A))  denote  the  set  of  all  analytic  (respectively  entire) 
elements  of  A.  Then  each  of  CW(A)  and  Ceu(A]  is  a  *-subalgebra  of  A  and 
Ce"(A)  C  CW(A)  C  C°°(A).  For  each  /  >  0  and  x  G  Cn(A\  define 


Then  ||  \\n  and/£(  )  are  equivalent  norms.  Hence  P*  =  (pfn(  ))andp  =  (||  HJ  define 
the  same  C°°-topology  r  on  C°°(A).  Let  A,  =  {jc  G  C°°(A)  :  pf(x)  =  sup*p£(jt)  <  oo},  a 
*-subalgebra  of  C°°(A),  which  is  a  Banach  "-algebra  with  norm  p?(  ),  and  which 
consists  of  elements  of  C°°(A)  whose  numerical  ranges  defined  with  respect  to  P*  are 
bounded.  For  /  <  s9  the  inclusion  A^  —>  At  is  norm  decreasing.  Thus 

C"(A)  =  f|{Af  :  t  >  0}  =  f)  An  =  proj  lim  An, 


n=l 


a  Frechet  m-convex,  *-algebra  with  the  topology  reuj  defined  by  the  family  of  norms 

{/?<(     )  :*GN}  (setting p°(    )  =  ||   ||).  Further, 


C"(A)  =  U  Ar  =       A1/n  =  ind  lim  A1/n 


with  the  linear  inductive  limit  topology  r^  By  ([21],  Corollary  10.2,  Lemma  10.2,  p.  317) 
and  ([32],  Proposition  6.6,  p.  59),  (CtJ(A),  rw)  is  a  complete  m-com  ex  *-algebra  which  is 
a  g-algebra.  Thus  Ca;(A)  is  an  algebra  with  a  C*  -enveloping  algebra.  Further  if  each  A,  is 
dense  and  spectrally  invariant  in  C°°(A),  then  Ce(JJ(A)  is  an  algebra  with  a  C*-enveloping 
algebra  and  ^(C^A))  =  ^(C^A))  =  A. 


The  smooth  crossed  product 

We  recall  the  smooth  Frechet  algebra  crossed  product  [29].  Let  B  be  a  Frechet  *-algebra. 
Let  (pn)  be  a  sequence  of  submultiplicative  *-seminorms  defining  the  topology  of  B.  Let 
(3  be  a  strongly  continuous  action  of  R  by  continuous  "-automorphisms  of  B.  Then  0  is 
called  m-tempered  (respectively  isometric)  if  for  each  m  £  N,  there  exists  a  polynomial 


P(X)  such  that  pm(pr(x))  <  P(r)pm(x)  for  all  jc  €  B,  r  <£  R  (respectively  for  each  m  €  N, 
Pm(Pr(x))  =  />«(*)  for  all  *  €  #,  all  r  E  R).  Let  5(R)  be  the  Schwartz  space  consisting  of 
the  rapidly  decreasing  C°°-functions  on  R.  It  is  a  Frechet  space  with  the  Schwartz 
topology.  The  completed  projective  tensor  product  5(R)  ®B  =  5(R,5)  consists  of 
B-valued  Schwartz  functions  on  R.  If  j3  is  m-tempered,  then  5(R,5)  becomes  an  m- 
convex  Frechet  algebra  with  twisted  convolution 

(/**)(')  = 

This  Frechet  algebra  is  called  the  smooth  Schwartz  crossed  product  of  B  by  the  action  /? 
of  R,  and  is  denoted  by  S(R,  #,/?).  In  general,  S(R,5,/3)  need  not  be  a  "-algebra  ([34], 
§  4).  If  /?  is  isometric,  then  the  completed  projective  tensor  product 


=  {/  :  R  —  >  #  measurable  function  :  /  pm(/(r))dr  <  oo  for  all  /w  G  N} 

JR 

is  a  Frechet  *  -algebra  with  twisted  convolution  and  the  involution  /*  (r)  =  /?r  (/(-/*)*), 
denoted  by  L^R,!?,/?).  One  has  S(R,B,/3)  C  L!(R,  #,/?). 
The  following  is  closely  related  with  ([29],  Lemma  1.1.9). 

Lemma  5.3.  Let  A  be  a  dense  Frechet  *~subalgebra  of  a  Frechet  *  -algebra  B.  Assume  that 
A  and  B  can  be  expressed  as  inverse  limits  ofBanach  *  -algebras  An  and  Bn  respectively, 
where  An  is  dense  in  Bnfor  all  n;  the  inclusions  A  —  >  An,  B  —  >  Bn  have  dense  ranges  for 
all  n;  and  each  An  is  spectrally  invariant  in  Bn.  Then  A  is  spectrally  invariant  in  B  and 


Proof.  By  ([15],  Theorem  4.3),  E(A)  =  proj  lim^A,,)  and  E(Bn)  =  proj  ]imE(Bn). 
Since  An  —  >  Bn  is  spectrally  invariant  with  dense  range,  An  is  a  <2-normed  algebra  in  the 
nonxi  of  Bn.  Hence  every  C*-seminorm  on  An  is  continuous  in  the  norm  of  En\  and 
extends  uniquely  to  Bn.  Thus  An  and  Bn  have  the  same  collection  of  C*-seminorms.  It 
follows  that  E(An)  =  E(Bn)  for  all  /i;  and  so  E(A)  =  E(B). 

PROPOSmON  5.4 

Let  a  be  an  m-  tempered  strongly  continuous  action  0/R  by  continuous  *  -automorphisms 
of  a  Frechet  *  -algebra  B  contained  as  a  dense  *-subalgebra  of  a  C*  -algebra  A  such  that 
E(B)=A.  ThenE(C°°(B))=A. 

Proof.  Let  \\  \\  denote  the  C*-norm  on  A.  Let  (pn)  be  an  increasing  sequence  of 
submultiplicative  *-seminorms  defining  the  topology  of  B.  In  view  of  the  continuity  of  the 
inclusion  B  —  *  A,  the  increasing  sequence  qn(  )  =  pn(  )  +  \\  \\  of  norms  also  deter- 
mines the  topology  of  B.  Let  Bn  =  (B,  qn)  be  the  completion,  which  is  a  Banach  *-algebra. 
Then  B  =  proj.  lim£n  =  {}Bn.  Now,  for  any  n  €  N,  r  €  R,  and  x  €  B, 


=  \\ar(x)\\ 

=  I  Wl  +  Poly  (r)pn(x)  =  pol/  (r)qn(x) 


for  some  polynomial  poly'(  ).  It  follows  that  a  is  m-tempered  for  (qn(    ))  also;  and  it 
induces  an  action  a^  of  R  by  continuous  "-automorphisms  of  Bn.  Let  Bn,m  be  the  Banach 


86  S  J  Bhatt 

*-a!gebra  consisting  of  all  Cm-vectors  in  Bn  for  a(n).  By  ([33],  Theorem  2.2),  #n,m  -+  Bn 
are  spectrally  invariant  embeddings  with  dense  ranges.  Also,  C°°(B)  =  proj  limn?m 
Bn,m  =  pr°j  limnBrtin.  Now  Lemma  5.2  implies  that  C°°(B)  is  spectrally  invariant  in  B 

and£(C°°(J9))=A'. 

PROPOSITION  5.5 

Let  a  be  a  strongly  continuous  action  of  R  by  *  -automorphisms  of  a  C*  -algebra  A.  The 
following  hold. 

(a)  The  Frechet  algebras  S(R,A,  a)  and  S(R,  C°°(A),  a)  are  Q-algebras. 

(b)  77*e  embeddings  S(R,  C°°(A),  a)  -+  S(R,  A,  a)  -»  C*(R,  A,  a)  are  continuous,  spec- 
trally invariant  and  have  dense  ranges. 

(c)  The   Frechet  algebra   5(R,C°°(A),a)    is   *~algebra   and  £(S(R,C°°(A),a)  = 
C*(R,A,a). 


By  ([34],  Theorem  A.2),  a:  leaves  C°°(A)  invariant.  In  ([34],  Corollary  4.9),  taking 
the  scale  a  to  be  the  weight  w(r)  =  1  +  r\  on  G  =  R  =  H  ,  it  follows  that  5(R,  C°° 
(A),  a),  is  a  Frechet  '-algebra.  Now  &s(f)(r)  =  oij(/(r))  defines  an  action  a  of  R  on  the 
Frechet  algebra  5(R,A,a)  for  which,  by  ([29],  p.  189),  C°°(S(R,A,a))  =5(R, 
C°°(A),a)  homeomorphically.  Note  that  the  embeddings 

S(R,C°°(A),aO  -»  S(R,A,a)  -»Ll(R,A,a)  ->  C*(R,A,a) 

are  continuous;  S(R,C°°(A),a)  is  dense  in  5(R,A,a)  by  ([34],  Theorem  A.2);  and 
S(R,A,a)  is  dense  in  L^RjAja);  which,  in  turn,  is  dense  in  C*(R,A,a). 

Now  let  (|  |rt}  be  an  increasing  sequence  of  submultiplicative  seminorms  defining  the 
topology  of  5(R,A,a).  Let  (Brt,  |  |n)  be  the  Hausdorff  completion  of  5(R,A,a)  in  |  |n. 
Then  Bn  is  a  Banach  algebra  and  5(R,A,a)  =  proj.limBn.  Since  ||ar(jc)||  =  ||jc||,  the 
action  d  of  R  on  5(R,A,Q:)  extends  to  a  strongly  continuous  action  o:^  of  R  by 
automorphisms  of  Bn.  Let  Cm(Bn)  be  the  Banach  algebra  of  all  Cm  -vectors  in  Bn  fo/  the 
action  of  fiW.  As  noted  in  ([29],  p.  189),  Cn(Bn)  is  dense  and  spectrally  invariant  in  Bn\ 
and  5(R,  C°°(A),  a)  =  proj  lim  Cn(Bn).  Let  x  E  S(R,  C°°(A),  a),  x  =  (xn)  being  a  cohe- 
rent sequence  with  xn  G  Cn(Bn)  for  all  n  G  M.  Now 

sps(R,c°*(AM(x)  =  \Jspcn(Bll)(xn)  =  \JspBn(xn)  =  sp5(M^)  W-' 

n  n 

Thus  5(R,C°°(A),a)  is  spectrally  invariant  in  5(R,A,a);  which  in  turn  is  spectrally 
invariant  in  C*(R,A,a)  by  '([33],  Corollary  7.16).  Thus  each  of  S(R,C°°(A),a)  and 
S(R,A,a)  are  Q-normed  algebras  in  the  C*-norm  of  C*(R,A,o.j;  and  hence  are  Q- 
algebras  in  their  respective  Frechet  topologies.  Using  Lemma  2.10,  £(5(R,  C°° 


Proof  of  Theorem  1.5.  Since  C°°(B)  =  B,  the  Frechet  m-convex  algebra  5(R,B,a)  is  a 
*-algebra  by  ([34],  Corollary  4.9).  Since  B  is  Frechet  and  sits  in  the  C*  -algebra  A,  B  is 
*-semisimple.  Similarly,  since  the  inclusion  S(U,B,a)  —  >  C*(R,A,a)  is  continuous  and 
one-one,  5(R,  B,  a)  is  also  *-semisimple.  To  prove  that  £(5(R,  B,  a))  =  C*(R,  A,  a),  it  is 
sufficient  to  prove  that  any  "-representation  cr  :  5(R,B,a)  —  ^  B(Ha)  extends  to  a 
^representation  (or)  :  C*(R,A,a)  -*  B(Ha).  This  would  imply  that  the  C*-norm  on 
5(R,  B,  a)  induced  by  the  C*-algebra  norm  on  CB*(R,  A,  a)  is  the  greatest  (automatically 


C*  -enveloping  algebras  87 

continuous)  C*-seminorm  on  S(R,  J5,  a).  This  is  shown  below  by  arguments  analogous  to 
those  in  ([25],  Proposition  7.6.4,  p.  255). 

Let  (x\)  be  a  bounded  approximate  identity  for  A  contained  in  B  and  which  is  also  a 
bounded  approximate  identity  for  B.  For  each  n  G  M,  let  fn  G  C£°(R)  be  such  that 
0  <fn  <  !,/«(*)  =  1  for  all  x  G  [-«,«],  and  supp/n  C  [-«  -  l,n  4-  1].  Then  (/„)  is  a 
bounded  approximate  identity  for  S(R)  (pointwise  multiplication)  contained  in  C£°(R). 
The  inverse  Fourier  transforms  gn  of  /„  constitute  a  bounded  approximate  identity  for 
5(R)  with  convolution.  Thus  vn?A  =  gn  ®x\  constitute  a  bounded  approximate  identity 
for  S(R,jE?,a).  Given  a  *  -representation  cr  :  S(R,  /?,<*)  -^  B(Hcr)  automatically  contin- 
uous, let  It  (Ho)  be  the  group  of  all  unitary  operators  on  E0.  Define  TT  :  B  —»  B(Ha)  and 
U  :  R  ->  Z/(/fff)  by 


(n,A) 

The  limits  are  taken  in  the  weak  sense;  and  they  exist.  As  in  ([25],  §  7.6,  p.  256),  it  is 
verified  that  TT  is  a  *  -representation  of  B;  U  is  a  unitary  representation  of  R; 
Utn(x)U;  =  7r(a,(jt))  for  all  t  G  R,  all  x  G  B;  and  for  all  y  G  S(R,5,  a),  a(y)  =  /7r(v(?)) 
£/,df.  Now,  since  E(B)  =  A,  TT  extends  to  a  *  -representation  TT  :  A  —  »•  ^(/f^)  so  that 
(TT  ,  t/,  /fa)  is  a  covariant  representation  of  the  C*-dynamical  system  (R,A,a).  Then 
a(j)  =  /7f(v(r))t//dr  defines  a  non-degenerate  *  -representation  of  the  Banach  *-algebra 
L^RjJ?,  a);  and  hence  extends  uniquely  to  a  *  -representation  <j  of  C*(R,£,  a).  This  <j  is 
the  desired  extension  of  <j.  This  shows  that  E(S(R,B,a))  =  C*(R,A,a). 

Further,  suppose  that  the  action  a  of  R  on  B  is  isometric.  Then  by  [29],  L1  (R,  5,  a)  is  a 
*-algebra,  which  is  a  Frechet  m-convex  *-algebra;  and 

S(R,B,a)-+Ll(R,B,a)^>Ll(R,A,a)  -*  C*(R,A,a) 

are  continuous  embeddings  with  dense  ranges.  It  follows  that  E(Ll(R,B,a)  = 
C*(R,  A,  a).  This  completes  the  proof  of  the  theorem. 

Actions  on  topological  spaces 

(a)  Let  M  be  a  locally  compact  Hausdorff  space.  Let  a  :  M  —*  [0,  oo)  be  a  Borel  function, 
&(m)  >  1  for  all  m  £  M.  Assume  that  a  is  bounded  on  compact  subsets  of  M.  Following 
([34],  §5),  let 


Ca(M)  =  {f  G  C0(M)  :  I  (a4/)  I  <  oo  for  all  d  G  N}, 

called  the  algebra  of  continuous  functions  on  M  vanishing  at  infinity  cr-rapidly.  It  is 
shown  in  [34]  that  Cff(M}  is  a  Frechet  m-convex  "-algebra  with  the  topology  defined  by 
seminorms 

Ho4/)!  =  sup{|(cr(jt))rf/(jc)|  :  jc  G  Af},      d  G  N; 

and  that  CC(M)  — >  Ca(M)  — »  CQ(M)  are  continuous  embeddings  with  dense  ranges.  Thus  ^ 

E(Ca(M))  =  C0(M).  In  fact,  Ca(M)  is  an  ideal  in  C0(M);  hence  inverse  closed  in  C0(M); 

and  so  is  a  g-algebra. 

(b)  Let  G  be  a  Lie  group  acting  on  M.  Iff  G  Ca(M),  define  ag(f)(m)  =f(g-lm).  By 

([34],  §  5),  if  a  is  uniformly  G-translationally  equivalent  (in  the  sense  that  for  every 


88  S  J  Bhatt 

compact  K  C  G,  there  exists  /  G  M  and  C  >  0  such  that  cr(gm)  <  Ccr(m)1  for  all  g  €  G, 
m  G  M),  then  g  -^  otg  defines  a  strongly  continuous  action  of  G  by  continuous 
'-automorphisms  of  Cff(M).  Then  the  space  C°°(Cff(M))  consisting  of  C°°-vectors  for  the 
action  a  of  G  on  Ca(M)  is  an  m-convex  Frechet  * -algebra  with  a  C* -enveloping  algebra 
and  £(C°°(Ca(M))  =  C0(Af). 

(c)  In  particular,  let  G  =  R,  M  be  a  compact  C°° -manifold,  and  let  the  action  of  R  on  M 
be  smooth.  Then  the  induced  action  a  on  C(M)  is  smooth,  so  that  ar(C°°(M))  C  C°°(M) 
for  all  r  €  R.  It  follows  from  Theorem  5.1  that  £(S(R,  C°°(M),  a)  =  C*(R,  C(M),  a)  the 
covariance  C* -algebra. 

6.  The  Pedersen  ideal  of  a  C* -algebra 

Let  A  be  a  non-unital  C*-algebra.  Let  KA  be  its  Pedersen  ideal.  It  is  a  hereditary,  minimal 
dense  *-ideal  of  A.  For  a  G  A,  let  La  =  (Aa)~9  Ra  =  (oA)~ ,  la  be  the  closed  *-ideal  of  A 
generated  by  aa*.  Since  a  G  Lfl  f|jRa,  aa*  G  7fl.  Let  JSj  =  KA  P|A+  be  the  positive  part  of 
&A  endowed  with  the  order  relation  induced  from  that  of  A+.  Let  K™  =  \J{Ia  -  a  G  K%}. 

Lemma  6.1.  jKJc  w  a  dense  * -ideal  of  A  containing  KA,'  and  A  —  C*-ind  lim  {Ia  :  a  £  KA}- 

Proof.  Let  a  G  Kf.  Then  a2  =  aa*  G  la\  and  7a  being  a  C* -algebra,  a  =  (a2}{/2  G  7a. 
Thus  ^  C  £™.  Observe  that  for  any  x  =  x*  £KA,  x£  Ix.  Indeed,  ;t2  €  K+\  hence 
;t2  €  Ix  and  bc|  G  4.  But  than  taking  the  Jordan  decomposition  x  =  x+  -  x~~  in  A, 
(x+)2  =  (*+)*  +  jc+^-  =  jc+|x|  e  Ix\  so  that  ^+  €  Ix,  x~  €  4,  and  x  £  Ix.  In  particular, 
x2  G  42  and  \x  G  7^.  By  repeating  this  argument,  x  G  7^2  C  K™  for  any  jc  =  jc*  G  KA.  It 
follows  that  A:A  c  Klc.  Now,  by  ([28],  Lemma  1),  0  <  a  <  b  in  A  implies  La  C  L^,, 
^  C  Rb  and  7a  C  7^;  and  KA  =  (J{La  :  a  G  K+}  =  \J{Ra  :  a  G  K%}.  The  family  {Ia  : 
fl  G  K%}  forms  an  inductive  system  of  C*-algebras;  and  C*-ind  lim{7a  :  a  G  K%}  = 
(\J{Ia  -  a  G  KA}}~  =  A,  (  )"  denoting  the  norm  closure.  This  proves  the  lemma. 

Let  t\  (respectively  ti)  be  the  finest  locally  convex  linear  topology  (respectively  finest 
locally  m-convex  topology)  on  Kf  making  continuous  the  embeddings  Ia  —>  A^c,  where 
a  e  K%.  Then  (A^Vi)  (respectively  (^,r2))  is  the  linear  topological  inductive  limit 
(respectively  topological  algebraic  inductive  limit)  of  {Ia  :  a  G  K%}  ([21],  ch.  IV). 

Proof  of  Theorem  1.6.  In  the  present  set  up,  ([21],  p.  115,  118,  125)  implies  that  t\  =  fe, 
equal  to  r  say,  and  (K™,  r)  is  a  complete  m-barrelled  locally  m-convex  *-algebra;  and  the 
||  1 1 -topology  on  Kff  is  coarser  than  r.  Since  K%€  is  an  ideal,  it  is  inverse  closed  in  its 
1 1  j  j-completion  A,  and  hence  (7^c,  1 1  1 1)  and  (KA ,  1 1  1 1)  are  Q-algebras.  This  implies  that 
any  *-homomorphism  from  K%  into  B(H)  for  a  Hilbert  space  H  is  ||  |  (-continuous  and 
extends  uniquely  to  A.  Thus  1 1  1 1  is  the  greatest  C*-seminorm  on  K%.  To  show  that  1 1  1 1  is 
the  greatest  r-continuous  C*-seminorm  on  K£c  so  that  E(K%)  =  A,  it  is  sufficient  to  show 
that  (7^c,  r)  is  a  g-algebra.  To  that  end,  in  view  of  ([23],  Lemma  E.2),  we  show  that  0  is 
a  r-interior  point  of  the  set  (Kf)^  of  quasiregular  elements  of  K%.  Note  that,  by  ([21], 
p.  114),  basic  r-neighbourhoods  of  0  in  K.%  are  precisely  of  the  form  V  =  \co\ 
{\J(Ua  :  a  G  Jfjf")},.  where  \co\  denotes  the  absolutely  convex  hull  and  Ua  denotes  a 
convex  balanced  neighbourhood  of  0  in  (/fl,  ||  ||).  For  any  a  G  K%,  (7flJ  ||  ||)  is  a  Q- 
algebra,  and  being  an  ideal  in  A,  (Ia)_i  =  A^i  f]Ia.  Hence,  for  the  zero  neighbourhood 
Ua  =  {x  £la:  \\x\\  <l}  in  (Ia,\\  ||), 


C*  -enveloping  algebras  89 

ua  c  (/<,)_,  =  (*r)-i  f>  c  (^)-i;  ^ 

/a  :  a  €  #)}  =  {*  €  K~  :  ||x||  <  l    =  U  (say) 


is  a  zero  neighbourhood  in  (A^c,r)  contained  in  (K™)_{.  It  follows  that  (K™,r)  and 
(KA,T)  are  £>-algebras.  Now,  as  in  the  proof  of  ([28],  Theorem  4),  K£c  =  U{/eJ,  (e\) 
being  a  bounded  approximate  identity  for  A  contained  in  A^.  Thus  if  A  has  countable 
bounded  approximate  identity,  then  K™  is  an  LFQ-algebra;  and  r  is  the  finest  (unique) 
locally  convex  topology  on  K™  such  that  for  each  A,  r|7  is  the  norm  topology. 

7.  The  groupoid  C*  -algebra 

We  follow  the  terminology  and  notations  of  [31].  Let  G  be  a  locally  compact  groupoid, 
i.e.,  a  locally  compact  space  G  with  a  specified  subset  G2  C  G  x  G  so  that  two  conti- 
nuous maps  G  —  >  G,  *  —  ^jc"1,  and  G2  —  >  G,  (*,}>)—  »*y  are  defined  satisfying 
(,ry)z  =  x(yz),  x~~l(xy)=y  and  (zx)jc"1  =  z.  The  unit  space  of  G  is  G°  =  {xx~l  : 
x  e  G}  =  {x~lx  :  x  G  G}.  Let  r(jc)  =  xx~l  and  d(x)  =  x~lx.  Assume  that  there  exists  a 
left  Haar  system  {\u  :  u  €  G0}  on  G,  i.e.,  a  family  of  measures  A"  on  G  such  that 
supp  A"  =  r-1(u);  for  each/  C  Q(G),  u  —  >  //dAM  is  continuous;  and  for  all  jc  €  G  and 
/€CC(G),  //(jcy)dAdW(y)  =  //Cv)dArW(y).  Let  a  be  a  continuous  2-cocycle  in 
Z2(G,  T).  Let  ?  denote  the  usual  inductive  limit  topology  on  CC(G).  Then  (CC(G),  r)  is  a 
topological  *  -algebra  with  jointly  continuous  multiplication 


and  the  involution/*^)  =  (f(x'l)a(x,x-l)}~  ([31],  Proposition  II.l.l,  p.  48).  The  7- 
norm  on  Cc(G,cr)  is  ||/||7  =  max(||/||/ir,  ||/||7>/),  where 

\\f\kr  =  sup{|  |/|dA"  :  u  €  G°  j,      ||/||A/  =  supjj  |/|dAtt  :  u  €  ( 

AM  =  (A")'1  being  the  image  of  A"  by  the  inverse  map  x  -»  x"1  ([31],  p.  50).  Then  ||  ||7 
is  a  submultiplicative  *-norm  on  Cc(G,<j).  The  L1 -algebra  of  (G,  a)  is  the  completion 
A  =  (CC(G,  a) ,  1 1  1 17),  a  Banach  *-algebra.  For/  in  CC(G,  cr),  define  1 1/| |  =  sup{ | |TT(/)  1 1 }, 
TT  running  over  all  weakly  continuous,  non-degenerate  ^representations  TT  :  (CC(G,  cr), 
t)  -»  5(7^^)  satisfying -|  |TT(/)  1 1  <  1 1/|  |7  for  all/.  Then  1 1  1 1  defines  a  C*-norm  on  Cc (G,  cr) ; 
and  the  groupoid  C*-algebra  of  (G,  cr)  is  C*(G,  a)  =  (CC(G,  cr),  ||  ||)~,  the  completion. 
The  following  can  be  proved  using  cyclic  decomposition  and  ([31],  Corollary  II.  1.22, 
p.  72). 

PROPOSITION  7.1 

Let  G  be  second  countable  having  sufficiently  many  non-singular  G-Borel  sets.  Then 


8.  The  universal  *  -algebra  on  generators  with  relations 

Let  G  be  any  set.  Let  F(G)  be  the  free  associative  *-algebra  on  generators  G,  viz.,  the 
*-algebra  of  all  polynomials  in  non-commuting  variables  G]JG*  where  G*  = 
{x*  :  x  £  G}.  Let  R  be  a  collection  of  statements  about  elements  of  G,  called  relations 


90  SJ  Bhatt 

on  G,  assumed  throughout  to  be  such  that  they  make  sense  for  elements  of  a  locally 
m-convex  *-algebra.  A  Banach  (respectively  C*-)  representation  of  (G,R)  is  a  function  p 
from  G  to  a  Banach  *-algebra  (respectively  a  C* -algebra)  p  :  G  — >  A  such  that 
{p(&)  :  8  £  G}  satisfies  the  relations  R  in  A.  Let  RepB(G,#)  (respectively  Rep(G,#)) 
be  the  set  of  all  Banach  representations  (respectively  C* -representations)  of  (G,R). 
Motivated  by  ([27],  Definition  1.3.4),  it  is  assumed  that  R  satisfies  the  following. 

(i)  The  function  p  :  G  — »  {0}  is  a  Banach  representation  of  (G,/?). 
(ii)  Let  p  :  G  — >  A  be  a  representation  of  (G,  R)  in  a  Banach  *-algebra  A.  Let  B  be  a 
closed  *-subalgebra  of  A  containing  p(G).  Then  p  is  a  representation  of  (G,  R)  in  B. 
(iii)  Let  p  be  a  representation  of  (G,/?)  in  a  complete  locally  m-convex  * -algebra  A.  Let 
<f> :  A  — >  B  be  a  continuous  *-homomorphism  into  a  Banach  *-algebra  B.  Then  </)  o  p  is 
a  representation  of  (G,  /?)  in  J3. 

(iv)  Let  A  be  a  complete  locally  m-convex  *-algebra  expressed  as  an  inverse  limit  of 
Banach  *-algebras  viz.  A  =  proj.  HmAp.  Let  TTP  :  A  — >  Ap  be  the  natural  maps.  Let 
p  :  G  —» A  be  a  function  such  that  for  all  p,  irp  o  p  is  a  representation  of  (G,  R).  Then 
p  is  a  representation  of  (G,jR). 

DEFINITION  8.1 

(a)  (Blackadar)  (G,./?)  is  C* -bounded  if  for  each  g  in  G,  there  exists  a  scalar  M(g)  such 
that  \\p(g)\\  <  M(g)  for  all  p  G  Rep  (G,R). 

(b)  (Blackadar)  (G,  R)  is  C* -admissible  if  it  is  C* -bounded  and  the  following  holds. 
(bC*)  If  (pa)  is  a  family  of  representations  pa  :  G  — »  £(Ha)  of  (G,  R)  on  Hilbert  spaces 
#a,  then  ©PQ,  :  G  — >  B(®Ha)  is  a  representation  of  (G,/?). 

(c)  (G,  £)  is  weakly  Banach  admissible  if  given  finitely  many  representations  pi  :  G  — »  A; 
(1  <  /  <  n)  of  G  into  Banach  *-algebras,  the  map  g  — >  p\(g)  ©  P2(g)  ©  •  -  -  ©  Pn(g)  is  a 
representation  of  (G,/?)  in  ©  A,.  (G,R)  is  vmzfc/y  C* -admissible  [27]  if  this  holds  with 
Banach  algebras  replaced  by  C*-algebras. 

The  class  of  relations  making  sense  for  elements  of  a  Banach  *-algebra  is  smaller  than 
the  class  of  relations  making  sense  for  elements  of  a  C*-algebra.  The  usual  algebraic 
relations  involving  the  four  elementary  arithmetic  operations  on  elements  of  G  and  G*  do 
make  sense  for  Banach  *-algebras;  but  relations  like  jc+  >  x~  for  x  =  x*  in  G,  or  like 
x\  >  \y\  for  elements  jc,  y  in  G,  which  make  sense  for  C*-algebras,  fail  to  make  sense  for 
Banach  *-algebras.  We  refer  to  [27]  for  relations  satisfying  (i)-(iv)  except  (ii).  The 
relation  (suggested  by  the  referee).  "The  elements  a,  b  and  c  generate  A"  fails  to  satisfy 
Definition  8.1(c).  Our  definition  of  weakly  Banach  admissible  relations  is  very  much 
ad  hoc  aimed  at  exploring  a  method  of  constructing  non-abelian  locally  m-convex 
* -algebras. 

Lemma  8.2.  (a)  Let  (G,R)  be  weakly  Banach  admissible.  Then  there  exists  a  complete 
m-convex  * -algebra  A(G,R)  and  a  representation  p  :  G  — >  A(G,R)  such  that  given  any 
representation  a :  G  — » B  into  a  complete  m-convex  * -algebra  B,  there  exists  a 
continuous  * -homomorphism  $  :  A(G,R)  — >  B  satisfying  <j>  o  p  —  a. 
(b)  ([27],  Proposition  1.3.6).  Let  (G,R)  be  weakly  C* -admissible.  Then  there  exists  a  pro- 
C* -algebra  C*(G,J?)  and  a  representation  p^  :  G  — >  C*  (G,  R)  such  that  given  any 
representation  a :  G  -»  B  of  G  into  a  pro-C* -algebra  B,  there  exists  a  continuous 
* -homomorphism  <j> :  C*(G,R)  —+  B  such  that  <j>  o  p^  =  a. 


C*  -enveloping  algebras  91 

Proof,  (a)  Let  K  =  K(F(G))  be  the  set  of  all  submultiplicative  *-seminorms  p  on  F(G)  of 
the  form/?(jt)  =  ||CT(JC)||,  a  running  through  all  Banach  representations  of  G.  For/?  G  AT, 
let  Np  =  {x  €  F(G)  :  p(x)  =  0}  and  Na  =  r}{Np  :p  G  K}  a  "-ideal  of  F(G).  Let 
B  =  F(G)/Na.  Take  K*  +  #a)  =  p(*).  Let  t  be  the  Hausdorff  topology  defined  by 
{pipeK}.  Let  A(G,R)  be  the  completion  of  (B,t).  Let  p:G-*A(G,R)  be 


Cto'w  7.  p  is  a  representation  of  G  in  A(G,  J?). 

Let  g  be  any  f-continuous  submultiplicative  *-seminorm  on  A(G,/?).  Let  A9  be  the 
Banach  *-algebra  obtained  by  the  Hausdorff  completion  of  (A(G,  R),q).  By  (iv)  above,  it 
is  sufficient  to  prove  that  TT^  o  p  :  G  —  >  Aq  is  a  representation  of  (G,  R).  Since  q  is  t- 
continuous,  there  exists  p\,p2,  -  •  •  ,Pk  in  K  such  that  q(x)  <  c  max  /?,-(*)  for  all  x  G  F(G)\ 
and  each  pt  is  of  form  /7,(jt)  =  ||<jj(jc)||,  cr/  :  G  —  >  A(/)  being  a  representation  into  some 
Banach  algebra  A(/).  By  (c)  of  Definition  8.1,  there  exists  a  Banach  "-algebra  B  and  a 
representation  a  :  G  —>  #  such  that  #(*)  <  ||<T(JC)||  for  all  x  G  F(G).  In  view  of  (ii),  we 
assume  that  B  is  generated  by  <r(G).  Let  <f>:B—>Aq  be  0(<r(x))  =  (x  +  Nfl)-h 
ker^  =  7rq(p(x)).  Then  0  is  well  defined,  continuous  and  0o  a  =  Kq  o  p.  By  the 
assumption  (iii)  above,  0  o  a  is  a  representation  of  G. 

Claim  2.  Given  any  representation  <7  :  G  —  >  C  into  a  complete  m-convex  *  -algebra  C, 
there  exists  a  unique  continuous  *-homomorphism  $  :  A(G,R)  —  >  C  such  that  0  o  p  =  a. 

Let  C  =  proj.  lim  Cai  an  inverse  limit  of  Banach  *-algebras  CQ,,  7ra  :  C  —  »•  Ca  being  the 
projection  maps.  By  (iii)  of  above,  TTOCT  is  a  Banach  representation  of  (G,/?).  By  the 
construction  of  A(G,/?),  there  exist  continuous  *-homomorphisms  </>a  :A(G,R)  —  >  Ca 
such  that  <j)a  o  p  =  na  o  a.  Hence  by  the  definition  of  an  inverse  limit,  there  exists  a 
continuous  *-homomorphism  0  :  A(G,  /?)—»•  C  such  that  0  o  p  =  a. 
(b)  We  only  outline  the  (needed)  construction  of  C*(G,R)  from  [27].  Let  S  be  the  set  of 
all  C*-seminorms  on  F(G)  of  form  q(x)  =  ||p~(^)||,  &  running  over  all  representations  of 
G  into  C*-algebras.  Let  Nq  =  {x  G  F(G)  :  q(x)  =  0}  and  Af  =  n{A^  :  q  G  5}.  Let  r  be 
the  pro-C*  -topology  on  F(G)/N  defined  by  ^(^:  +  A/r)  =  q(x\  q  G  5.  Then  C*(G,/?)  is 
the  completion  of  (F(G)/Af,r).  The  map  p^  :  G  -»  C*(G,/?)  where  p^  (jc)  =  ^  -f-  A^  is 
the  canonical  representation. 

The  following  brings  out  the  essential  point  in  arguments  in  claim  1  above. 

Lemma  8.3.  TTzere  exists  a  natural  one-to-one  correspondence  between  RepB(G,/?) 
(respectively  Rep(G,/?))  <2«d  t-continuous  Banach  ""-representations  (respectively  C*- 
algebra  representations)  ofA(G,R). 

Lemma  8.4.  srad  (A(G,  R))  ^](F(Cf)/Na)  =  srad  (F(G)/Na)  =  {x  +  Na:xeN}. 


Proof.  Let  C  =  F(G)/Na.  Let  Jc  +  ATa  G  CfjsradA.  Then  7r(jc  +  JVfl)=0  for  all 
continuous  *-homomorphisms  TT  :  A  —  >  B(HV).  By  Lemma  8.3,  /?(*)  =  0  for  all  p  G  5. 
Hence  jc  G  N9  and  A:  4-  Na  G  srad  (F(G)/Na).  Conversely,  let  xeN.  Then  ^(jc)  =  0  for  all 
q  E  S.  Again  by  Lemma  8.3,  |  |TT(JC  -f  Na)  \  \  =  0  for  all  TT  G  R(A),  hence  x  +  Na  G  srad  A. 

Proof  of  Theorem  1.7.  (1)  Let  A  =  A(G,R).  Let  0  :  (F(G)/Na,t)  -»  (F(G)/A/rfl,r)  be 
Then  0  is  a  well  defined,  continuous  *-homomorphism;  hence 


92  SJ  Bhatt 

extends  as  a  continuous  surjective  *-homomorphism  0  :  A  — >  C*(G,/?).  The  universal 
property  of  C*(G,#),  Lemma  8.3  and  weak  Banach  admissibility  of  R  imply  the  follow- 
ing whose  proof  we  omit. 

Assertion  1.  Given  any  continuous  *-homomorphism  TT  :  A(G,/?)  — >  #  to  a  pro-C*- 
algebra  #,  there  exists  a  continuous  *-homomorphism  TT  :  C*(G,R)  — >  #  such  that 

7T  =  7T  O  <^>. 


C*(G ,  R) 


By  applying  the  above  to  the  maps  <J>  and  j  :  A  — >  E(A)J(x)  =x  +  srad  (A),  it  follows 
that  there  exist  continuous  *-homomorphisms  0  :  E(A)  — »•  C*(G,J?)  and  j  :  C*(G,/?)  — > 
E(A)  such  that  the  following  diagrams  commute. 

A 

4> 


7 


E(A)    ^  -  1    C  (G,R) 
j 

Assertion  2.  The  maps  <^  and  j  are  inverse  of  each  other. 
Indeed,  j  is  one-one  on  F(G)/N.  For  given  ;t  €  F(G), 


which  implies    (jc  +  Na)  +  srad  (A)  =  0  and    (jc  +  A^a)  €  srad  (A).   Hence  x  e  N  by 
Lemma  8.4,  so  that  x  +  N  =  0.  Similarly  0  is  one-one  on  F(G)/N.  Also, 


which  implies  that  0  ==  j-1  on  F(G)/Na;  and  ]  =  0-1  on  F(G)/Na  +  sradA.  By 
continuity  and  density,  <£  establishes  a  homeomorphic  *-isomorphism  ^  :  £(A)  ->  C* 
(G,J?)  withf"1  =]. 

(2)  Let  (G,«)  be  C*-admissible.  Then  sup{||cr(jc)||  :  a  G  Rep(G,#)}  <  oo;  and  TT  = 
®  (a  :  a-  E  Rep  (G,R)}  €  Rep  (G,  J?).  Thus  q(x)  =  ||TT(JC)||  defines  the  greatest  member 
of  5(F(G)j,  ^  is  a  C*-norm,  and  it  is  the  greatest  f-continuous  C*-seminorm  on  F(G)/N. 
Thus  the  topology  r  on  C*(G,  R)  is  determined  by  #.  Conversely  suppose  that  C*(G,  R)  is 
a  C*-algebra  so  that  \\Z\\QQ  =  sup{q(z)  :  q  is  a  continuous  C*-seminorm  on 
C*(G,«)}  <  oo  for  all  z  €  C*(G,/J),  and  r  is  determined  by  the  C*-norm  ||  1^.  Let 
Poo  (x)  =  1  1*  +  N\  |M  =  sup{^(jc)  :  4  e  5}  for  all  x  €  F(G).  Then  p^  6  S  and  ker  p^  =  N. 
There  exists  a  (^-representation  a  :  G  -+  C  such  thatpoo(g)  =  ||flr(g)||  for  all  g  6  G;  and 
this  defines  a  continuous  C*  -representation  a  :  C*(G,/?)  ~>  C.  It  is  clear  that  R  is  C*- 
bounded.  We  verify  (bC*)  of  Definition  8.1.  Let  {pa}  C  Rep  (G,fi)  with  pa:G->  B(Ha) 


C*  -enveloping  algebras  93 

for  some  Hilbert  space  Ha.  Let  H  =  ®HQ.  For  x  G  F(G),  let  A(JC)  =  ®pa  (x).  By  the  C*- 
boundedness  of  (G,fl),  A(x)  6  #(#).  This  defines  a  *-homornorphism  A  :  F(G)  ->  S(/f) 
satisfying  ||A(jc)||  =  sup||pa(jt)||  <p<x>(x)  for  all  x  E  F(G).  Since  ker^  =  Af,  A  factors 
to  a  "-representation  X:  F(G)/N  -*  B(H)  satisfying  ||A(z)||  <  HZ^.  As  ||  (^  is  r- 
continuous,  so  is  A.  By  lemma  8.3,  (A(g)  :  g  €  G]  satisfies  the  relations  R  in  B(H).  Thus 
(G,#  is  C*-admissible. 


Acknowiedgemetits 

The  author  sincerely  thanks  Prof.  M  Fragoulopoulou  for  fruitful  correspondence  on  [5] 
and  for  providing  [16,17];  as  well  as  Prof.  N  C  Phillips  and  Prof.  L  B  Schweitzer  for 
providing  pre-publication  versons  of  [29,33,34].  Section  5  has  been  significantly 
influenced  by  suggestions  from  the  referee.  Besides  suggesting  the  reference  [18], 
Proposition  5.1  and  the  actions  on  C°°  -manifolds,  he  has  also  suggested  the  problem 
discussed  in  Theorem  1.5.  Further,  he  has  critically  read  the  manuscript  and  made  several 
detailed  suggestions  on  appropriately  presenting  the  material  in  readable  form.  Thank  you 
very  much,  referee. 

References 

[1]  Allan  G  R,  A  spectral  theory  for  locally  convex  algebras,  Proc.  London  Math.  Soc.  3  (1965) 

399-421 
[2]  Bhatt  S  J,  Representability  of  positive  functional  on  abstract  *  -algebra  without  identity  with 

applications  to  locally  convex  *  -algebras,  Yokohana  Math.  J.  29  (1981)  7-16 
[3]  Bhatt  S  J,  An  irreducible  representation  of  a  symmetric  *  -algebra  is  bounded,  Trans.  Am. 

Math.  Soc.  292  (1985)  645-652 
[4]  Bhatt  S  J,  Bounded  vectors  for  unbounded  representations  and  standard  representations  of 

polynomial  algebras,  Yokohama  Math.  J.  41  (1993)  67-83 
[5]  Bhatt  S  J  and  Karia  D  J,  Topological  *-algebras  with  C*  -enveloping  algebras,  Proc.  Indian 

Acad.  Sci.  (Math.  Sci.)  102  (1992)  201-215 
[6]  Bhatt  S  J  and  Karia  D  J,  On  an  intrinsic  characterization  of  pro-C*  -algebras  and  applications, 

J.  Math.  Anal.  Appl.  175  (1993)  68-80 
[7]  Bonsall  F  F  and  Duncan  J,  Complete  Normed  Algebras  (New  York:  Springer-  Verlag,  Berlin 

Heidelberg)  (1973) 
[8]  Bhatt  S  J  and  Dedania  H  V,  On  seminorm,  spectral  radius  and  Ptak's  spectral  function  in 

Banach  algebras,  Indian  J.  Pure.  Appl  Maths.  27(6)  (1996)  551-556 
[9]  Brooks  R  M,  On  locally  m-convex  *-algebras,  Pacific  J.  Math.  23  (1967)  5-23 
[10]  Brooks  R  M,  On  representing  F*  -algebras,  Pacific  J.  Math.  39  (1971)  51-69 
[11]  Bratteli  O,  Derivations,  dissipations  and  group  actions  on  C*  -algebras  -,  Lecture  Notes  in 

Mathematics  1229  (Springer-  Verlag)  (1986) 
[12]  Dixon  P  G,  Automatic  continuity  of  positive  functional  on  topological  involution  algebras, 

Bull.  Australian  Math.  Soc.  23  (1981)  265-283 

[13]  Dixon  P  G,  Generalized  B*  -algebras,  Proc.  London  Math.  Soc.  21  (1970)  693-715 
[14]  Dixmier  J,  C*-algebras  (North  Holland)  (1977) 
[15]  Fragoulopoulou  M,  Spaces  of  representations  and  enveloping  l.m.c.  *  -algebras,  Pacific  J. 

Math.  95  (1981)  16-73 
[16]  Fragoulopoulou  M,  An  introduction  to  the  representation  theory  of  topological  *  -algebras, 

Schriften  Math.  Inst.  Univ.  Munster,  2  Serie,  Heft  48,  June  1988 
[17]  Fragoulopoulou  M,  Symmetric  topological  *  -algebras  II:  Applications,  Schriften  Math.  Inst. 

Univ.  Munster,  3  Sen,  Heft  9,  1993 

[18]  Green  P,  The  local  structure  of  twisted  covariance  algebras,  Acta.  Math.  140  (1978)  191-250 
[19]  Inoue  A,  Locally  C*-algebras,  Mem.  Fac.  Sci.  (Kyushu  Univ.)  (1972) 
[20]  La/3ner  G,  Topological  algebras  of  operators,  Rep.  Math.  Phys.  3  (1972)  279-293 


94  S  J  Bhatt 

[21]  Mallios  A,  Topological  algebras:  Selected  topics,  (Amsterdam:  North  Holland  Publ.  Co.) 

(1985) 

[22]  Mathot  F,  On  decomposition  of  states  of  some  *-algebras,  Pacific  J.  Math.  90  (1980)  41 1-424 
[23]  Michael  E  A,  Locally  multiplicatively  convex  topological  algebras,  Mem.  Am.  Math.  Soc.  11, 

1952 
[24]  Palmer  T  W,  Algebraic  properties  of  * -algebras ;  presented  at  the    13th.  International 

Conference  on  Banach  Algebras  97,  Heinrich  Fabri  Institute,  Blaubeuren,  July-August  1997 
[25]  Pedersen  G  K,  C* -algebras  and  their  automorphism  groups,  London  Math.  Soc.  Monograph 

14  (Academic  Press)  (1979) 

[26]  Phillips  N  C,  Inverse  limits  of  C* -algebras,  J.  Operator  Theory  19  (1988)  159-195 
[27]  Phillips  N  C,  Inverse  limits  of  C* -algebras  and  applications  in:  Operator  Algebras  and 

Applications  (eds.)  D  E  Evans  and  M  Takesaki)  (1988)  London  Math.  soc.  Lecture  Note  135 

(Cambridge  Univ.  Press) 
[28]  Phillips  N  C,  A  new  approach  to  the  multipliers  of  Pedersen  ideal  Proc.  Am.  Math.  Soc.  104 

(1988)  861-867 
[29]  Phillips  N  C  and  Schweitzer  L  B,  Representable  AT-theory  for  smooth  crossed  product  by  R 

and  Z,  Trans.  Am.  Math.  Soc.  344  (1994)  173-201 
[30]  Powers  RT,  Self  adjoint  algebras  of  unbounded  operators  I,  II,  Comm.  Math.  Phys.  21  (1971) 

85-125;  Trans.  Am.  Math.  Soc.  187  (1974)  261-293 
[31]  Renault  J,  A  grouped  approach  to  C* -algebras,  Lecture  Notes  in  Maths.  793  (Springer- Verlag) 

(1980) 

[32]  Schaefer  H  H,  Topological  Vector  Spaces  (MacMillan)  (1967) 
[33]  Schweitzer  L  B,  Spectral  invariance  of  dense  subalgebras  of  operator  algebras,  Int.  J.  Math.  4 

(1993)289-317 
[34]  Schweitzer  L  B,  Dense  w-convex  Frechet  algebras  of  operator  algebra  crossed  product  by  Lie 

groups,  Int.  J.  Math.  4  (1993)  601-673 

[35]  Schmudgen  K,  Lokal  multiplikativ  konvexe  0p*-Algebren,  Math.  Nach.  85  (1975)  161-170 
[36]  Schmudgen  K,  The  order  structure  of  topological  * -algebras  of  unbounded  operators,  Rep. 

Math.  Phys.  7  (1975)  215-227 

[37]  Schmudgen  K,  Unbounded  operator  algebras  and  representation  theory  OT  37,  (Basel- 
Boston-Berlin:  Birkhauser-Verlag)  (1990) 
[38]  Sebestyen  Z,  Every  C*-seminorm  is  automatically  submultiplicative,  Period.  Math.  Hungar. 

10  (1979)  1-8 
[39]  Sebestyen  Z,  On  representability  of  linear  functional  on  * -algebras,  Period.  Math.  Hungar. 

15(3)  (1984)  233-239 
[40]  Stochel  J  and  Szafranieo  F  H,  Normal  extensions  of  unbounded  operators,  /.  Operator  Theory 

14(1985)31-55 
[41]  Traves  F,  Topological  vector  spaces,  distributions  and  kernals  (New  York  London:  Academic 

Press)  (1967) 
[42]  YoodB,  C*-seiDmom&,,StudiaMath.  118  (1996)  19-26 


Proc.  Indian  Acad.  Sci.  (Math.  ScL),  Vol.  Ill,  No.  1,  February  2001,  pp.  95-106. 
©  Printed  in  India 


On  the  equisummability  of  Hermite  and  Fourier  expansions 


E  K  NARAYANAN  and  S  THANGAVELU       • 

Statistics  and  Mathematics  Division,  Indian  Statistical  Institute,  8th  Mile,  Mysore 

Road,  Bangalore  560059,  India 

E-mail:  naru@isibang.ac.in;  veluma@isibang.ac.in 

MS  received  10  March  2000 

Abstract.  We  prove  an  equisummability  result  for  the  Fourier  expansions  and 
Hermite  expansions  as  well  as  special  Hermite  expansions.  We  also  prove  the  uniform 
boundedness  of  the  Bochner-Riesz  means  associated  to  the  Hermite  expansions  for 
polyradial  functions. 

Keywords.    Hermite  functions;  special  Hermite  expansions;  equisummability. 


1.  Introduction 

This  paper  is  concerned  with  a  comparative  study  of  the  Bochner-Riesz  means  associated 
to  the  Hermite  and  Fourier  expansions.  Recall  that  the  Bochner-Riesz  means  associated  to 
the  Fourier  transform  on  (Rrt  are  defined  by 

Sf/W  =  (27T)-"/2 

where 


is  the  Fourier  transform  on  Rn.  Let  3>a,  a  £  Nn  be  the  n-dimensional  Hermite  functions 
which  are  eigenfunctions  of  the  Hermite  operator  H  =  —A  +  |;c|2  with  the  eigenvalue 

(2|a|  +  n)  where  |a|  =  ai  H h  cnn.  Let  P*  be  the  orthogonal  projection  of  L2(1R/I) 

onto  the  kih  eigenspace  spanned  by  $a,  |a|  =  k.  More  precisely, 


Then  the  Bochner-Riesz  means  associated  to  the  Hermite  expansions  are  defined  by 


For  the  properties  of  Hermite  functions  and  related  results,  see  [6]. 

In  our  study  of  the  Bochner-Riesz  means  associated  to  Hermite  and  special  Hermite 
expansions  we  make  use  of  a  transplantation  theorem  of  Kenig-Stanton-Tomas  [2].  Let  us 

95 


96  E  K  Narayanan  and  S  Thangavelu 

briefly  recall  their  result.  Let  P  be  a  differential  operator  acting  on  C^(Un)  C  L2((R") 
which  is  self  adjoint.  Let 


/y  =      XdEx 
be  the  spectral  resolution  of  P.  Let  m  be  a  bounded  function  on  R  and  define 


Let  AT  be  a  subset  of  R"  with  positive  measure  and  define  the  projection  operator  Qk  on 

L2(r)  by 


where  XAT(*)  is  the  characteristic  function  of  K.  Let  />(*,  f  )  be  the  principal  symbol  of  P. 
Since  P  is  symmetric  p  is  real  valued.  Then  we  have  the  following  theorem. 

Theorem  1.1.  Assume  1  <  p  <  oo  an  J  that  there  is  a  set  of  positive  measure  KQ  for 
which  the  operators  QKQ^R(P}QKO  cire  uniformly  bounded  on  Lp(Rn).  If  XQ  in  KQ  is  any 
point  of  density,  then  m(p(jto,f))  w  a  Fourier  multiplier  of  Lp(Rn). 

Let  B  be  any  compact  set  in  Rn  containing  origin  as  a  point  of  density  and  let  XB  be  the 
operator 

XBf(x)  =X*  (*)/(*)• 


Then  from  Theorem  1.1  it  follows  that  the  uniform  boundedness  of  XB^RXB  on  Lp(Un) 
implies  the  uniform  boundedness  of  Sf  on  Lp(Rn).  Thus  once  we  have  the  local 
summability  theorem  for  Hermite  expansions  then  a  global  result  is  true  for  the  Fourier 
transform.  At  this  point  a  natural  question  arises,  to  what  extend  the  converse  is  true?  In 
this  paper  we  answer  this  question  in  the  affirmative  in  dimensions  one  and  two  and 
partially  in  higher  dimensions.  We  also  study  the  equisummability  of  the  special  Hermite 
expansions,  namely  the  eigenfunction  expansion  associated  to  the  operator 


on  C".  In  this  case  we  show  that  the  local  uniform  boundedness  of  the  Bochner-Riesz 
means  for  the  special  Hermite  operator  is  equivalent  to  the  uniform  boundedness  of  Sf  on 
IR2".  Using  a  recent  result  of  Stempak  and  Zienkiewicz  [4],  on  the  restriction  theorem  we 
study  the  Bochner-Riesz  means  associated  to  the  Hermite  expansions  on  R2n  for  functions 
having  some  homogeneity.  We  also  prove  a  weighted  version  for  the  Hermite  expansions 
which  slightly  improves  the  local  estimates  proved  in  [5].  Eigenfunction  expansions 
associated  to  special  Hermite  operator  L  has  been  studied  by  Thangavelu  [6]. 

2.  Hermite  expansions  on  Rn 

The  Hermite  functions  hk  on  R  are  defined  by 


Equisummability  of  Hermite  and  Fourier  expansions  97 

In  the  higher  dimensions  the  Hermite  functions  are  defined  by  taking  tensor  products: 


Given  /  G  LP((R)  consider  the  Hermite  expansion 
/W  = 


Let  Sfff(x}  =  X)f=o(/'  nk)hk(x)  be  the  partial  sums  associated  to  the  above  series.  In 
1965,  Askey-Wainger  [1]  proved  the  following  celebrated  theorem. 


Theorem  2.1.  S^f  — >  /  in  the  Lp  norm  iff  |  <  p  <  4. 

Let  St  be  the  partial  sum  operator  associated  to  the  Fourier  transform  on  R.  Then  it  is 
well  known  that  Stf  — »  /  in  Lp  norm  for  all  1  <  p  <  oo.  In  this  section  we  show  that  on  a 
subclass  of  LP(R)  the  same  is  true  for  the  Hermite  expansions. 

In  the  higher  dimensions  it  is  convenient  to  work  with  Cesaro  means  rather  than  Riesz 
means.  These  are  defined  by 

1     W 


where  A{  are  the  binomial  coefficients  defined  by  Ai  =  r^iitrl/^A-  It  is  well  known  that 

f  C  L     ~"~      /       I        '        / 

<TN  are  uniformly  bounded  on  Lp(Rn)  iff  S°R  are  uniformly  bounded.  We  have  the 
following  equisummability  result.  Let  E  stand  for  the  operator  Ef(x)  =  e~sM  f(x). 

Theorem  2.2.  Ea6NE  are  uniformly  bounded  on  Lp(Rn)  iff  S6t  are  uniformly  bounded, 
provided  6  >  max{0,|  —  1}. 

As  a  corollary  we  have  the  following. 

COROLLARY  2.3 

Let  1  <  p  <  oo.  Then  for  the  partial  sum  operators  associated  to  the  one  dimensional 
Hermite  expansion  we  have  the  uniform  estimate 

-/ 

Thus  for  f  G  Lp(e^2dv),  1  <  p  <  oo  the  partial  sums  converge  to  fin  Lp(t~ 
For  a  general  weighted  norm  inequality  for  Hermite  expansions,  see  Muckenhoupt's 

paper  [3]. 
The  celebrated  theorem  of  Carleson-Sjolin  for  the  Fourier  expansion  on  R2  says  that  if 

S  >  2(p  ~"  2)  ~  2»  *  -  P  <  3  ^Gn  St  are  uniformly  bounded  on  L/7(R2).  As  a  corollary  to 

this  we  obtain  the  following  result  for  the  Cesaro  means  o-6N  on  R2. 

COROLLARY  2.4 

;  =  2,  1  <  p  <  |  and  S  >  2(±  -  i)  -  \.  Then  for  f  €  LP(U2) 


<  CJ 


98  E  K  Narayanan  and  S  Thangavelu 

It  is  an  interesting  and  more  difficult  problem  to  establish  the  above  without  the 
exponential  factors. 

We  now  proceed  to  prove  Theorem  2.2.  It  is  a  trivial  matter  to  see  that  uniform 
boundedness  of  Ecr^E  implies  the  same  for  XB^XB  for  any  compact  subset  B  of  IRn.  In 
fact,  if  Ecr6NE  are  uniformly  bounded  then 

\XBcr6NXBf\P&c 


<C  f  ^E 


which  proves  the  one  way  implication,  by  the  transplantation  theorem  [2].  To  prove  the 
converse  we  proceed  as  follows.  Let 


be  the  kernel  of  the  projection  operator  Pk.  Then  the  kernel  o^  (*,)>)  of  the  Cesaro  means 
is  given  by 


"/v  *=0 

We  first  obtain  a  usable  expression  for  this  kernel  in  terms  of  certain  Laguerre  functions. 
Let  L%(t]  be  the  Laguerre  polynomials  of  the  type  a.  >  —  1  defined  by 

e~rfaL£(f)  =  (-1)*T7T"E (e~'**+a)>     l  >  0- 
We  have  the  following  expression. 

PROPOSITION  2.5 

4(^)=^|>^ 

Proof.  The  generating  function  identity  for  the  projection  kernels  $*(jc,y)  reads 


Since 


Equisummability  of  Hermite  and  Fourier  expansions  99 

the  generating  function  for  cr£(;t,y)  is  given  by 


The  right  hand  side  of  the  above  expression  can  be  written  as 

(1  _  r)-H-»e-ifcl*->l'(i  +  r)-f  e-^l*< 
Now  the  generating  function  for  the  Laguerre  polynomials  L%  is 


Therefore,  we  have 


Equating  the  coefficients  of  r*  on  both  sides  we  obtain  the  proposition. 

The  Laguerre  functions  Lf  are  expressible  in  terms  of  Bessel  functions  Ja.  More 
precisely,  we  have  the  formula 


Using  this,  the  kernel  e~iW  a6N(x,y)  e~iM   of  the  operator  Eo*NE  is  given  by. 

eHW^yJeHW1  = 

(r-  ,f 
~~ 


C    /-  /-  (r-  ,f  6  „  ,  JH  (x^l^  -  y|)  JHi  (  V5|»  +  y|) 

4/0    7o  ~^!~r  (V2t\X-y\)^  (v^|.  +  y|)^1         ' 


where  C  depends  only  on  <5.  Now  the  kernel  of  the  Bochner-Riesz  means  Sf  on  Rn  is 
given  by 


When  n  =  1, 


H       W 

and  hence 


=    —       cos  t 


100  E  K  Narayanan  and  S  Thangavelu 

where 

7*f(x)  = 

and  C  an  absolute  constant. 
By  Minkowski's  integral  inequality  we  get 

\\E<SNEf\\f  <  C± 

<  ell/11 

since 

/ 
. 


which  proves  the  theorem  in  one  dimension. 

When  n  >  2  we  have  the  Bessel  functions  J«_i  inside  the  integral.  If  d/j,  is  the  surf  a 
measure  on  the  unit  circle  \x\  =  1  in  Rn  then  we  have 


where  C  is  an  absolute  constant.  If  we  use  this  in  the  above  we  get  Ecr8NEf(x)  equals 


As  before,  using  Minkowski's  inequality  we  get 

\\EasNEf\\< 


since 


/*oo     /»oo 

/      /     e-'e-'Ir-jV^dftfc 
Jo     Jo 

<  /°°e-f/f  Te^^-^-F  f 
JQ  \JQ  Jt 

<C  f" e-'f Vdf  +  T f AT  +  J)  / 
7o  V       2/  70 


provided  8  >  %  -  1.  This  completes  the  proof. 


Equisummability  of  Hermite  and  Fourier  expansions  101 

3.  Special  Hermite  expansions 

Let  $Q,/3,  a,  (3  €  Nn,  be  the  special  Hermite  functions  on  Cn  which  form  an  orthonormal 
basis  for  L2(Crt).  The  special  Hermite  expansion  of  a  function/  in  Lp(Cn)  is  given  by 


The  functions  $a/3  are  the  eigenfunctions  of  the  operator  L  with  eigenvalues  (2|/3|  -f-  n). 
Let 


be  the  projection  onto  the  Ath  eigenspace.  Then  we  have 


x 


where  <#k(z)  =L2"1(5|z|2)e"i'zl    are  the  Laguerre  functions  and  /  x  g  is  the  twisted 
convolution 


cn 
The  special  Hermite  expansion  then  takes  the  compact  form 


Jt=0 

The  Cesaro  means  are  then  defined  by 


In  this  section  we  prove  the  following  theorem. 
Let  5f  be  the  Bochner-Riesz  means  for  the  Fourier  transform  on  IR2rt  =  Cn. 

Theorem  3.1.  Let  B  be  any  compact  subset  of  Cn  containing  the  origin.  Then  ; 

are  uniformly  bounded  on  Lp,  1  <  p  <  oo  if  and  only  ifS^  are  uniformly  bounded  on  the 

same  Lp. 

Proof.  The  kernel  0#(z)  of  &6N  is  given  by 


Using  the  formula 


we  have 


102  E  K  Narayanan  and  S  Thangavelu 

As  in  the  previous  section  we  can  express  the  Laguerre  function  in  terms  of  the  Bessel 
functions,  thus  getting 


J0 


. 

(V2t\z\)s+n 


Now,  a^f  =  /  x  o^  so  that 


where 


Writing  |z  -  w\2  =  \z\2  +  \w\2  +  2Rez  •  w  we  have 


|w 


where  (z  •  w)a  =  (ziw,)"1  •  •  •  (znwn)a".  Therefore, 


where 


If  we  assume  that  Sf  are  uniformly  bounded  we  get 


when  B  is  contained  in  the  ball  {z  :  \z\  <  R}>  Using  this  in  the  above  equation  we  get 

\\XBONXBf\\p  <  Call/Up- 

The  converse  is  the  transplantation  theorem  of  Kenig-Stanton-Tomas. 

In  [5],  Thangavelu  has  established  the  following  local  estimates  for  the  Cesai 
means. 


Theorem  3.2.  Let  ^-^  <  p  <  oo  and  S  >  S(p]  =  ln(l-  -  i)  -  £  then  for  any  compa 
subset  B  of  Cn 


Equisummability  of  Hermite  and  Fourier  expansions  103 

Recently  Stempak  and  Zienkiewicz  have  proved  the  global  estimate 

\a6Nf(z)\pdz<C       \f(z}\pdz 


cn 
for  the  above  range.  The  key  point  is  the  restriction  theorem  namely,  the  estimate 


which  they  established  in  the  range  1  <  p  <  2{%^.  In  the  next  section  we  use  this 
restriction  theorem  in  order  to  prove  a  positive  result  for  the  Hermite  expansions  on  [R2n. 


4.  Hermite  expansions  on  R2n 

In  this  section  we  consider  the  operator  -A  +  1  \z\2  rather  than  the  operator  -  A  -f  \z  2.  If 
$n(x,y),iJ,  e  N2n  are  the  eigenfunctions  of  the  operator  -A  +  |z|2  then  vj>M(z)  = 
$M(^  ,  -4)  are  the  eigenfunctions  of  -A  +  \  \z\2  with  eigenvalues  (|/x|  -f  n).  The  operator 
-A  -f  |]z  has  another  family  of  eigenfunctions  namely  the  special  Hermite  functions. 


In  fact,  $a/3  are  eigenfunctions  of  the  operator  —  A  +  ||z|    with  eigenvalue  (|a|+ 


In  this  section  we  study  the  expansion  in  terms  of  ^  for  functions  having  some 
homogeneity.  The  torus  T(n)  =  {(e1'*1  ,  e''^2,  .  .  .  ,  e1'*")  :  0  €  Rn}  acts  on  functions  on  C"  by 
ref(z)  =  /(e'*z)  where  ewz  =  (e^zi,e^2z2,  .  .  .  ,e^zn).  We  say  that  a  function  is  m- 
homogeneous  if  ref(z)  =  tim'ef(z),  here  meZn  and  m.O  =  m\.6i-\  -----  h  mn-0n.  It  is  a 
fact  that  $a/3  is  (fi  —  a)  homogeneous.  0-homogeneous  functions  are  also  called 
polyradial. 

The  operator  —A  +  \  \z\2  commutes  with  re  for  all  0,  therefore  P^rof  =  roPkf  which 
shows  that  P^f  is  m-homogeneous  if  /  is  .  In  particular,  Pkf  is  polyradial  if  /  is. 
Therefore,  for  such  functions  L(Pkf)  =  (-  A  +  \  \z\2}Pkf  =  (k  +  n)Pkf.  This  shows  that 
P^/  is  an  eigenfunction  of  L  with  eigenvalue  k  +  n.  But  the  spectrum  of  L  is 
{2^  +  n  :  k  =  0,  1,  .  .  .}  which  forces  />*/  =  0  when  k  is  odd. 

PROPOSITION  4.1 

Let  f  be  polyradial  on  Cn.  Then  Pijt+i/  =  0  and  P2kf  =  f  x  </?*. 


/  We  show  that  when/  is  polyradial  the  operators  ?ikf  and/  x  ipk  have  the  same 
kernel.  Let 


be  the  kernel  of  Pk.  Then  by  Mehler's  formula 

]TY#fc(z,  w)  =  7r~n(l  -  *2)~ne 
so  that 


00 


Jt=0 


104  E  K  Narayanan  and  S  Thangavelu 

Let  Wy  =  My  +  ivy  =  rye**.  When  /  is  polyradiai  /(w)  =  /o(n,r2, . . .  ,rn)  and  so  we 
have 


where  s  =  (51,  52,  .  .  .  ,  J«),  £y  =  |zy|  and  *  is  given  by 
*(*,  r)  =  (1  -  rT* 


Now  Re  zy  •  Wj  =  rjSjCQ$(0j  -  (pj)  where  z/  =  J/e1^,  wy  =  rye'*.  Consider  the  integral 
* 


which  equals,  if  we  recall  the  definition  of  the  Bessel  functions,  ./oCiir^fyty)-  Thus  we 
have  proved 

*(,,  r)  =  (1  -  O- 


On  the  other  hand  when/  is  polyradiai  /  x  tpk  reduces  to  the  finite  sum 


where  we  have  written 


as  it  is  polyradiai.  Then  /  x  ^  is  given  by  the  integral  operator 

/•oo  /»o 

f*Vk(z)=         ••• 

J°  J° 


i,  .  .  .  ,  rn)ri,  .  .  .  ,  rBdri,  .  .  .  ,drn. 
We  have  the  formula  (see  [6]) 


Recalling  the  generating  function  identity  for  the  Laguerre  polynomials  of  type  0, 


Equiswnmability  of  Hermite  and  Fourier  expansions 
we  get,  if  S*(r,  s)  is  the  kernel  for  /  x  <pk 

(r,s)  -  (1  -rpe 


105 


Comparing  the  two  generating  functions  we  see  that 


from  which  follows  \J>2/t(r,.s)  =  Sk(r,s)  and  this  proves  the  proposition. 

Consider  now  the  Bochner-Riesz  means  associated  to  the  expansions  in  terms  of  ^ 
defined  by 


pi  '   H- 

For  these  means  we  have  the  following  result. 


Theorem  4.2.  Let  \<p< 
polyradial.  Then 


,  S  >  6(p]  = 


/  €  L*(Cn) 


where  C  is  independent  off  and  R. 
The  key  ingredient  in  proving  the  above  theorem  is  the  Lp  -  L2  estimates 


which  now  follows  from  the  corresponding  estimates  for  /  x  (pk.  We  omit  the  details. 

We  conclude  this  section  with  the  following  remarks.  As  we  have  observed,  P^f 
is  m-homogeneous  whenever/  is  and  so  Pkf  can  be  obtained  in  terms  of  /  x  <pk  when 
/  is  m-homogeneous.  So  an  analogue  of  the  above  theorem  is  true  for  all  m-homo- 
geneous functions.  More  generally,  let  us  call  a  function/  of  type  N  if  it  has  the  Fourier 
expansion 


where 


Note  that/m  is  m-homogeneous.  We  can  show  that  when  /  is  of  type  Af  then 


under  the  conditions  of  the  above  theorem  on  p  and  S  where  now  C#  depends  on  N.  We 
leave  the  details  to  the  interested  reader.  It  is  an  interesting  problem  to  see  if  the  theorem 
is  true  for  all  functions. 


106          •   E  K  Narayanan  and  S  Thangavelu 

Acknowledgement 

The  authors  thank  the  anonymous  referee  of  an  earlier  version  of  this  paper,  whose 
suggestions  improved  the  presentation  of  the  paper  and  for  pointing  out  ref.  [3].  This 
research  was  supported  by  NBHM,  India. 

References 

[1]  Askey  R  and  Wainger  S,  Mean  convergence  of  expansions  in  Laguerre  and  Hermite  series,  Am. 

J.  Math.  87  (1965)  695-708 
[2]  Kenig  C  E,  Stanton  R  J  and  Tomas  P  A,  Divergence  of  eigenfunction  expansions,  /.  Funct. 

Anal.  46  (1982)  28-44 
[3]  Muckenhoupt  B,  Mean  Convergence  of  Hermite  and  Laguerre  series  II,  Trans.  Am.  Math.  Soc. 

147  (1970)  433-460 
[4]  Stempak  K  and  Zienkiewicz  J,  Twisted  convolution  and  Riesz  Means,  J.  Anal  Math.  16  (1998) 

93-107 
[5]  Thangavelu  S,  Hermite  and  Special  Hermite  expansions  revisited,  Duke.  Math.  J.  94  (1998) 

257-278 
[6]  Thangavelu  S,  Lectures  on  Hermite  and  Laguerre  expansions,  Mathematical  Notes,  (Princeton: 

Princeton  Univ.  Press)  (1993)  vol.  42 


Proc.  Indian  Acad  Sci.  (Math.  Sci.),  Vol.  Ill,  No.  1,  February  2001,  pp.  107-125. 
©  Printed  in  India 


Periodic  and  boundary  value  problems  for  second  order  differential 
equations 

NIKOLAOS  S  PAPAGEORGIOU  and  FRANCESCA  PAPALINI* 

Department  of  Mathematics,  National  Technical  University,  Zografou  Campus, 
Athens  15780,  Greece 

*Department  of  Mathematics,  University  of  Ancona,  Via  Brecce  Bianche, 
Ancona  60131,  Italy 

MS  received  4  February  2000;  revised  22  June  2000 

Abstract.  In  this  paper  we  study  second  order  scalar  differential  equations  with 
Sturm-Liouville  and  periodic  boundary  conditions.  The  vector  field  f(t,x,y)  is 
Caratheodory  and  in  some  instances  the  continuity  condition  on  x  or  y  is  replaced  by  a 
monotonicity  type  hypothesis.  Using  the  method  of  upper  and  lower  solutions  as  well 
as  truncation  and  penalization  techniques,  we  show  the  existence  of  solutions  and 
extremal  solutions  in  the  order  interval  determined  by  the  upper  and  lower  solutions. 
Also  we  establish  some  properties  of  the  solutions  and  of  the  set  they  form. 

Keywords.  Upper  solution;  lower  solution;  order  interval;  truncation  map;  penalty 
function;  Caratheodory  function;  Sobolev  space;  compact  embedding;  Dunford- 
Pettis  theorem;  Arzela-Ascoli  theorem;  extremal  solution;  periodic  problem;  Sturm- 
Liouville  boundary  conditions. 


1.  Introduction 

The  method  of  upper  and  lower  solutions  offers  a  powerful  tool  to  establish  the  existence 
of  multiple  solutions  for  initial  and  boundary  value  problems  of  the  first  and  second  order. 
This  method  generates  solutions  of  the  problem,  located  in  an  order  interval  with  the 
upper  and  lower  solutions  serving  as  bounds.  In  fact  the  method  is  often  coupled  with  a 
monotone  iterative  technique  which  provides  a  constructive  way  (amenable  to  numerical 
treatment)  to  generate  the  extremal  solutions  within  the  order  interval  determined  by  the 
upper  and  lower  solutions. 

In  this  paper  we  employ  this  technique  to  study  scalar  nonlinear  periodic  and  boundary 
value  problems.  The  overwhelming  majority  of  the  works  in  this  direction,  assume  that 
the  vector  field  is  continuous  in  all  variables  and  they  look  for  solutions  in  the  space 
C2(0,&).  We  refer  to  the  books  by  Bernfeld-Lakshmikantham  [2]  and  Gaines-Mawhin 
[6]  and  the  references  therein.  The  corresponding  theory  for  discontinuous  (at  least  in  the 
time  variable  t)  nonlinear  differential  equations  is  lagging  behind.  It  is  the  aim  of  this 
paper  to  contribute  in  the  development  of  the  theory  in  this  direction.  Dealing  with 
discontinuous  problems,  leads  to  Caratheodory  or  monotonicity  conditions  and  to 
Sobolev  spaces  of  functions  of  one  variable.  It  is  within  such  a  framework  that  we  will 
conduct  our  investigation  in  this  paper.  We  should  mention  that  an  analogous  study  for 
first  order  problems  can  be  found  in  Nkashama  [18]. 

107 


108  Nikolaos  S  Papageorgiou  and  Francesco,  Papalini 

2.  Sturm-Liouville  problems 

Let  T=  [0,6].  We  start  by  considering  the  following  second  order  boundary  value 
problem: 

/-^(0  =  /(rXO 
\  (Box)(0)  =  ID,  (B 

Here  (#oJc)(0)  =  a0;c(0)  -  coJc'(O)  and  (Bijc)(6)  =  ai*(6)  -f  c\x!(b\  with  a0,c0,ai; 
ci  >  0  and  a$(a\b  +  c\)  +  CQ^I  7^  0.  Note  that  if  c0  =  c\  =  z^o  =  ^i  =  0,  then  we  have 
the  Dirichlet  (or  Picard  in  the  terminology  of  Gaines-Mawhin  [6])  problem.  The  vectoi 
field/(f  ,  jc,  y)  is  not  continuous,  but  only  a  Caratheodory  function;  i.e.  it  is  measurable  in 
t  G  T  and  continuous  in  (jc,  y)  G  (R  x  R  (later  the  continuity  in  y  will  be  replaced  by  a 
monotonicity  condition).  Hence  jc"(-)  is  not  continuous,  but  only  an  L1(r)-function. 
Recently  Nieto-Cabada  [17]  considered  a  special  case  of  (1)  with/  independent  of  y. 
Also  there  is  the  work  of  Oman  [19]  where/  is  continuous. 

We  will  be  using  the  Sobolev  spaces  W1'1^)  and  W2'!(r).  It  is  well  known  (see  for 
example  Brezis  [3],  p.  125),  that  Wlfl(T)  is  the  space  of  absolutely  continuous  functions 
and  W2^(T)  is  the  space  of  absolutely  continuous  function  whose  derivative  is  absolutely 
continuous  too. 

DEFINITION 

A  function  ty  £  W2*l(T)  is  said  to  be  a  'lower  solution'  for  problem  (1)  if 

</^        ".  ^  T 


\ 
\' 


A  function  (f>  G  W2'1  (T)  is  said  to  be  an  'upper,  solution'  for  problem  (1)  if  the  inequalities 
in  (2)  are  reversed. 

For  the  first  existence  theorem  we  will  need  the  following  hypotheses: 
H(f)j:  /:TxRx(R->[Risa  function  such  that 

(i)  for  every  x,y  6  R,  t  —  »/(f,*,3>)  is  measurable; 
(ii)  for  every  t  €  T,  (x,y)  —»/(?,#,  y)  is  continuous; 

(iii)  for  every  r  >  0  there  exists  jr  €  LJ(r)  such  that  '\f(t,x,y)\  <  jr(t)  a.e.  on  T  for  all 
x,y€  (R  with  |jc|,|y|  <  r. 

HQ:  There  exists  an  upper  solution  <f>  and  a  lower  solution  ^  such  that  ^(r)  <  </>(t)  foi 
every  r  e  r  and  there  exists  h  G  C(R+,  (0,  oo))  such  that  |/(r,  jc,y)|  <  A(|y|)  for  all  r  G  7 
and  all  jc,y  G  [R  with  ^(f)  <  x  <  <f>(i)  and  J?°rrT  >  max,6r  0(r)  -  minr€r^(f),  with 

>  _  max[|V>(0)-^(fc)|,|^(f>)-0(0)n  u 

A~  b 

Remark.  The  second  part  of  hypothesis  HO  (the  growth  condition  on  /),  is  known  as  the 
'Nagumo  growth  condition'  and  guarantees  an  a  priori  L°°  -bound  for  jc'(-).  More 
precisely,  if  H0  holds,  then  there  exists  N\  >  0  (depending  only  on  </>,  ?/>,  h)  such  that  foi 
every  x  G  W2>l(T)  solution  of  -*"(*)  =  f(t,x(t),xf(t))  a.e.  on  T  with  ^(0  <  ^(0  <  </>(t) 
for  all  f  G  T,  we  have  \x!  (t)  \  <  NI  for  all  r  G  T  (the  proof  of  this,  is  the  same  (with  minoi 
modifications)  with  that  of  Lemma  1.4.1,  p.  26  of  Bernfeld-Lakshmikantham  [2]). 


Periodic  and  boundary  value  problems  109 

We  introduce  the  order  interval  K  =  [^,  0]  =  {x  e  Wl>2(T)  :  ^(t)  <  x(t)  <  0(0  for  all 
t  €  T}  and  we  want  to  know  if  there  exists  a  solution  of  (1)  within  the  order  interval  K. 
Also  we  are  interested  on  the  existence  of  the  least  and  the  greatest  solutions  of  (1)  within 
K  ('extremal  solutions').  The  next  two  theorems  solve  these  problems.  In  theorem  1  we 
prove  the  existence  of  a  solution  in  K  and  in  theorem  2  we  prove  the  existence  of 
extremal  solutions  within  K.  Although  the  hypotheses  in  both  theorems  are  the  same,  we 
decided  to  present  them  separately  for  reasons  of  clarity,  since  otherwise  the  proof  would 
have  been  too  long. 

Theorem  1.  If  hypotheses  H(f)1  and  HO  hold,  then  problem  (1)  has  a  solution  x  €  W2>1 
(T)  within  the  order  interval  K  =  [0,  </>]. 

Proof.  As  we  already  mentioned  in  a  previous  remark,  the  Nagumo  growth  condition  (see 
HO)  implies  the  existence  of  N\  >  0  (depending  only  on  -0,  0,  h)  such  that  1^(01  <  M  for 
all  /  6  T,  for  every  x  €  W2'l(T)  solution  of  (1)  belonging  in  K.  Set  N  =  1  4-  max 

{M,  Halloo*  Halloo}-  Also  define  the  truncation  operator  r:  Wl>l(T]  ->  Wl>l(T)  by 

(0(0   if   0«<*(0 

r(x)(t}={  x(t)     if     <KO<*(0<0(0- 
[V(0     if    x(t)<^(t) 

The  fact  that  r(x)  €  Wl>l(T)  can  be  found  in  Gilbarg-Trudinger  [8]  (p.  145)  and  we 
know  that 

f(t)   if   #0<*« 

(0     if     ^(0<*(0<*W- 

(0   if  *(0<^W 

Also  we  define  the  truncation  at  N  function  q#  G  C(R)  by 


and  the  penalty  function  u  :  T  x  R  —  >  R  by 

0(0     if 
0         if 
if 


Then  we  consider  the  following  Sturm-Liouville  problem 

f  -y(r)  =  /(r,T(x)(0,«v(rW;(r))).-  «(^(0)  a-e.  on  T  \ 

vi  y 


Denote  by  S  the  solution  set  of  (3). 

Claim  #7.  5  C  K  =  [^,  0].  Let  x  €  5.  Then  we  have 

-^(0  =  /(f,  r(x)(i),qN  (r(x)\t)}}  -  ii(r,  *(0)  a.e.  on  T.  (4) 


1  10  Nikolaos  S  Papageorgiou  and  Francesco  Papalini 

Also  since  j>  e  W2>l(T)  is  a  lower  solution  of  (1),  we  have 

/(f)  >  -/(*,  V>W,  ^(0)      a-e-  on  r- 

Adding  (4)  and  (5),  we  obtain 


'W)  -  «(*>*(*))      a-e-  on  r- 

Multiplying  with  (i/>  -  *)+(0  and  integrating  over  T  =  [0,  ft],  we  have 

/'V(0-A 

JO 

>  fb 

Jo 


From  the  integration  by  parts  formula  (Green's  identity),  we  have 

[Vw-A 

JO 


-    ^' 

JO 

Using  the  boundary  conditions  for  x  and  i/>  at  (  =  0,  we  have 


<  "o  = 


If  Co  =  0,   then 


and   so    (tf-*)+(°)  =  °-   Therefore   -(^- 


-  ^  (0))  <  -  * 

(0)  <  _  a  (^(o)  -  x(0))(^  -  *)+(0).  Thus  if  (</>(0)  -  x(0))  >  0,  we  have  -(V  - 
-*<  0   and   if    W0)-x(0))<0,    we   have    (^-*)+(0)  =  0   and    so 
=  0.  Therefore  we  always  have 


From  the  boundary  condition  at  t  =  fo,  we  have 

<v\  =  aix(b) 


Then  arguing  as  above,  we  infer  that 


Periodic  and  boundary  value  problems  111 

Finally  recall  that 

-*)'W   if 
0         if 

(see  Gilbarg-Trudinger  [8],  p.  145).  Hence  it  follows  that 

/  V  -  *')«(</<  -  *)'+(0<fc  =   "     [(V-  -  *)'«]2<fc  >  o.  (10) 

JO 

Using  (8),  (9),  (10)  in  (7),  we  deduce  that 


*)+«<fc<o.  (ii) 

Also  note  that 

(f(t,r(x)(t),qN(r(x)'(f)-))  -  f(t^(i)^'(t))}(^  -  x)+(t}& 
o 

=  f        (f(t,r(x)(t},qN(T(x}'(t)})  -  /M«,  </>'«)]«>  -*)(*)& 

J{x<^} 

=  f     [/(',  ^(0,^(0)  -/M(0^(0)](tf-*)«*  =  o          (12) 
*<v» 


since  on  the  set  {>  e  T  :  jt(f)  <  ^(f)}.  we  have  r(jc)(f)  =  i/>(t)  and  r(jc)'(f)  =  ^/(f).  Using 
(11)  and  (12)  in  (6),  we  have  that 


0<   /"  i 

JO 


(recall  the  definition  of  M(?,JC)).  So  ^(^)  <  jc(f)  for  all  f  €  T.  In  a  similar  way  we  show 
that  jc(f)  <  </>(t)  for  all/  G  T.  Therefore  5  C  K  as  claimed. 

Claim  #2.  S  is  nonempty.  This  will  be  proved  by  means  of  Schauder's  fixed  point 
theorem.  To  this  end  let  D  =  {x  €  W"2)1(r)  :  (fio*)(0)  =  z/o,  (^i^)(*)  =  ^1}  and  let 
L  :  D  C  L1  (T)  ~>  L1  (J)  be  defined  by  Lx  =  -jc77  for  every  x  £  D.  First  note  that  for  every 
h  €  ^(r)  the  boundary  value  problem 


.c.anTl 
4)  =  i/i  J 


\ 

has  a  unique  solution  jc  G  W^2)1(r).  Indeed  uniqueness  of  the  solution  is  clear.  For  the 
existence,  note  that  if  h  G  C(T),  then  it  follows  from  corollary  3.1  of  Monch  [15].  In  the 
general  case,  let  h  e  L1  (r)  and  take  hn  e  C(T)  such  that  hn  -~>  h  in  L1  (T)  as  n  -+  oo.  For 
each  hn,  n  >  1,  the  solution  *„(•)  of  (13)  is  given  by  xn(t)  =  u(t)  +  $G(t,s)(xn(s)- 
hn(s))As,  where  M  e  C2(T)  is  the  unique  solution  of  jc"(r)  =0  r  e  T,  (Bo*)(0)  =  ^o, 
(BIJC)(&)  =  1/1  and  G(f,j)  is  the  Green's  function  for  the  problem  x"  =  g(t}  teT, 
(Bo*)(0)  =  0,  (BiJc)(fc)  =  0  for  g  E  C(r)  given.  From  the  proof  of  corollary  3.1  (b)  of 
Monch  [15],  we  know  that  sup,^  H^H^  <  sup^  Halloo,  where  rjn  £  C2(T]  is  the 
unique  solution  ofrf'(t)  =  ~hn(t)  t  €  T,  (B<w)(0)  =  |^o|,  (*i  ??)(*)  =  |i/i|.  We  know  that 


112  Nikolaos  S  Papageorgiou  and  Francesco.  Papalini 

r;n(r)  ==  u(t)  -  Jo  G(r,  s)hn(s)ds  and  so  it  follows  that  supn>j  H^IL  <  oo.  Hence  {*„}„>! 
is  bounded  in  C(T).  Since  -xf^t)  =hn(t) -xn(t),  t  e  T,  it  follows  that  {*£}„>!  is 
uniformly  integrable.  From  Brezis  [3]  (p.  132)  we  know  that  the  norm  ||  •  ||W2,i(r\  is 
equivalent  to  the  norm  ||jc||  =  \\x\\l  +  H^'lli-  Therefore  {^}n>i  is  Bounded  in  W2^(T). 
Since  W2'1  (r)  embeds  continuously  in  C1  (r)  and  compactly  in  Ll  (T)  and  by  the  Dunford- 
Pettis  compactness  criterion,  by  passing  to  a  subsequence  if  necessary,  we  may  assume  that 
xn  ^  x  in  C1  (T)  (hence  xfn (t}  -»  xf  (t)  for  all  t  <G  T\  xn  -*  x  in  Ll  (T)  and  <  -^  y  in  L1  (7) 
as  «  — »•  oo.  Evidently  )>  =  Jt".  So  in  the  limit  as  n  — *•  oo,  we  have  —  xf'(t)  4-  x(t)  =  h(t)  a.e. 
on  I,  (Bo*)(0)  =  "o,  (#!*)(£)  =  vi.  Therefore  we  have  proved  that  R(I  +  L)  =  L{(T). 
Next  let  x\ ,  X2  €  />  and  ^  =  jci  —  #2-  Define 

r+  =  {t  e  T  :  jc(f)  >  0}     and     T_  =  {t  €  T  :  jc(0  <  0} 
both  open  sets  in  T.  For  A  >  0  we  have 

jc(r)-A*f/(0|dr+  /  \x(t)-Xx"(t)\dt 
T+  7r_ 


c(f)df-  f  x(t)dt-X  f  x"(t)dt  +  \  I  x"(t)dt 
T+  JT,  JT+  JT. 

=  /   \x(t)\dt-X\[  x"(t)dt-  f  x"(t)di\. 
Jo  UT+  JT,  J 


Let  (a,  c)  be  a  connected  component  of  r+.  Then  x(a)  =  *(c)  =  0  and  jc(r)  >  0  for  all 
te  (a,c).  Thus  xf(a)  >  0  and  x/(c)  <  0  and  from  this  it  follows  that  /flV(0<fr  = 
xf(c)  -x?(a)  <  0.  Therefore  we  deduce  that  /r  x"(/)df  <  0.  Similarly  we  show  that 
/r_  x?'(t)dt  >  0.  So  finally  we  have  -A[/r+  x?'(t)dt  -  JT_  xff(t)dt]  >  0  and  thus  we  obtain 

\x(t)-Xx"(t)\dt>      "  x(t)\dt 


o  o 

=»  l^i  +AI*!  -  (jc2  4- 

This  last  inequality  together  with  the  fact  that  R(I  -f  L)  =  L1  (r),  implies  that  (7  +  L)"1  : 
L1  (r)  —  >  Z)  C  L1  (r)  is  well-defined  and  nonexpansive  (is  the  resolvent  of  the  m-accretive 
operator  L;  see  Vrabie  [21],  Lemma  1.1.5,  p.  20).  For  k  >  0  consider  the  set 


Recalling  that  IWIj  +  11^11!  is  an  equivalent  norm  on  W2'1^)  see  Brezis  [3],  p.  132),  it 
follows  that  rk  is  bounded  in  W2>1  (T)  and  since  the  latter  embeds  compactly  in  L1  (r),  we 
conclude  that  Tk  is  relatively  compact  in  L1  (T).  So  from  Vrabie  [21]  (Proposition  2.2.1,  p. 
56),  we  have  that  (/  +  L)"1  is  a  compact  operator.  If  C  C  L1  (r)  is  bounded  and  u  e  C,  let 
x  =  (7  -h  L)~~  (u).  Then  -x!1  +x=  u  and  from  what  we  proved  we  have 


IWIi  <  II  -  *"  +*||i  <  supOHl!  :  u  e  C]  =  |C|  <  oo. 

So  |  |x"|  |  !  <  2|C|  and  thus  we  conclude  that  (/  -f-  L)"1  (C)  is  bounded  in  W2'1  (T).  Since 
the  latter  embeds  compactly  in  WU(T),  we  infer  that  (/  +  L)-1(C)  is  relatively  compact 
in  W1'1^).  Moreover,  if  un  ~~>  u  in  Ll(T)  as  n  -*  oo  and  xn  =  (I  +  Lr\Un),  then 


Periodic  and  boundary  value  problems  113 

xn  -+  x  =  (/  +  L)"1  (M)  in  L1  (7)  as  n  ->  oo  (recall  that  (/  +  L)"1  is  continuous  on  L1  (T)) 
and  {*«}„>!  is  bounded  in  W2'l(T).  Exploiting  the  compact  embedding  of  W2'l(T)  in 
W1'1  (7),  we  have  that  xn  -»  *  in  W1'1  (7),  i.e.  (7  +  L)""1  :  L1  (7)  ->  Z>  C  W1'1  (7)  is  conti- 
nuous, hence  a  compact  operator. 
Now  let  H  :  Wl>l(T)  -»  Ll(T)  be  defined  by 


We  will  show  that  //(•)  is  bounded  and  continuous.  Boundedness  is  a  straightforward 
consequence  of  hypothesis  H(f)1  (iii)  and  of  the  definition  of  the  penalty  function  u(t,x). 
So  we  need  to  show  that  H(-)  is  continuous.  To  this  end  let  xn  —  *  x  in  W1'1  (T)  as  n  —  >  oo. 
By  passing  to  a  subsequence  if  necessary,  we  may  assume  that  xn(t)  —  >  x(t)  and 
x!n(t)  —>  xf(t)  a.e.  on  T  as  n  —  *  oo.  Hence  we  have  r(xn)(t)  —  >  r(;c)(r)  for  every  r  G  7  and 
#AKT(*n)'(0)  "*  ^(TW/(0)  a-e-  on  7  as  ft  —>  oo.  Note  that  {xn}n>i  is  bounded  in  C(7) 
(since  Wli!(7)  embeds  continuously  in  C(7))  and  so  by  virtue  of  hypotheses  H(f){,  the 
continuity  of  w(f  ,  •)  and  the  dominated  convergence  theorem,  we  have  that  H(xn)  ->  /f  (jt) 
in  L*(7)  as  n  -»  oo  and  so  we  have  proved  the  continuity  of  H  :  W1'1(7)  —  >  L[(7). 

Then  consider  the  operator  (I  +  L)~1H  :  Wl*l(T)  ->  WM(7).  Evidently  this  operator 

is  continuous  (in  fact  compact),  (/  +  L)~1H(D)  C  D  and  (I  +  L)~1H(D)  is  compact  in 


Wl>l(T)  (since  for  every  x  €  Wl*l(T),  \\H(x)\\{  <  If  with  **  =  l 
Halloo)  and  r  =  maxdl^H^,  ll^iL^})-  since  D  £  ^ljl(7)  is  closed,  convex,  we  can 
apply  Schauder's  fixed  point  theorem  (see  Gilbarg-Trudinger  [8],  Corollary  10.2,  p.  222), 
to  obtain  x  =  (/  +  L)"1^).  Then  -jc77  -j-  x  =  H(JC),  jc  G  D;  i.e.  x  G  W2'1  (7)  is  a  solution 
of  (3).  This  proves  the  nonemptiness  of  S. 

To  conclude  the  proof  of  the  theorem,  note  that  if  x  G  S,  then  from  claim  #  1  we  have 
$(t)  <x(t)  <  <j)(t)  for  all  f  €  7.  So  we  have  r(x)(t)  =*(0.  r(x)'(t)  =xf(t)  and  w(?, 
x(t))  =  0.  Also  recalling  that  ^(/Jl  <  A^  for  all  t  G  7,  we  also  have  that  qi^(xf(t))  =xf(t). 
Therefore  finally 

r-AO=/(',*«y«)        a.e.on7\ 
\  (Box)(0)  =  i^),  (Bi*)(*)  =  1/1  J 

i.e.,  x  G  W2'l(T)  solves  problem  (1)  and  x  G  [?/>,  </>]. 

Now  we  will  improve  the  conclusion  of  theorem  1,  by  showing  that  problem  (1)  has  I 

extremal  solutions  in  the  order  interval  K  —  [?/>,  </>];  i.e.  there  exist  a  least  solution  x*  G  K 
and  a  greatest  solution  jc*  G  K  of  (1),  such  that  if  jc  G  W2)1(7)  is  any  other  solution  of  (1) 
in  K,  we  have  x*(t)  <  x(t)  <  jc*(r)  for  all  t  G  7. 


Theorem  2.  If  hypotheses  H(f)t  <?nd  HO  fto/d,  then  problem  (1)  /z<2s  extremal  solutions  in 
the  order  interval  K  =  [?/>,  </>]. 


Let  S\  be  the  set  of  solutions  of  (1)  contained  in  the  order  interval  K  =  [?/>,  (/>]. 
From  theorem  1  we  have  that  S\  ^  (p.  First  we  will  show  that  S\  is  a  directed  set  (i.e.  if 
*i,*2  €  S\,  then  there  exists  jc  G  Si  such  that  xi(t)  <  x(t)  and  X2(t)  <  x(t)  for  all  t  G  7). 
To  this  end  let  xi,xi  G  Si  and  let  x^  =  max{^i,jc2}.  Since  xi,X2  G  W2il(7),  we  have  that 
;c3  G  Wl>l(T)  (see  Gilbarg-Trudinger  [8],  Lemma  7.6,  p.  145).  Let  rk  :  W1'1(7)  -* 
W1'1  (7)  be  defined  by 


if 

if  '*  (0  <  XO  <  0(0      *  =  i,  2,  3  . 

if    jc<^ 


1  14  Nikolaos  S  Papageorgiou  and  Francesca  Papalini 

Also  we  introduce  the  penalty  function  u3  :  T  x  R  -»  R  and  the  truncation  function 

qN  :  R  -*  R  (N  =  1+  max{M,  IhA'lL'  110'lloJ)  defined  bY 


-  <j>(t)     if 

=  {  0  if    *3(0  <*<  0(0 

kx-*3(0     if    Jt<*3(0 

and 

{#      if    #<;c 
x        if     -N<x<N. 
-N    if    *<-# 

Then  we  consider  the  following  boundary  value  problem: 

2 


-f(t,T3(x}(t),qN(T3(x)'(t)})\-  «3(r,*(0)        a.e.  on  T 
[  (Bo*)(0)  -  n>,  (Bi*)(&)  =  1/1 

(14) 

Arguing  as  in  the  proof  of  theorem  1,  we  establish  that  problem  (14)  has  a  nonempty 
solution  set.  We  will  show  that  this  solution  set  is  in  the  order  interval  [*3,0].  So  let 
x  €  W2il(T)  be  a  solution  of  (14).  We  have 


(0, 

-  w3(r,jc(r))     a.e  on  T. 


Multiply  with  (x\  —  x)+(t)  and  then  integrate  over  T  =  [0,  fc].  Using  the  definition  of  the 
truncation  functions  rk(k=  1,2,  3),  ##  a&d  boundary  conditions,  we  obtain 


rb 

=»   /    (jci  —  jc)i(r)df  =  0        (recall  the  definition  of  MS) 
JQ 

^xi(t)<x(t)     foraUrer.        a.e.  on  /. 

In  a  similar  way  we  show  that  x2(t)  <  x(t)  and  x(f)  <  (j)(t)  for  all  t  G  T.  Therefore  we 
conclude  that  every  solution  jc(-)  €  W2'1  (J)  of  (14)  is  located  in  the  order  interval  [xs,  ^]. 
Hence  rfe(x)(r)  =  jc(0  and  ^(^'(r)  =  xf(t)  for  aU  reT  and  all  jfc  €{1,2,3}  and 
u3(t,x(t))  =0.  Thus 

a.e.  on 


As  we  already  mentioned  the  Nagumo  growth  condition  (see  (Ho))  guarantees  that 
\xf(t)\  <  N  for  all  t  €  T  and  so  ##  (j</(f))  =  x'(f).  Therefore  x  G  Si  and  we  have  proved 
that  Si  is  a  directed  set. 


Periodic  and  boundary  value  problems  115 

Now  let  C  be  a  chain  in  Si .  Then  since  C  C  L1  (r),  according  to  Dunford-Schwartz  [5] 
(Corollary  IV.IL7,  p.  336),  we  can  find  {xn}n^  C  C  such  that  sup  C  =  sup^  xn.  Then  by 
the  monotone  convergence  theorem,  we  have  that  xn  — >  x  in  Ll(T)  as  n  -»  oo  and  so 
^(t)  <  x(t)  <  (j)(t)  a.e.  on  T.  For  every  n  >  I  we  know  that  H-x^l^  <  max-dl^H^, 
Halloo}  =  ro  and  supn>!  H^'JI^  <  N\.  So  if  r  —  max{r0, N\},  by  virtue  of  hypothesis 
H(f){  (iv)  we  have  that  |K(OII  <  7r(0  a.e.  on  T.  Thus  {*„}„>!  is  bounded  in  W2>l(T) 
and  {^/}n>1  is  uniformly  integrable.  So  as  before  exploiting  the  compact  embedding  of 
W2'{(T]  in  Wlsl(r),  the  continuous  embedding  of  W2^(T)  in  Cl(T)  and  invoking  the 
Dunford-Pettis  theorem,  we  may  assume  that  xn  — »  x  in  Wljl(r),  xn(t)  — >  jc(f),  jc^(r)  — » 
*'(r)  for  all  t  £  T  and  ^  -^  v  in  L!(r)  as  n  — >  oo.  It  is  easy  to  see  that  y  —  x"  and 
(Box)(Q)  =  r/o,  (BiJt)(fe)  =  i/i.  Also  from  the  dominated  convergence  theorem,  we  have 
that  -x"(-)=f(.,x(-),x'(-))  in  L1^).  Hence  ~x"(t)  =  f(t,x(t),xf(t))  a.e.  on.  T, 
(5o^)(0)  =  VQ,  (B\x)(b)  =  z/i.  Thus  .x  =  supC  €  Si.  Using  Zorn's  lemma,  we  infer  that 
Si  has  a  maximal  element  x*  &  S\.  Since  Si  is  directed,  it  follows  that  x*  is  unique  and  is 
the  greatest  element  of  Si  in  [?/?,  </>].  Similarly  we  can  prove  the  existence  of  a  least 
solution  #*  of  (1)  in  [-0,  (/>].  Therefore  (1)  has  extremal  solutions  in  K  =  [-0, 0], 

3.  Periodic  problems 

In  this  section,  we  focus  our  attention  on  the  'periodic  problem': 
f  -x"(t]  =  f(t,x(t),x?(i))        a.e.  on  T  ) 


This  problem  was  studied  using  the  method  of  upper  and  lower  solutions  by  Gaines- 
Mawhin  [6],  Leela  [14],  Lakshmikantham-Leela  [13],  Nieto  [16],  Cabada-Nieto  [4], 
Omari-Trombetta  [20]  and  Gao-Wang  [7].  From  these  works  only  Gaines-Mawhin, 
Cabada-Nieto,  Omari-Trombetta  and  Gao-Wang  had  a  vector  field  depending  also 
on  x1  and  moreover,  among  these  papers  only  Cabada-Nieto  and  Gao-Wang  used 
Caratheodory  type  conditions  on/(f,jc,y)  with  Lipschitz  continuity  in  the  y- variable  in 
Cabada-Nieto  (see  Theorem  2.2  in  Cabada-Nieto  [4]).  Theorem  3  below  extends 
all  these  results.  A  similar  result  using  a  different  method  of  proof,  was  obtained  by 
Gao-Wang  [7]. 

DEFINITION 

A  function  ^  €  W2}1(r)  is  said  to  be  a  'lower  solution'  of  (18)  if 

a.e.  on  T 


A  function  </>  £  W2'l(T)  is  said  to  be  an  'upper  solution'  of  (18)  if  it  satisfies  the 
reverse  inequalities. 

Theorem  3.  If  hypotheses  H(f)1  and  HO  hold,   then  problem  (18)  has  a  solution 
x  e  W2>l(T)  within  the  order  interval  K  =  ty>,  $. 

w 

Proof.  The  proof  is  the  same  as  that  of  theorem  1,  with  some  minor  modifications.  Note  || 

that  in  this  case  D  =  {x  G  W2'l(T]  :  Jt(0)  =  x(b),xf(Q)  =  xf(b)}  and  L  :  D  C  Ll(T)  -> 
L1  (r)  is  defined  by  Lx  =  —  xf/  for  all  x  G  D.  The  rest  of  the  proof  is  identical  and  only  in 


116  Nikolaos  S  Papageorgiou  and  Francesca  Papalini 

the  applications  of  the  integration  by  parts  formula  (Green's  identity),  we  use  the  periodic 
conditions  instead  of  the  Sturm-Liouville  boundary  conditions. 

Next  we  look  for  the  extremal  solutions  in  the  order  interval  [fa  0]  of  the  periodic 
problem  (18).  For  this  we  introduce  a  different  set  of  hypotheses  on  the  vector  field 

f(*,x,y). 

H(f  )2:  /:rxRxR-+[Risa  function  such  that 

(i)  for  every  x,y  <E  R,  t  ->f(t,x,y]  is  measurable; 

(ii)  there  exists  M>0  such  that  for  almost  all  t£T  and  all  y€[-N,N],  x-> 

f(t,x,y}+Mx  is  strictly  increasing  (recall  that  N=l+  max{M,  H^ILi  Halloo}); 
(iii)  there  exists  k  E  Ll  (T)  such  that  \f(t,x,yi)  -  f(t,x,y2)  \  <k(t)\yi  -y2|  a.e.  on  Tfor 

aUjc,yi,)?2eR; 
(iv)  for  every  r  >  0,  there  exists  7,  E  Ll  (T}  such  that  \f(t,x,y)\  <  7r(f)  a.e.  on  T  for  all 

*,yeR,|x|,  |y|<r. 

Remark.  Hypothesis  H(f)2  (ii)  allows  for  jump  discontinuities  (countably  many)  in  the  x- 
variable.  However  note  that  for  every  x  :  T  -»  R  measurable,  t  -+f(t,x(i),y)  is 
measurable.  This  is  an  immediate  consequence  of  Theorem  1.9,  p.  32  of  Appell- 
Zabrejko  [1].  Moreover  since  (t,y)  -*f(t,x(t),y)  is  a  Caratheodory  function,  is  jointly 
measurable  and  so  in  particular  superpositionally  measurable;  if  y  :  T  —  >  R  is  measurable, 


Theorem  4.  If  hypotheses  H(f)2  and  HO  hold,  then  problem  (18)  has  extremal  solutions 
in  the  order  interval  K  =  [fa  </>]. 

Proof.  Without  any  loss  of  generality,  we  may  assume  that  M  >  1.  Then  for  any  z  €  K  = 
[fa  <£],  we  consider  the  following  periodic  problem 

*"W  =  f(t,z(t),qN(r(x)'(t))}  -  u(t,x(i))  +M(z(t)  -x(t))    a.e.  on  T  1 

o)=*(6),y(o)=y(fe)  J* 

(16) 

We  will  establish  the  existence  of  solutions  for  problem  (18).  So  let  D  =  {x  e  W2'1 
(r)^(0)=x(6),y(0)=^(6)}  and  let  L  :  D  C  Ll(T)  ->  Ll(T)  be  defined  by  Lx  = 
-J^'  +  (M  -  l)x.  As  in  the  proof  of  theorem  1,  we  can  check  that  L  is  invertible  and 
L  :Lii^  ~~*D~  wM(r)  is  a  compact,  linear  operator.  Also  as  before  we  define 


TMs  map  is  bounded  and  continuous.  Note  that  x  e  D  solves  (19)  if  and  only  if 
x  -  L    H(x).  As  in  the  proof  of  theorem  1,  the  existence  of  a  fixed  point  of  L'1//  is 
mighed  by  corollary  10.2,  p.  222  of  Gilbarg-Trudinger  [8],  since  L~l(D}CD  and 
L~H(D)  is  compact  in  Wl>l(T).  So  problem  (19)  has  solutions. 
Now  we  will  show  that  any  solution  of  (19)  is  within  K  =  [fa  $.  Indeed  we  have: 


a.e.  on  T 


Periodic  and  boundary  value  problems 


117 


Multiplying  the  above  inequality  with  (^  -  x)+(t)  and  integrating  over  T  =  [0,  b]  as  in 
the  proof  of  theorem  2,  using  the  definitions  of  r,  q^  and  the  boundary  conditions  for  ip 
and  jt,  we  obtain  that 


0<  /  «(,, 
Jo 


=  -  /  [(^- 
Jo 


=»  V>(0  <  *W        for  all  r  G  T. 

In  a  similar  fashion,  we  show  that  jc(f)  <  0(r)  for  all  r  €  T.  Therefore  every  solution 
x  G  W^(T)  of  (19)  is  located  in  K  =  [^,  </>].  Thus  recalling  the  definitions  of  T(JC),  ^  and 
M,  we  see  that  -*"(*)  ==  /(*,*(/),  x'(f))  +  M(x(t)  -x(i))  a.e.  on  T,  jc(0)  =  x(b\  ^(0)  = 
xf(b).  Now  we  will  show  that  this  solution  is  unique.  To  this  end,  on  L1  (r)  we  consider  an 
equivalent  norm  |  •  \l  given  by 

fb       (        fl  \ 

|*|  i  =  /    exp    -A  /   k(s}ds    \x(t}\dt,     A  >  0. 

Jo         V      Jo  / 

Similarly  on  W2)l(r)  we  consider  the  equivalent  norm  given  by 

W2,i  =  Wi  +  Mi  +  Mi- 

Suppose  that  JCi,jC2  6  W2il(r)  are  two  solutions  of  (19).  Then 
)     and    x2  =  L^lH0(xi), 


where  L^1  =  (MI  4-  L)"1  with  Lx  =  -/'  for  all  jc  €  D  =  {x  €  W2'1  (r)  :  ;c(0)  =  x(b), 


and  ffoW(-)=/(-,z(0»^(r(jc)/(-))).  Recall  that  Z^ 


is  linear  compact.  So  L^1  :  (Ll(T],  \ 


(IV2'1  (r),  | 


2|1 


is  linear  contin- 


| 
uous.  Moreover,  using  hypotheses  H(f)2  we  can  easily  check  as  before  that  HQ  : 

(W^l(T),  |  •  |2jl)  -*  (Ll(T),  |  •  h)  is  continuous.  Then  we  have 


exp 


-A  r 
Jo 


<  ||L^||£  Texpf 
Jo         V 


exp-A 


Tll^1ll 
A  Jo 


<Til^MlL  / 
A  Jo 


£l 


2l' 


So  if  A  >  IIL 
all  r  €  T,  with 


II^,  we  infer  that  xf{(t)  =  x%(t)  a.e.  on  r.  Hence  ^(t)  -  xf2(t)  =  d  for 
G  R.  Since  ^(0)  =  xf^b)  and  4(0)  =4(fc),  from  the  mean  value 


118 


Nikolaos  S  Papageorgiou  and  Francesca  Papalini 


theorem,  we  deduce  that  there  exists  £  G  (0,  b)  such  that  x\  (f  )  =  4(£)-  Therefore  c\  =  0 
and  so  xfl  (t)  =  j£(0  for  all  t  G  T,  which  implies  that  x\  (t)  -  x2(t)  =  c2  for  all  t  G  T,  with 
c2  €  R.  But  for  almost  all  f  €  T,  we  have 

0  -*i  (0)  =  /(',z( 


a.e.  on  T;  i.e.  c2  =  0  and  so  #1  =  x2. 


=>  *i      = 


Then  define  /?  :  [?/>,  </>]  —  >  [V7,  </>]  where  /?(z)(-)  is  the  unique  solution  of  (19).  We  claim 
that  R(-)  is  increasing.  Indeed  let  zi,z2  £  [&<t>]i  Zi  ^  *2,  Zi  7^  Zi  and  set  jci  =  #(zi), 
jc2  =  /?(z2).  We  have 


a.e.  on 


and 


*2(0) 


.  on  7. 


Suppose  that  maxrej[zi  (?)  —  JC2(0]  =  ^  >  0  and  suppose  that  this  maximum  is  attained 
at  to  G  T.  First  we  assume  that  0  <  fy  <  b.  Then  we  have  j^  (to)  =  ^2(^o)  =  ^o  and  we  can 
find  <5  >  0  such  that  for  every  t  G  T^  =  [?o,  fe  4-  ^]  we  have  *2(f)  <  ^i  (r).  So  we  obtain 


a.e.  on  r5,  with  w(t)  =  z2(0  - 


and 


a.e.  on 


Since  ^(^o)  =  ^(fy)  =  z/o»  froni  a  well-known  differential  inequality  (see  for  example 
Hale  [9],  theorem  6.1,  p.  31),  we  obtain  that  0  <  xfl(t)  -  4(0  for  a11  r  ^  TS-  So  after 
integration  we  see  that  x\  (r0)  -  *2(r0)  <  ^i  (0  ~  *2(0  for  everY  f  ^  T6-  Since  ?0  G  T  is  the 
point  at  which  (x\  -  ^c2)(*)  attains  its  maximum  on  T,  we  have  that  jci  (0  =  ^(0  +  ^  for 
every  t  €  7>  and  so  xfl  (t)  =  xf2(t)  for  every  t  €  7$.  Thus  we  have 


4-  Af  (jci  (0  -  JC2(0)  >  0     a.e.  on  Ts, 

a  contradiction. 

Next  assume  f0  =  0.  Then  £  ==  ^i(O)  -  ;c2(0)  >  xi(h)-x2(h)  for  all  h  e  [0,5]  and 
e  =  x\(b)  -x2(b)  >  xi(h)  ~x2(h)  for  all  h€[b-6,b].  From  the  first  inequality  we 
infer  that  (x\  -x2)f(0)  <  0  while  from  the  second  we  have  (*i  -x2)'(b)  >  0  and  so 
(*i  -  ^2)'(0)  >  0.  Therefore  xfl  (0)  =  xf2  (0)  =  z/0  and  so  we  can  proceed  as  in  the  previous 
case  and  derive  a  contradiction.  Similarly  we  treat  the  case  to  =  b.  Therefore  x\  <  x2  and 
so  R(-)  is  increasing  as  claimed. 

Now  let  {yn}n>i  be  an  increasing  sequence  in  [^,0].  Set  xn  =R(yn),  n>l.  The 
sequence  {xn}n^l  C  [t/j,  0]  is  increasing.  From  the  monotone  convergence  theorem,  we 
have  that  yn  ~>  y  and  *„  ->  x  in  L\T)  as  n  -*  oo.  Also  by  hypothesis  H(f)2  (iii), 
K(OI  <  7r(0  a.e.  on  T  with  r  -  max{^,  H^,  1^1^},  with  7r  G  Ll(T).  So  {jc^}^!  is 
bounded  in  W2>1  (T)  and  {X,'}n>i  is  uniformly  integrable.  From  the  compact  embedding  of 


Periodic  and  boundary  value  problems  119 

W2'1  (r)  in  Wl>1  (T)  and  the  Dunford-Pettis  theorem,  we  have  that  xn  ->  x  in  W1'1  (J)  and 
at  least  for  a  subsequence  we  have  xf£  -^  g  in  L1  (r)  as  /i  —  >  oo.  Clearly  jc"  =  g  and  so  for 
the  original  sequence  we  have  jc^  ^xff  in  L^T)  as  n  —  »  oo.  So  finally  *„  -^>;t  in  W2)1(r). 
Invoking  theorem  3.1  of  Heikkila-Lakshmikantham-Sun  [10],  we  deduce  that  /?(•)  has 
extremal  fixed  points  in  K  =  [-0,  <£].  But  note  these  extremal  fixed  points  of  jR(-),  are  the 
extremal  solutions  in  K  =  [T/;,  (/>]  of  the  periodic  problem  (19). 

Next  we  consider  the  situation  where  the  vector  field/  is  independent  of  V.  This  is  the 
case  studied  by  Nieto  [16].  However  here  we  are  more  general  than  Nieto,  since  the 
dependence  of/  on  x  can  be  splitted  into  a  continuous  and  a  discontinuous  part.  So  we 
will  be  studying  the  following  periodic  problem: 


x'(0)=x'(*) 

HQ:  There  exist  ip  G  W2)1(r)  a  lower  solution  and  0  G  W2'1^)  an  upper  solution  such 
that  i/)(t)  <  4(t)  for  all  t  €  T. 

H(f)3:  /  :TxIRx[R-»Risa  function  such  that 

(i)  for  every  ;y  G  W2>l(T)  and  every  x  G  R,  f  —  ^/(^,Jt,XO)  is  measurable; 
(ii)  for  almost  all  t  G  T  and  all  y  G  R,  x  —>f(t,x,y)  is  continuous; 
(iii)  there  exists  M  G  L!(r)+  such  that  for  almogt  all  t  G  T  and  all  x  G  W>(f)><KO]» 

3^  —*f(t,x,y)  +M(t)y  is  increasing; 

(iv)  for  every  r  >  0  there  exists  7r  G  L^J)  such  that  if  \f(t,x,y)\  <  7r(r)  a.e.  on  T  for 
all^yG  Rwith  \x\,\y\  <  r. 

Remark.  The  superpositional  measurability  hypothesis  H(f)3  (i)  is  satisfied,  if  for  every 
x  G  K,  there  exists  gx:T  xU—*U  a  Borel  measurable  function  such  that  gx(t,y)  = 
f(t,x,y)  for  almost  all  te  T  and  all  y  G  R.  This  follows  from  the  monotonicity 
hypothesis  H(f)3  (iii)  and  theorem  1.9  of  Appell-Zabrejko  [1]. 

Theorem  5.  If  hypotheses  HQ  and  H(f)3  hold,  then  problem  (23)  has  a  solution 
x  G  Wz>1  (T)  in  the  order  interval  K  =  [^,  <f>]. 

Proof.  Let  y  G  K  =  ty>,  0]  =  {y  G  W2^(T]  :  ^(f)  <  y(t)  <  <f>(t)  for  all  t  G  T}  and 
consider  the  following  periodic  problem 

xt      a.e.onr 


Problem  (24)  has  at  least  one  solution  in  K  (see  Nieto  [16]).  By  S(y)  we  denote  the 
solutions  of  (24)  in  K.  Let  yi,)>2  G  K,  y\  <  y%,  x\  G  S(y\)  and  y\  <x\.  Consider  the 
following  problem: 

-ui(t,x(t})     a.e.  on  T  \.  (19) 


(the  truncation  function)  is  defined  by 

if  <£(/).  <  x 
x  if  jci(r)<jc 
i  (0  if  x<xi(0 


120 


Nikolaos  S  Papageorgiou  and  Francesca  Papalini 


and  MI  :  T  x  R  -+  R  (the  penalty  function)  is  defined  by 

*-0(0     if     0(0  <* 

0          if    *i(0<*<0(0- 
*-jci(0     if    *<*i(0 

Both  are  Caratheodory  functions.  As  before  we  let  D  =  {x  €  W2>l(T)  :  x(0)  =  x(b), 
xf(0)  =  Jc7^)}  and  define  L  :  D  C  Ll(T)  ~>  L^J)  by  Lx  =  ~x"  for  all  x  G  D.  Again  we 
can  check  that  L=  (7  +  L)  is  invertible  and  L"1  :Ll(T}-+DC  Wl^l(T)  is  compact, 
Also  H  :  Wl>l(T)  ~>  Ll(T)  is  given  by 

#(jc)(0=/M(^j^ 

This  map  is  continuous  and  there  exists  fc*  >  0  such  that  H//MH!  <  &*  for  all  x  G  IV1'1 
(r).  So  L~1H(D)  is  relatively  compact  in  W1'1^)  and  thus  we  can  apply  corollary  10.2 
p.  222,  of  Gilbarg-Trudinger  [8]  and  obtain  x  6  D  such  that  x  =  L~{H  (x).  Therefore 
problem  (25)  has  a  solution. 

Note  that  by  virtue  of  hypothesis  H(f)3  (iii)  and  the  fact  that  n(t,x\(t))  =  xi(t)  anc 
Mi(r,jci(0)  =0,  we  have 


a-e-  on 


So  ^i  G  W2?1(r)  is  a  lower  solution  of  (25).  Similarly  since  y2  <  0,  we  have 


a.e.  on 


and  so  we  see  that  <f>  €  W2>l(T)  is  an  upper  solution  of  (25). 

Now  we  will  show  that  the  solutions  of  (25)  are  within  the  order  interval  £j  =  [jcj  ,  <f>] 
Indeed  we  have 

J[(t)  -x"(t)  =f(t,Tl(t,x(t)),y2(t))  +M(t)y2(t)-f(t,Xl(t),yi(t)) 
-M(t}yi(t)  +M(t)(Xl(t)  -n(t,x(t)))  -ui(t,x(t)) 
a.e.  on  T. 

Multiply  the  above  equation  with  (x\  -  Jt)+(-)  and  then  integrate  over  T  =  [0,fc].  As  ii 
previous  proofs  we  obtain 


fb 

I 

JQ 


-  /    [(xi  -  x},(t)}2dt  >  0;     i.e.  xi(t)  <  x(t)    for  all  t  G  T. 
JQ 

Similarly  we  show  that  x(t)  <  <t>(t)  for  all  f  G  T.  Therefore  every  solution  of  (25)  is  ii 
the  order  interval  K\  =  [jti,  </>].  Because  of  this  fact,  equation  (25)  becomes 

-x!'(t)  =  f(t,x(t),y2(t))  +  M(t}(y2(t]  - *(*))     a.e.  on  T 
x(0)=x(b), 

and  so  jc  G  S(y2)  and  x\  <  x. 


Periodic  and  boundary  value  problems  121 

Next  we  will  show  that  for  every  3;  £  K  =  [^,  </>],  the  set  S(y)  is  compact  in  Ll(T).  To 
this  end  let  x  G  S(y).  Then  \\x\\,,  <  max{|  |</>|Uh/>|  L}  =  r.  Hence  ||*"(f)j|  < 
7r(f)  -f  2M(t)r  a.e.  on  71.  Hence  5(y)  is  bounded  in  W2)1(r)  and  since  the  latter  embeds 
compactly  in  Ll(T),  we  have  that  S(y)  is  relatively  compact  in  Ll(T).  Then  let 
{•*n}«>i  £  5(y)  and  assume  that  xn  -+  x  in  Ll(T)  as  rc  >  oo.  Since  (X[}n>1  is  uniformly 
integrable,  by  passing  to  a  subsequence  if  necessary  we  may  assume  that  jc^'  —  >•  g  in  Ll  (T) 
as  n  —  »  oo.  Because  W2'^!")  embeds  continuously  in  Cl(r),  {^}n>!  is  bounded  in  C(T) 
and  for  all  0  <  s  <  t  <  b  and  all  n  >  1,  ^(r)  -  xfn(s)\  <  £(7r(r)  +  2M(r)r)dr  from 
which  it  follows  that  {xfn}n>i  is  equicontinuous.  So  by  the  Arzela-Ascoli  theorem  we 
have  that  xfn  —  >  xf  in  C(T)  as  «  -»  oo  and  so  g  =  jc".  Then  via  the  dominated  convergence 
theorem,  as  before,  we  can  check  that 

-x"(0  =  f(t,x(t),y(t))+M(t)(y(t)  -*(r))    a.e.  on  r 


Hence  x  €  5(y)  and  this  proves  that  S(y]  is  closed,  hence  compact  in  Ll(T).  Since  the 
positive  cone  Ll(T}+  =  {x  €  Ll(T)  :  x(t)  >  0  a.e.  on  T}  is  regular  (in  fact  fully  regular; 
see  Krasnoselskii  [12]),  from  proposition  2  of  Heikkila-Hu  [11],  we  infer  that  S(-)  has  a 
fixed  point  in  K;  i.e.  there  exists  x  €  K  =  [ip,  <p]  such  that  x  €  S(x).  Therefore 


a.e.  on 


and  so  problem  (23)  has  a  solution  in  K  =  fyb,  </>}. 

4.  Properties  of  the  solutions 

For  problems  linear  in  jc7,  we  can  say  something  about  the  structure  of  the  solution  set  of 
the  periodic  problem.  Our  result  extends  theorem  4.2  of  Nieto  [16]. 
The  problem  under  consideration  is  the  following: 

a.e.  on  T 


Our  hypotheses  on  the  vector  field  f(t,x)  are  the  following: 
H(f)4:  /:TxlR-^Risa  function  such  that 

(i)  for  every  x  G  R,  t  — >  /(r,  jc)  is  measurable; 
(ii)  for  almost  all  t  £T,  x  —>f(t,x)  is  continuous  and  decreasing; 
(iii)  for  every  r  >  0  there  exists  7r  G  L°°(r)  such  that  \f(t,x)\  <  jr(t)  a.e.  on  T  for  all 

Jc  €  R,  |*|  <  r. 

Remark.  Under  these  hypotheses  the  Nagumo  growth  condition  is  automatically  satisfied 
since  for  k  =  maxflMI^,  ||^| L},  we  have  |/(f,x)  4-  My|  <  ^(r)  4-  M\y\  a.e.  on  T  for  all 
x  e  [ip(t),  </>(t)],  and  so  if  h(r)  =  Halloo  +Mr,  we  have  for  all 


roc      r  r 

>0A  W)dr'Jx 


Theorem  6.  If  hypotheses  HQ  a«^/  H(f)4  to/J  anJ  M  >  0,  then  the  solution  set  S  of  (30) 
in  K  =  [?/>,  <^>]  w  nonempty,  w-compact  and  convex  in  W2'1  (T). 


122  Nikolaos  S  Papageorgiou  and  Francesca  Papalini 

Proof.  From  theorem  1  we  know  that  S  ^  (/>.  Let  x  €  S  and  define  x(t  )  =  x(t)  -  \  jj  x(t)dt 
t  €  T.  Let  TO  =  {x  €  R  :  i  +  c  €  S}.  Note  that  7b  ^  </>,  since  c  =  £  jj  x(t)dt  £  T0.  We 
claim  that  TQ  is  an  interval.  Indeed  let  ci,c2  e  TO,  c\  <  ci  and  take  c  €  (ci,c2).  Set 
y  =  x  -f  c.  We  have 


=  /(*,  (*  +  c2)(0)  +  M(*  +  <*)'(')  a.e.  on  T. 
By  hypothesis  H(f)4  (ii),  we  have 

2  a.e.  on  T 


-y"(t)  =  f(t,y(t))  +  My'  (r)        a.e.  on  T. 


Also  it  is  clear  that  y(0)  =  y(fc)  and  y(0)  =  y'(b).  Therefore  y  €  5  and  so  c  6  TO, 
which  proves  that  TO  is  an  interval. 
Next  we  will  show  that  S={.x  +  c:c€7b}.  Indeed  if  v,  x  €  S,  then  we  have 


=  (/(/,  z/(0)  +MZ/(O  -  /MO)  -  M*'(0)  WO  -  i/(0) 

-  (f(*XO)  -/MO))  WO  -  KO)  +  M(i/(r)  -^(0)WO  -  K 

>  M(z/(0  -  ^(0)  WO  -  ^(0)     a-e"  °n  ^ 
Integrating  over  T  =  [0,  ft],  we  obtain 

rb  fb 

Jo  Jo 

>M 


=>  xf(t)  =  i/(t)  for  every  t  €  T 
=>  (x  -  z/)(-)  =  constant. 

So  indeed  S  =  {i  +  c  :  c  E7b}  and  since  as  we  saw  earlier  TO  is  an  interval,  we  deduce 
that  S  is  convex. 

Finally  we  will  prove  that  S  is  w-compact  in  W2'1  (T).  To  this  end,  let  v  G  5.  Then  there 
exists  k  e  TO  such  that  y  =  £  -f  k,  hence  ||y||2  i  =  I |i  +  *||2 1-  Since  y  £  K  =  [-0, 0],  we 
have  |*|  <  max{||^||00  -I- 1^1^,  H^  +  H^f^}  =  77.  Therefore  ||y||21  <  \\x\\ l  -f  b\k\ 
+ll^lli  +  ll^lli  <  ||*||2,i  +  ^  and  so  S  is  bounded  in  W2^(T).  We  will  show  that  S  is 
closed  in  W^l(T}.  Let  {yn}n>l  C  5  and  assume  that  vn  -*  y  in  W^\T).  We  have 

-^(0  =  /fcynW)  +^nW     a.e.  on  r,  n  >  1.  (21) 

Since  W2'1  (r)  embeds  continuously  in  C1  (r),  by  passing  to  a  subsequence  if  necessary, 
we  may  assume  that  y£(t)  ->  f(t]  a.e.  on  T,  ^(r)  ->  y  (r)  and  yn(r)  -*  y(t)  for  all  r  G  T. 
nW)  -*/(f>y(0)  a-e-  on  I.  Thus  passing  to  the  limit  as  n  ->  oo  in  (31),  we  obtain 

-  /W  =  /(',?«)  +  M/W     a.e.  on  r,  y(0)  =  y(b),  y1  (0)  =  y (i) 


Periodic  and  boundary  value  problems  123 

So  5  is  closed,  hence  weakly  closed  since  it  is  convex.  To  show  that  S  is  weakly 
compact  in  W2'l(T),  we  need  to  show  that  given  {xn}n>l  C  5,  we  can  find  a  weakly 
convergent  subsequence.  Since  {xn}n>{  is  bounded  in  W2*l(T)  and  the  latter  embeds 
compactly  in  W1'1^),  by  passing  to  a  subsequence  if  necessary,  we  may  assume  that 
*„  ->  jcin  Wl>l(T)  as  n  ->  oo.  Also<'  =  Sf'  and  so  \\^(t)\\  =  ||jt"(r)||  a.e.  on  T.  Therefore 
by  the  Dunford-Pettis  theorem,  we  may  assume  that  J^'-^g  in  Ll(T)  and  g  =xff.  So 
A:  €  W2'1  (r)  and  jcn  -^  z  in  W2'1  (7).  Since  5  is  weakly  closed  in  W2'1  (r),  x  €  S  and  so  5  is 
weakly  compact  in  W2>l(T). 

In  general  if  the  vector  field/  is  decreasing  in  the  x-  variable,  then  the  upper  and  lower 
solutions  of  the  problem,  as  well  as  the  solutions  exhibit  some  interesting  properties. 

First  we  consider  the  general  periodic  problem  (18),  with  the  following  hypotheses  on 
the  vector  field  /(f,  *,  v). 

H(f)5:  /:TxlRxR->[Risa  function  such  that 

(i)  for  every  x,y  €  R,  t  -*f(t,x,y)  is  measurable; 

(ii)  for  almost  all  t  €  T  and  all  3;  E  R,  x  —  >/(*,*,  y)  is  strictly  decreasing; 

(iii)  for  all  x,  v,/  €  ER  |/(f,  jc,y)  -  /(*,*,  y)|  <  k(t}\y  -  y'|  a.e.  on  T  with  jfc  €  Ll(T)\ 

(iv)  for  every  r  >  0  there  exists  7r  €  L^r)  such  that  |/(r,jc,y)|  <  7r(f)  a.e.  on  T  for  all 


PROPOSITION  7 


Jjf  H(f)5  /zoZA,  0  €  W2il(r)  w  fl«  wpper  solution  and  ip  £  W2)1(r)  a  /ow^r  solution  for 
problem  (18),  then  for  all  t  €  r,  V<0  <  (f>(t). 

Proof.  Suppose  not.  Let  tQ  €  T  be  such  that  maxteT(ip  -  </>)(t)  =  (^  -  (t>}(to)  =  £  >  0. 
First  assume  that  0  <  f<>  <  b.  Then  ^/(/o)  =  ^x(*o)  =  ^o  and  we  can  find  S  >  0  such  that 
for  all*  €  TS  =  [*o,  ^o  -h  <5],  we  have  0(?)  <  ip(t}.  Then  we  have 


»  a.e.  on 

and  -  0"r  >/r,  ^r,  ^;r       a.e.  on  T. 


Consider  the  following  initial  value  problem 

/«  =  f(t,  0(0»y(0)      a-e-  o 


,   . 
(22) 


Because  of  hypothesis  H(f)5  (iii),  problem  (32)  has  a  unique  solution  y  €  W 
Moreover,  from  the  definitions  of  upper  and  lower  solutions  and  a  well-known  differential 
inequality  (see  Hale  [9],  p.  31),  we  infer  that  <£'(?)  <  y(t)  <  ^f(t)  for  all  t  e  Ts  and  so 
(V?  -  cj))(t)  >  0  for  all  t  6  T6.  Integrating,  we  have  (-0  -  ^)(r0)  <  (^  -  <t>)(t)  for  all 
^  €  TS.  Recalling  the  choice  of  to,  we  see  that  (-0  —  0)(f)  =  constant  for  all  £  E  T^,  hence 
7//(f)  =  0x(r)  for  all  /  €  TS.  Thus  for  almost  all  t  6  TS,  we  have 


W)  <  ~^W, 
a  contradiction  to  the  fact  that  (T/>  -  </>)"(0  =  0  f°r  all  t  eTg. 

If  fo=0,  then  since  (-0-  0)(0)  =  (^  -  <t>)(b)9  we  can  find  ^>0  such  that 
(^  -  0)(0)  >  (V»  -  0)(/)  >  0  for  all  f  G  [0,  (5]  and  0  <  (</;  -  0)(f)  <  (^  -  ^)(fc)  for  all 
t  €  [i  —  8,b].  From  the  first  inequality  we  have  that  (T/;  —  <£)'(0)  <  0,  while  from  the 


124  Nikolaos  S  Papageorgiou  and  Francesca  Papalini 

second  it  follows  that  (^  —  </>)'(&)  >  0.  But  from  the  definitions  of  the  upper  and  lower 
solutions  we  have  (i/j  -  </>)'(0)  >  C0  ~  4>}'(b]  >  0»  therefore  we  conclude  that  ?//(0)  = 
0'(0)  =  i/o  and  we  can  proceed  as  in  the  previous  case. 

The  case  tQ  =  b  is  treated  in  a  similar  fashion. 

Our  second  observation  concerning  0,  -0,  refers  to  problem  (30)  where  the  vector  field 
depends  linearly  in  xf. 

PROPOSITION  8 

4f  H(f)4  holds,  $  €  W2>l(T)  is  an  upper  solution  of  (30),  ^  €  W2'1^)  w  0  lower  solution 
of  (30)  and/or  all  teT  </)(i)  <  ^(r),  tfien  (-0  -  <£)(•)  w  constant. 


Proof.  By  definition  we  have 

+W(r)        a.e.  on 
M<t>'(t)        a.e.  on 

Hence  we  have 

^W  -  /«  >/(',^W)  -  /(f^W)  +M(^'(r)  -  V'(0)     a.e.  on  T. 
Multiplying  with  (V>  -  ^)(r)  and  then  integrating  over  T  —  [0,b],  we  obtain 


t 

JO 


t) 

+  M  [  (^'-^)(0(^-^)(0dr.  (23) 

Jo 

By  Green's  formula,  we  have 


y-^XOpfc  (24) 

o  Jo 

Also  from  hypothesis  H(f  )4  (ii)  it  follows  that 

b 

Finally  note  that 

fb  (b 

^o  Jo 

_          fb 

=  —  M  I    (*0  —  0)(?)dr('0  —  0)(f)  =  — M(ip  —  4>)(^}  -\~ M(ib  —  ^)(0)  =::  0. 
Jo 

(26) 
Using  (34),  (35)  and  (36)  in  (33),  we  obtain 


o 
^/(r)  =  ^7(r)     for  all  t  G  T    and  so  (V>  -  ^)(-)  is  constant. 


Periodic  and  boundary  value  problems  125 

An  immediate  consequence  of  proposition  8,  is  the  following  result: 

COROLLARY  9 

If  H(f)4  holds  andxi,x2  €  W2*l(T)  are  two  solutions  of  (30)  such  that  x\(t)  <  x2(t)  for 
all  t  €  T,  then  (x\  —  #2)(-)  is  constant. 

Acknowledgement 

The  authors  wish  to  thank  the  anonymous  referee  for  his/her  remarks  that  improved  the 
presentation. 

References 

[1]  Appell  J  and  Zabrejko  P,  Nonlinear  superposition  operators  (Cambridge:  Cambridge  Univ. 

Press)  (1990) 
[2]  Bernfeld  S  and  Lakshmikantham  V,  An  Introduction  to  nonlinear  boundary  value  problems 

(New  York:  Academic  Press)  (1974) 
[3]  Brezis  H,  Analyse  Fonctionelle  (Paris:  Masson)  (1983) 
[4]  Cabada  A  and  Nieto  J,  Extremal  solutions  of  second  order  nonlinear  periodic  boundary  value 

problems,  Appl.  Math.  Comp.  40  (1990)  135-145 

[5]  Dunford  N  and  Schwartz  J,  Linear  Operators  I  (New  York:  Wiley)  (1958) 
[6]  Gaines  R  and  Mawhin  J,  Coincidence  degree  and  nonlinear  differential  equations  (Berlin: 

Springer- Verlag)  (1977) 
[7]  Gao  W  and  Wang  J,  On  a  nonlinear  second  order  periodic  boundary  value  problem  with 

Caratheodory  functions,  Ann.  Polon.  Math.  LXII  (1995)  283-291 
[8]  Gilbarg  D  and  Trudinger  N,  Elliptic  partial  differential  equations  of  second  order  (New  York: 

Springer- Verlag)  (1977) 

[9]  Hale  J,  Ordinary  differential  equations  (New  York:  Wiley)  (1969) 

[10]  Heikkila  S,  Lakshmikantham  V  and  Sun  Y,  Fixed  point  results  in  ordered  normed  spaces  with 

applications  to  abstract  and  differential  equations,  /.  Math.  Anal  Appl  163  (1992)  422-437 

[11]  Heikkila  S  and  Hu  S,  On  fixed  points  of  multifunctions  in  ordered  spaces,  Appl  Anal  51 

(1993)  115-127 
[12]  Krasnoselskii  MA,  Positive  solutions  of  operator  equations  (The  Netherlands:  Noordhoff, 

Groningen)  (1964) 
[13]  Lakshmikantham  V  and  Leela  S,  Remarks  on  first  and  second  order  periodic  boundary  value 

problems,  NonL  Anal.  -  TMA  8  (1984)  281-287 
[14]  Leela  S,  Monotone  method  for  second  order  periodic  boundary  value  problems,  NonL  Anal.  - 

TMA  7  (1983)  349-355 
[15]  Monch  H,  Boundary  value  problems  for  nonlinear  ordinary  differential  equations  of  second 

order  in  Banach  spaces,  NonL  Anal.  -  TMA  4  (1980)  985-999 
[16]  Nieto  J,  Nonlinear  second  order  periodic  value  problems  with  Caratheodory  functions,  Appl. 

Anal.  34  (1989)  111-128 
[17]  Nieto  J  and  Cabada  A,  A  generalized  upper  and  lower  solutions  method  for  nonlinear  second 

order  ordinary  differential  equations,  J.  Appl.  Math.  Stoch.  Anal  5  (1992)  157-166 
[18]  Nkashama  MN,  A  generalized  upper  and  lower  solutions  method  and  multiplicity  results  for 
nonlinear  first-order  ordinary  differential  equations,  J.  Math.  Anal.  Appl.  140  (1989)  381-395 
[19]  Oman  P,  A  monotone  method  for  constructing  extremal  solutions  of  second  order  scalar 

boundary  value  problems,  Appl.  Math.  Comp.  18  (1986)  257-275 
[20]  Oman  P  and  Trombetta  M,  Remarks  on  the  lower  and  upper  solutions  method  for  second  -and 

third-  order  periodic  boundary  value  problems,  Appl.  Math.  Comp.  50  (1992)  1-21 
[21]  Vrabie  I,  Compactness  Methods  for  Nonlinear  Evolutions,  (UK:  Longman  Scientific  and 
Technical,  Essex)  (1987) 


Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  Ill,  No.  1,  February  2001,  pp.  127-135. 
©  Printed  in  India 


Boundary  controllability  of  integrodifferential  systems  in  Banach 
spaces 

K  BALACHANDRAN  and  E  R  ANANDHI 

Department  of  Mathematics,  Bharathiar  University,  Coimbatore  641 046,  India 

MS  received  17  January  2000 

Abstract.  Sufficient  conditions  for  boundary  controllability  of  integrodifferential 
systems  in  Banach  spaces  are  established.  The  results  are  obtained  by  using  the 
strongly  continuous  semigroup  theory  and  the  Banach  contraction  principle.  Examples 
are  provided  to  illustrate  the  theory. 

Keywords.  Boundary  controllability;  integrodifferential  system;  semigroup  theory; 
fixed  point  theorem. 

1.  Introduction 

Controllability  of  nonlinear  systems  represented  by  ordinary  differential  equations  in 
Banach  spaces  has  been  extensively  studied  by  several  authors.  Balachandran  et  al  [1] 
studied  the  controllability  of  nonlinear  integrodifferential  systems  whereas  in  [2]  they  have 
investigated  the  local  null  controllability  of  nonlinear  functional  differential  systems  in 
Banach  spaces  by  using  the  Schauder  fixed  point  theorem.  Controllability  of  nonlinear  func- 
tional integrodifferential  systems  in  Banach  spaces  has  been  studied  by  Park  and  Han  [10]. 
Several  abstract  settings  have  been  developed  to  describe  the  distributed  control  sys- 
tems on  a  domain  fi  in  which  the  control  is  acted  through  the  boundary  T.  But  in  these 
approaches  one  can  encounter  the  difficulty  for  the  existence  of  sufficiently  regular 
solution  to  state  space  system,  the  control  must  be  taken  in  a  space  of  sufficiently  smooth 
functions.  Balakrishnan  [3]  showed  that  the  solution  of  a  parabolic  boundary  control  equa- 
tion with  L2  controls  can  be  expressed  as  a  mild  solution  to  an  operator  equation.  Fattorini 
[6]  discussed  the  general  theory  of  boundary  control  systems.  Barbu  and  Precupanu  [4] 
studied  a  class  of  convex  control  problems  governed  by  linear  evolution  systems  covering 
the  principal  boundary  control  systems  of  parabolic  type.  In  [5]  Barbu  investigated  a  class 
of  boundary-distributed  linear  control  systems  in  Banach  spaces.  Lasiecka  [8]  established 
the  regularity  of  optimal  boundary  controls  for  parabolic  equations  with  quadratic  cost 
criterion.  Recently  Han  and  Park  [7]  derived  a  set  of  sufficient  conditions  for  the 
boundary  controllability  of  a  semilinear  system  with  a  nonlocal  condition.  The  purpose  of 
this  paper  is  to  study  the  boundary  controllability  of  nonlinear  integrodifferential  systems 
in  Banach  spaces  by  using  the  Banach  fixed  point  theorem. 

2.  Preliminaries 

Let  E  and  U  be  a  pair  of  real  Banach  spaces  with  norms  1 1  •  1 1  and  |  •  |,  respectively.  Let  a 
be  a  linear  closed  and  densely  defined  operator  with  D(a)  C  E  and  let  r  be  a  linear 
operator  with  D(r)  C  E  and  R(r)  C  X,  a  Banach  space. 

127 


Integrodifferential  systems 


129 


f  u  is  continuously  differentiate  on  [0,  b],  then  z  can  be  defined  as  a  mild  solution  to  the 
I!auchy  problem 

z(t)  =Az(t)+<rBu(t)-Bit(t)  +f(t,x(t),      f  g(t,s,x(s))As\ 

V  jo  / 

z(0)  =  jc0  -  fiii(O) 
ind  the  solution  of  (1)  is  given  by 

x(t)  =  T(t)[x0-Bu(Q)]+Bu(t) 

\  JQ  '  /J 

Since  the  differentiability  of  the  control  u  represents  an  unrealistic  and  severe  require- 
nent,  it  is  necessary  to  extend  the  concept  of  the  solution  for  the  general  inputs  u  G  L1 
/,  U).  Integrating  (3)  by  parts,  we  get 

x(t)  =  T(t)xo  +  f  [T(t  -  s)a  -  AT(t  -  s)]Bu(s)ds 
JQ 

r         (       r  \ 

+  /    T(t  —  s)fls,x(s),  I    g(s, T,x(r))dT  \ds.  (4) 

Jo  \  Jo  ) 

rhus  (4)  is  well  defined  and  it  is  called  a  mild  solution  of  the  system  (1). 

DEFINITION 

Fhe  system  (1)  is  said  to  be  controllable  on  the  interval  /  if  for  every  XQ^XI  G  E,  there 
exists  a  control  u  €  L2(J,  U)  such  that  the  solution  jc(.)  of  (1)  satisfies  x(b)  =  x\. 

iVe  further  consider  the  following  additional  conditions: 

(vii)  There  exists  a  constant  K\  >  0  such  that  J0  v(t)dt  <  K\. 
yiii)  The  linear  operator  W  from  L2(7,  U)  into  E  defined  by 


Wu 


f 

=  / 

Jo 


induces  an  invertible  operator  W  defined  on  L2(J,  f/)/kerW  and  there  exists  a 
positive  constant  KI  >  0  such  that  ||  W""1)!  <  K^  The  construction  of  the  bounded 
inverse  operator  W     in  general  Banach  space  is  outlined  in  the  Remark. 
(ix)  M||jt0||  +  \bM\\aB\\  +  ft]  K2[\\x{\\  +M||jc0||  +tf\+N<r,  where  N  =  bM[M{[r+ 


+  ft)  be  such  that  0<q<l. 


(x)  Let  q  = 


5.  Main  result 

Fheorem.  If  the  hypotheses  (i)-(x)  are  satisfied,  then  the  boundary  control  integro- 
iifferential  system  (1)  is  controllable  on  J. 


130  K  Balachandran  and  E  R  Anandhi 

Proof.  Using  the  hypothesis  (viii),  for  an  arbitrary  function  jc(.)  define  the  control 

u(t]  =  W~l  L  - 


T(b-s]f(s,X(s),  \    g(s,r,x(r))dr)ds  (r). 

'0  Jo  J 

(5) 
Let  Y  =  C(y,J?r).  Using  this  control,  we  shall  show  that  the  operator  $  defined  by 

=  T(t)xQ  +  f  [T(t  -  s)a  -  AT(t  -  s)}BW~l  [x{  - 
Jo 


r 

)»  /  g(siQ>' 

JO 

has  a  fixed  point.  First  we  show  that  $  maps  Y  into  itself.  For  x  €  F, 

V(r  -  J)o-  -  AF(r  -  s)]BW'1  \Xl  -  T(b)x0 

(s)ds 


-  I  T(b-T)f(r,x(T),  /%(r,«,J 

JO  JO 

' T(t  -  s)f(s,x(s),  [* g(s,d, 
Jo 


+ 

fb\\T(b^r)\\\\f(rtx(r)JTg(T^ 
Jo  LI  Jo 

drlds 

iKiin^oii 

/(r,x(r),    r 


•/' 


i: 


\\T(t-s) 


10 

S,  0,0)  || 


M\\xo\\  +  bM\\trB\\K2\\\xl  \\  +  M\\XO\\ 
+  bM(Ml  (r  +  b(Lir  +  L^)}  +  Af2] 


+  bM[Mi  [r  +  b(Li  r  +  L2)}+  M2] 

M\\x0\\  +  [bM\\aB\\  +^1]Ar2[||jc1||  +M\\xo\\  +N}+N 


Integrodifferential  systems 
Thus  *  maps  Y  into  itself.  Now,  for  *,,  jc2  6  Y  we  have 


131 


,  r  g(T,e, 

JQ 


/r 
S(r,0, 

/>/ 

yo  w-*)ii  /(*,*,( 

/*J 

-/(J^2(J),  /    ^(j,fl,o:2 

«/  0 


dr 


contraction 


-X2(s)\\+bLl\\xl(e)  - 


hence  there  exists  a  unique  fixed 

' 


4.  Applications 


Consider  the  boundary  control  integrodifferential 


system, 


n     Y= 


y(t,  0)  =  u(t,  0),     on    S  =  (0,  &)  x  T,     /  e  [0,  b], 
y(0,x)=yo(x)!     for    j:€n>  ' 


where  „  €  L2(E),  yo  €  £2(fi),  M  €  £2(y)  ^  ^  g  y 

2  ~     (  ^'     '  =  7'  **  ldentlt    oerator  a 


and 


(Here  fl*(fl),  J5P(T)  and  ^(0)  are  usual  Sobolev  spaces  on  a  P.) 


132  K  Balachandran  and  E  R  Anandhi 

Let  us  assume  that  the  nonlinear  functions  //  and  77  satisfy  the  following  Lipschitz 
condition: 

<Ki[\\Vi  -V2\\  +  \\Wl  -Wi||], 

<  K2\\vl  -  v2\\, 

where  K\,  K2  >  0,  v\,  v2  G  Br  and  wi,  w2  G  fi. 

Define  the  linear  operator  £  :  L2(T)  ->  L2(fi)  by  Bu  =  ww  where  WM  is  the  unique 
solution  to  the  Dirichlet  boundary  value  problem, 

AwM  =  0    in    fl, 

vvu  =  u    in    F. 

In  other  words  (see  [9]) 

=  I  u^-&c,     for  all    ^  €  H^fl)  UJ¥2(fi),  (7) 

7r    ^w 

where  d^/dn  denotes  the  outward  normal  derivative  of  if)  which  is  well  defined  as  an 
element  of  H^(T}.  From  (7),  it  follows  that, 

K||L2(n)  <  CilHI^py     for  all 
and 

lK||/,1(Q)<C2||Wy(r)5     for  all    « 

where  Q,  /  =  1,  2  are  positive  constants  independent  of  u. 
From  the  above  estimates  it  follows  by  an  interpolation  argument  [12]  that 


)  <  C*~>     f°rall    r>0    with     i/(r)  =  c. 

Further  assume  that  the  bounded  invertible  operator  W"  exists.  Choose  b  and  other 
constants  such  that  the  conditions  (ix)  and  (x)  are  satisfied.  Hence,  we  see  that  all  the 
conditions  stated  in  the  theorem  are  satisfied  and  so  the  system  (6)  is  controllable  on 


Example  2.  Consider  the  boundary  control  system, 

dy(t  x]  fl 

,x)=f(t,y(t,x),J   g(t,s,y(s,x))ds    in 


in    (0,fe)xr,     re[0,fc],  (8) 


where  j>0  G  L2(ft),/  G  L2(Q),  g  G  Q  and  u  G  L2(T).  Here  /3  is  a  nonnegative  constant. 

Let  us  assume  that  the  nonlinear  functions  /  and  g  satisfy  the  Lipschitz  condition:  ii 


^ 

\\g(t,S,Vi)-g(t,S,V2)\\<M2\\Vi-V2\\, 

where  MI,  Af2  >  0,  ^i,  v2  G  5r  and  wi,  w2  G  J7. 


Integrodifferential  systems  133 

Take    £  =  L2(ft),    U  =  X  =  L2(T),    BI  =  /,    cry  =  A?,    ry  =  py  +  (9y / dn)    and 

(aH/f2(tt). 

The  operator  A  is  given  by 


Now  the  problem  (8)  becomes  an  abstract  formulation  of  (1). 

Define  the  linear  operator  B  :  L2(F)  -»  L2(fi)  by  J5w  =  zu  where  zu  G  Hl(ty  is  the 
unique  solution  to  the  Neumann  boundary  value  problem, 

zu  -  Azu  =0    in    Q, 
^+^  =  «    in    T. 

9/2 

Consider  on  the  product  space  Hl(Cl)  x  H^fl),  the  bilinear  functional 

h(y,  ^)  =  /  W  -1-  grady  grad  V>)d*  -  /  ("  -  $>0^da-,  (9) 

Jn  Jr 

where  w  €  #  ~2(F)  (here  Jr  w^  da  is  the  value  of  w  at  ^  E  H^  (F).  Since  /z  is  coercive,  there 
is  a  ZM  €  H  l  (£1)  satisfying  h(zu^)  =  0  for  all  ^  6  H1  (fi).  Hence  zw  =  5w  is  the  solution 
to  (8).  From  (9)  we  see  that 


Since  the  operator  —A  is  self-adjoint  and  positive,  we  have 
2  2 


/' 

Jo 


\\AT(t)yo\\2Ll(n)dt  <  C\\y0\\2D((_A^    for  all    y0  €  D((-A)i)  =  Hl  (12). 

(10) 
Let  <5  be  the  scalar  function  defined  by 

S(t)  =  Km  inf  ||AW7(0  ||I(//1  (n)itf  (n))  ,     t  €  [0,  ft]  , 

where  Aw  =  A(7  -f  n"^)"1  for  n  =  1,  2,  .  .  ..  Obviously, 

<5W    for    te(Q,b].  (11) 


Also  we  find  that  (10)  implies  that 

fb 

/    \\AnT(t)yG\\l(Hi(^L2mdt<C    for  all  n. 

Jo 

Therefore  by  Fatou's  lemma  it  follows  that  6  €  L2(0,  6)  and  hence  from  (10)  and  (11) 
we  have 

\\AT(t)Bu\\Lt<fl}  <  C6(t)\\U\\L2(T),     forall     »€(0,ft),     u  €  L2(F) 

with  i/(r)  =  C(5(r)  eL2(0,&).  Further  assume  that  the  bounded  invertible  operator  W 
exists.  Choose  b  and  other  constants  in  such  a  way  that  the  conditions  (ix)  and  (x)  are 
satisfied.  Thus  we  find  that  all  the  conditions  stated  in  the  theorem  are  satisfied.  Hence 
the  system  (8)  is  controllable  on  [0,ft]. 


134  K  Balachandran  and  E  R  Anandhi 

Remark  (see  also  [11]).  Construction  ofW~l. 

Let  Y  =  L2[/,  £7]  /ker  W.  Since  ker  W  is  closed,  Y  is  a  Banach  space  under  the  norm 

II  Wily  =  mf  \\u\\L2^u}  =  jnfjlw  -f 

where  [w]  are  the  equivalence  classes  of  u. 
Define  W  :  Y  ->  X  by 

W[u]  =  Wu,     u€(u}. 
Now  W  is  one-to-one  and 


We  claim  that  V  =  Range  W  is  a  Banach  space  with  the  norm 

\v\\v  =  \\W~lv\\Y. 

This  norm  is  equivalent  to  the  graph  norm  on  D(W~l)  =  Range  W,  W  is  bounded  and 
since  D(W)  =  Y  is  closed,  W'"1  is  closed  and  so  the  above  norm  makes  Range  W  =  V,  a 
Banach  space. 
Moreover, 


=  ||  M  ||=  inf.  ||«||  <  INI, 

«€[«] 


SO 


Since  L2|/,  17]  is  reflexive  and  ker  W  is  weakly  closed,  so  that  the  infimum  is  actually 
attained.  For  any  v  £  V,  we  can  therefore  choose  a  control  wGL2[J,  U]  such  that 
u  =  W~lv. 

References 

[1]  Balachandran   K,   Balasubramaniam   P  and  Dauer  J  P,   Controllability   of  nonlinear 

integrodifferential  systems  in  Banach  spaces,  J.  Optim.  Theory  Appl.  84  (1995)  83-91 
[2]  Balachandran  K,  Dauer  J  P  and  Balasubramaniam  P,  Local  null  controllability  of  nonlinear 

functional  differential  systems  in  Banach  spaces,  J.  Optim.  Theory  Appl  88  (1995)  61-75 
[3]  Balakrishnan  A  V,  Applied  functional  analysis  (New  York:  Springer)  (1976) 
[4]  Barbu  V  and  Precupanu  T,  Convexity  and  optimization  in  Banach  spaces,  (New  York:  Reidel) 

(1986) 
[5]  Barbu  V,  Boundary  control  problems  with  convex  cost  criterion,  S1AM  7.  Contr.  Optim.  18 

(1980)  227-243 

[6]  Fattorini  H  O,  Boundary  control  systems,  SIAM  J.  Contr.  Optim.  6  (1968)  349-384 
[7]  Han  H  K  and  Park  J  Y,  Boundary  controllability  of  differential  equations  with  nonlocal 

condition,  /.  Math.  Anal.  Appl.  230  (1999)  242-250 
[8]  Lasiecka  I,  Boundary  control  of  parabolic  systems;  regularity  of  solutions,  Appl.  Math.  Optim. 

4(1978)301-327 
[9]  Lions  J  L,  Optimal  control  of  systems  governed  by  partial  differential  equations,  (Berlin: 

Springer-Verlag)  (1972) 
[10]  Park  J  Y  and  Han  H  K,  Controllability  of  nonlinear  functional  integrodifferential  systems  in 

Banach  space,  Nihonkai  Math.  J.  8  (1997)  47-53 


Integrodifferential  systems 


135 


[11]  Quinn  M  D  and  Carmichael  N,  An  approach  to  nonlinear  control  problem  using  fixed  point 
methods,  degree  theory,  pseudo-inverse,  Numer.  Funct.  Anal  Optim.  7  (1984-1985)  197-219 

[12]  Washburn  D,  A  bound  on  the  boundary  input  map  for  parabolic  equations  with  application  to 
time  optimal  control,  SI  AM  J.  Contr.  Optim.  17  (1979)  652-671 


Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  Ill,  No.  1,  February  2001,  pp.  137. 
©  Printed  in  India 


Errata 

Steady-state  response  of  a  micropolar  generalized  thermoelastic  half- 
space  to  the  moving  mechanical/thermal  loads 

RAJNEESH  KUMAR  and  SUNITA  DESWAL 

(Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  110,  No.  4,  pp.  449-465,  November  2000) 


1.  On  page  451,  in  eq.  (13)  following  two  expressions  have  been  left  out  and  should  be 
included: 


2.  On  page  454,  a  typographical  error  has  been  found  out  in  eq.  (48).  The  expression  for 
AO  should  read  as: 


All  the  analytical  expressions  and  numerical  results  do  not  change  due  to  these  errors. 


137 


Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  Ill,  No.  2,  May  2001,  pp.  139-149. 
Printed  in  India 


Descent  principle  in  modular  Galois  theory 


SHREERAM  S  ABHYANKAR  and  PRADIPKUMAR  H  KESKAR* 

Mathematics  Department,  Purdue  University,  West  Lafayette,  IN  47907,  USA 
*Mathematics  Department,  University  of  Pune,  Pune  411  007,  India 
E-mail:  ram@cs.purdue.edu;  keskar@math.unipune.ernet.in 

MS  received  24  July  2000;  revised  1  December  2000 

Abstract.  We  propound  a  descent  principle  by  which  previously  constructed  equa- 
tions over  GF(qn)(X)  may  be  deformed  to  have  incarnations  over  GF(g)(X)  without 
changing  their  Galois  groups.  Currently  this  is  achieved  by  starting  with  a  vectorial 
(=  additive)  g  -polynomial  of  ^-degree  m  with  Galois  group  GL(m,  q)  and  then,  under 
suitable  conditions,  enlarging  its  Galois  group  to  GL(m,  qn)  by  forming  its  general- 
ized iterate  relative  to  an  auxiliary  irreducible  polynomial  of  degree  rc.  Elsewhere 
this  was  proved  under  certain  conditions  by  using  the  classification  of  finite  simple 
groups,  and  under  some  other  conditions  by  using  Kantor's  classification  of  linear 
groups  containing  a  Singer  cycle.  Now  under  different  conditions  we  prove  it  by 
using  Cameron-  Kantor's  classification  of  two-transitive  linear  groups. 

Keywords.     Galois  group;  iteration;  transitivity. 

1.  Introduction 

In  this  paper  we  make  some  progress  towards  understanding  which  finite  groups  are  Galois 
groups  of  coverings  of  the  affine  line  over  a  ground  field  of  characteristic  p  •£  0,  having 
at  most  one  branch  point  other  than  the  point  at  infinity.  We  are  specially  interested  in  the 
case  when  the  ground  field  is  not  algebraically  closed.  In  particular  we  realize  some  of  the 
matrix  groups  GL(m,  qn),  where  q  =  pu  >  1  is  a  power  of  p  and  m  >  0  and  n  >  0  are 
integers,  over  smaller  fields  of  characteristic  p  than  had  previously  been  accomplished.  For 
a  tie-up  with  the  geometric  case  of  an  algebraically  closed  ground  field  and  the  arithmetic 
case  of  a  finite  ground  field  see  Remark  5.1  at  the  end  of  the  paper.  Likewise,  for  a  tie-up 
with  Drinfeld  module  theory  see  Remark  5.2  at  the  end  of  the  paper. 

To  describe  the  contents  of  the  paper  in  greater  detail,  henceforth  let  q  =  pu  >  1  be  a 
power  of  a  prime  p,  let  m  >  0  and  n  >  0  be  integers,  and  let  GF(<?)  C  kg  C  K  C  Q  be 
fields  where  £2  is  an  algebraic  closure  of  K\  note  that  there  are  no  assumptions  on  the  field 
.kq  other  than  for  it  to  contain  GF(#).  Also  let  E  =  E(Y)  be  a  monic  separable  vectorial 
^-polynomial  of  ^-degree  m  in  Y  over  K,  i.e., 

m 

E  =  E(Y)  =  Yqm  +   T  XiYqm~l      with     X;  €  K  and  Xm  +  0,  (1.1) 


where  the  elements  X\,  .  .  .  ,  Xm  need  not  be  algebraically  independent  over  kq.  When 
we  want  to  assume  that,  for  a  subset  7*  of  {1,  .  .  .  ,  m},  the  elements  {Xi  :/€/*}  are 
algebraically  independent  over  kq  and  K  =  kq({Xi  :  i  e  7*})  with  Xi  =  0  for  all  i  £  7*, 
we  may  express  this  by  saying  that  we  are  in  the  generic  case  of  type  7*,  and  we  may 
indicate  it  by  writing  E*  for  E  and  K*  for  K.  When  7*  is  the  singleton  7b  =  {m} 


140  Shreeram  S  Abhyankar  and  Pradipkumar  H  Keskar 

we  may  say  that  we  are  in  the  binomial  case.  When  J*  is  the  pair  7;{  =  {m  -  JJL,  m] 
with  1  <  /z  <  m  we  may  say  that  we  are  in  the  [L-lrinomial  case.  When  J*  is  the  set 
y^rrfm  —  v:v  =  0ori»=a  divisor  of  m},  we  may  say  that  we  are  in  the  divisorial  case. 
NotethattheF-derivativeof£(y)is  Xm  andhenceifra  6  7*  then  in  the  generic  case  of  type 
/*,  the  equation  E(Y)  =  0  gives  a  covering  of  the  affine  line  over  kq  ({X/  :  m  ^  i  e  /*}) 
having  Xm  =  0  as  the  only  possible  branch  point  other  than  the  point  at  infinity. 

In  the  general  (=  not  necessarily  generic)  case,  let  V  be  the  set  of  all  roots  of  E  in 
Q,  and  note  that  then  V  is  an  m-dimensional  GF(g)-vector-subspace  of  fi.  Moreover, 
since  GF(#)  is  assumed  to  be  a  subfield  of  kq  and  hence  of  K,  every  #  -automorphism 
of  the  splitting  field  K(V)  of  E  over  K  induces  a  GF(#)-linear  transformation  of  V. 
Consequently  Gal(£,  K)  <  GL(V),  i.e.,  the  Galois  group  of  E  over  K  may  be  regarded 
as  a  subgroup  of  GL(  V)  (see  [Ab3]).  If  we  do  not  assume  GF(g)  c  kq  then  we  only  get 
Gal(E,  K)  <  FL(V),  where  FL(V)  is  the  group  of  all  semilinear  transformations  of  V 
(see  [Ab6]).  By  fixing  a  basis  of  V  we  may  identify  GL(V)  with  GL(wz,  q\  and  FL(V) 
with  FL(m,  q).  If  //  C  J*  then  in  the  generic  case  of  type  /*,  as  shown  in  [Ab2]  to 
[Ab4],  we  have  Gal(£*^,  K*)  =  GL(m,  q)  but  over  GF(p),  as  shown  in  [Ab6],  we  have 
Gal(£^,  GF(/>)({*/  :/  e  J*}))  =  TL(/n,  q)\  for  applications  of  these  results  see  [Abl] 
and  [Ab5].  To  mitigate  this  bloating  we  take  recourse  to  generalized  iteration  as  defined 
in  Remark  3.30  of  [Ab7]  and  repeated  below.  Here  bloating  refers  to  the  fact  that  a  more 
direct  approach  would  give  a  Galois  group  which  is  larger  than  desired,  when  working  over 
a  smaller  ground  field,  and  the  goal  is  to  modify  the  covering  in  order  to  shrink  the  group 
from  semilineai  to  general  linear. 

DEFINITION  1.2 

For  every  nonnegative  integer  j  we  inductively  define  the  yth  iterate  E^^  of  E  by 
putting  E™  =  E^(Y)  =  F,  £«!«  -  E™(Y)  =  E(Y),  and  E^  =  E^(Y)  = 
£(£[[;-!]]  (y))  for  all  j  >  1.  Next  we  define  the  generalized  rth  iterate  E[r]  of  E  for  any 
r  =  r  (T)  =  JT  n  T  €  &[T]  with  n  G  S2  (and  n  =  0  for  all  except  a  finite  number  of  i\ 
where  T  is  an  indeterminate,  by  putting  E[r]  =  E[r](Y)  =  £r,-E[ll']J(y).  Note  that,  for 
the  F-derivative  E[y](Y)  of  E^(Y)  we  clearly  have 


and  hence  if  r(Xm)  ^  0  then  E^  is  a  separable  vectorial  ^-polynomial  over  Q  whose  q- 
degree  in  7  equals  m  times  the  T-degree  of  r  .  Also  note  that  the  definition  of  £M  remains 
valid  for  any  vectorial  E  without  assuming  it  to  be  monic  or  separable.  Moreover,  in  such 
a  general  set-up,  this  makes  the  additive  group  of  all  vectorial  ^-polynomials  E  =  E(Y) 
in  Y  over  £2  into  a  ti[T]-premodule  having  all  the  properties  of  a  module  except  the  left 
distributive  law  and  the  associativity  of  multiplication,  i.e.,  for  all  r,  r'  e  Q[T]  we  have 
E[r+r>]  ==  £M+£[r']?butforall£,  E'  over  &  we  need  not  have  (£  +  £0[r]  =  E[r]  +  Ef[r\ 
and  in  general  E[rr>]  need  not  be  equal  to  (£M)[r']  .  Reverting  to  the  fixed  monic  separable 
vectorial  E  exhibited  in  (1.1),  the  said  premodule  structure  makes  Q  into  a  GF(q)[T]- 
module  when  for  every  r  G  GP(q)[T]  and  z  €  Q  we  define  the  'product'  of  r  and  z  to  be 
E[r](z)',  we  denote  this  GFO?)[r]-module  by  QE.  Now  let  us  fix 

s  =  s(T)  zR  =  GF($)|T]  of  7-degree  n  with  s(Xm)  +  0  (1.2.2) 

and  note  that  then  E^  is  a  separable  vectorial  ^-polynomial  of  ^-degree  mn  in  Y  over  K, 
and  the  coefficient  of  its  highest  degree  term  equals  the  coefficient  of  the  highest  degree 


Modular  Galois  theory  141 

of  s(T).  Let  Vls]  be  the  set  of  all  roots  of  E[s]  in  ft,  and  note  that  then  V[s]  is  an  (mri)- 
dimensional  GF(<7)-vector-subspace  of  £2.  Let  GF(#,  5)  =  /?/£/?  where  sR  is  the  ideal 
generated  by  s  in  R  =  GF(q)[T],  and  let  o> :  ^  — >  GF(g,  s)  be  the  canonical  epimoiphism. 
Now  V^  is  a  submodule  of  £2#  and  as  such  it  is  annihilated  by  sR  and  hence  we  may 
regard  it  as  a  GF(<7,  .sO-module;  note  that  then,  for  every  r  6  R  and  z  €  £2,  the  'product'  of 
o>(r)andzisgivenby<y(r)z  =  E[r]U)  =  I>/E[/1  (z),  and  for  every  £  e  Gal(£(V[jl),  #) 
we  have  g(to(r)z)  =  £]£(>"/ )El/J(g(z))  =  (&>(>"))g(z);  also  note  that  for  all  r  e  /?  and 
Z  €  £2  we  have  r-  =  <y(r)z  =  E^Cz)  =  0(r,  z)  with  0(r,  z)  e  (GF($)[Xi, ....  Xm])U]. 
It  follows  that,  in  a  natural  manner, 

Gal(£M,  A:)  <GL(V[S]),  (1.2.3) 

where  GL(  V^)  is  the  group  of  all  GF(#,  s)-linear  automorphisms  of  V^,  by  which  we 
mean  all  additive  isomorphisms  <r  :  V^  ->  V^  such  that  for  all  77  €  GF(/y,5)andz  6  V'^1 
we  have  a(rjz)  =  rjcr(z).  Note  that 

s  irreducible  in  R  =»  GL(V[s])  *  GL(m,  qn),  (1-2.4) 

where  ^  denotes  isomorphism.  Also  note  that  the  F-derivative  of  E^(Y)  is  s(Xm)  and 
hence  if  m  e  J*  and  s  is  irreducible  in  /?  then  in  the  generic  case  of  type  J*,  the  equation 
£[s](Y)  =  0  gives  a  covering  of  the  affine  line  over  kq([Xi  :  m  ^  i  e  J*})  having 
s(Xm)  =  0  as  the  only  possible  branch  point  other  than  the  point  at  infinity;  this  branch 
point  is  rational  if  and  only  if  n  —  1 . 

Now  part  of  what  was  proved  in  [Ab7]  can  be  stated  as  follows: 

Trinomial  Lemma  1.3.  IfJj  C  J  *  then  in  the  generic  case  of  type  J  *  we  have  Gal  ( E^  q ,  K  * ) 
=  GL(m,q). 

In  Note  3.37  of  [Ab7]  the  following  problem  about  generalized  iterations  was  posed. 

Problem.  Show  that  if  J*  =  (1,  2, . . . ,  m)  then  in  the  generic  case  of  type  J*  we  have 
,  ^*)  =  GL(V^). 


In  [AS1]  this  was  proved  when  s  =  Tn  and  in  Theorem  3.25  of  [Ab7]  that  result  was 
semilinearized.  Likewise  in  [AS2]  it  was  proved  under  the  assumptions  that  s  is  irreducible 
and  m  is  a  square-free  integer  with  GCD(m,  n)  =  1  arid  GCD(mnu,  2p)  =  1,  where  we 
recall  that  u  is  the  exponent  of  p  in  q,  i.e.,  u  is  the  positive  integer  defined  by  q  =  pu . 
Actually,  what  was  proved  in  ( 1 . 1 8)  of  [ AS2]  was  the  following  slightly  more  general  result. 

Weak  divisorial  Theorem  1.4.  Assume  that  s  is  irreducible  in  R,  and  J*  C  /*.  Also 
assume  that  m  is  a  square-free  integer  with  GCD(m,  n)  =  1,  and  GCD(mnu,  2p)  =  1. 
Then  in  the  generic  case  of  type  /*  we  have  Gal(E$$,  K*)  =  GL(V[5j)  «  GL(m,  qn). 

Now  CPT  (=  the  classification  of  projectively  transitive  permutation  groups,  i.e.,  sub- 
groups of  GL  acting  transitively  on  nonzero  vectors)  is  a  remarkable  consequence  of  CT 
(=  the  classification  theorem  of  finite  simple  groups).  The  implication  CT  =$  CPT  was 
mostly  proved  by  Hering  [Hel,  He2];  it  is  also  discussed  by  Cameron  [Cam],  Kantor  [Ka2], 
and  Liebeck  [Lie].  The  proof  of  (L4)  given  in  [AS2]  makes  essential  use  of  the  follow- 
ing weaker  version  of  CPT,  which  follows  by  scanning  the  list  of  projectively  transitive 
permutation  groups  given  in  [Ka2]  or  [Lie]. 

Weak  CPT  1.5.  Let  d  be  an  odd  positive  integer,  and  let  G  <  GL(d,  p)  be  transitive  on 
the  nonzero  vectors  GF(/?)^  \  {0}.  Then  there  exist  positive  integers  b,  c  with  be  —  d  and 
a  group  GO  with  SL(i,  pc)  <  GQ  <  TL(6,  pc)  such  that  G  «  GO. 


142  Shreeram  S  Abhyankar  and  Pradipkumar  H  Keskar 

The  m  =  I  case  of  (1.4),  without  the  hypothesis  GCD(mnw,  2p)  =  1,  was  proved  by 
Carlitz  [Car]  (also  see  Hayes  [Hay])  in  connection  with  his  explicit  class  field  theory.  In 
our  proof  of  (1 .4)  we  used  the  following  variation  of  Carlitz's  result  which  we  reproved  as 
Theorem  1.20  in  [AS2];  recall  that  a  univariate  polynomial  F(Y)  =  YlfLv  FI  Yl  of  positive 
degree  N  in  Y  is  said  to  be  Eisenstein  relative  (/?,  M),  where  M  is  a  prime  ideal  in  a  ring 
£,  if  FN  €  R  \  M,  Fi  e  M  for  1  <  i  <  N  ~  1,  and  FQ  e  M  \  M2. 

Carlitz  irreducibility  lemma  1.6.  Assume  that  s  is  irreducible  in  R,  and  Jb  C  /*.  Let 
s*(T)  be  a  nonconstant  irreducible  factor  of  s(T)  in  kq[T],  and  let  M*  be  the  ideal  in 
R*  =  kq[{Xi  :  i  e  /*}]  generated  by  {Xt  :  i  €  J*  \  /b}  U  (s*(Xm)}.  Then,  form  =  1, 
in  the  generic  case  of  type  /*  we  have  that  M*  =  s*(Xm)R*  is  a  maximal  ideal  in 
R*  =  kq[Xm],  Y~lEl[sq](Y)  is  Eisenstein  relative  to  (/?*,  M*),  F"1  E^sq](Y)  is  irreducible 
in  K*[Y],  and  Gzl(E^q\  K*)  =  GL(V[j])  «  GL(l,qn).  Moreover,  without  assuming 
m  =  1,  but  assuming  GCD(m,  n)  —  1,  m  f/i£  generic  case  of  type  J*  we  have  that  M* 
is  a  maximal  ideal  in  R*,  Y~1E^(Y)  is  Eisenstein  relative  to  (/?*,  M*),  Y~1E$$(Y)  is 
irreducible  in  K*[Y],  and  Gal(£*[^],  K*)  has  an  element  of  order  qmn  -  1. 

In  proving  (1.4),  in  addition  to  items  (1.5)  and  (1.6),  we  also  used  the  first  part  of  the 
following  well-known  versatile  lemma  which  was  initiated  by  Singer  in  [Sin]  and  which 
was  stated  as  Lemma  1.23  in  [AS  2];  for  an  elementary  proof  of  a  supplemented  version  of 
this  see  Lemma  5.13  and  §6  of  [Ab8]. 

Singer  cycle  lemma  1.7.  Let  A  e  GL(m,  q)  have  order  e  =  qm  —  1.  Then  det(A)  has 
order  €  =  q  —  1,  and  A  acts  transitively  on  the  nonzero  vectors  GF(#)m  \  {0},  i.e.,  it  is  an 
e-cycle  in  the  symmetric  group  Se  (and  as  such  it  is  called  a  Singer  cycle).  Moreover,  in 
GL(m,  q)  all  subgroups  generated  by  such  elements,  i.e.,  all  cyclic  subgroups  of  order  e, 
form  a  nonempty  complete  set  of  conjugates. 

Now  the  last  assertion  of  (1.6)  says  that  if  s  is  irreducible  in  R  and  7b  C  J*  with 
GCD(m,  n)  =  1  then  Gal(£^fJ,  K),  as  a  subgroup  of  GL(w,  qn),  contains  a  Singer  cycle. 
In  his  1980  paper  [Kal],  without  using  CT,  Kantor  proved  the  following  variation  (1.8)  of 
( 1 .5)  by  replacing  the  hypothesis  of  G  acting  transitively  on  nonzero  vectors  by  the  stronger 
hypothesis  that  G  contains  a  Singer  cycle. 

Kantor's  Singer  cycle  theorem  1.8.  If  G  <  GL(m,  qn)  contains  an  element  of  order 
qmn  -  1  thenforsome  divisor  m1  ofm  we  have  GL(m',  qnm/m>)<G,  where  GL(m',  qnmfm') 
is  regarded  as  a  subgroup  ofGL(m,  q)  in  a  natural  manner. 

As  a  consequence  of  (1.6)  and  (1.8),  but  without  using  (1.5),  and  hence  without  using 
CT,  in  (5.18)  of  [Ab8]  we  proved  the  following  stronger  version  (1.9)  of  (1.4)  in  which  the 
assumption  GCD(mnu,  2p)  =  1  is  replaced  by  the  weaker  assumption  GCD(m,  p)  =  1. 

Strong  divisorial  theorem  1.9.  Assume  that  s  is  irreducible  in  R,  and  7*  C  7*.  Also 
assume  that  m  is  a  square-free  integer  with  GCD(m,  n)  =  1,  and  GCD(m,  p)  =  1.  Then 
in  the  generic  case  of  type  7*  we  have  Gal(£^3,  K*)  =  GL(V[s])  ^  GL(m,  qn). 

In  ( 1 . 1 4)  of  [ Ab9]  we  settled  another  case  of  the  above  Problem  by  proving  the  following 
Theorem  without  using  the  above  results  (1.4)  to  (1.9). 


Modular  Galois  theory  143 

Two  step  theorem  1.10.  Assume  thats  is  irreducible  in  R,  and  J^  —  J*.  Also  assume  that 

C^, 


=  n  =  2.  Then  in  the  generic  case  of  type  /*  we  have  Gal(£*C^,  #*)  =  GL(V[s]) 


GL(m,qn). 

The  proof  of  (1  .  10)  was  based  on  the  following  lemma  which  was  stated  as  Lemma  1.16 
in  [Ab9]  and  established  in  §3  of  that  paper. 

Packet  throwing  lemma  1.11.  Let  M  be  the  maximal  ideal  in  a  regular  local  domain  R 
of  dimension  d  >  0  with  quotient  field  K  .  Let  F(Y)  =  ^o<i<N  F^1  be  a  polynomial 
of  degree  N  >  0  in  Y  which  is  Eisenstein  relative  to  (R,  M).  [Note  that  then  for  some 
elements  FI,  .  .  .  ,  Fd  in  R  we  have  (F<3,  FI,  .  .  .  ,  Fd)R  =_M.]  Let  K  —  K(rj)  where  rj  is 
an  element  in  an  overfield  of  K  with  FJj))^==  0,  and  let  R  =  R[n]  and  M  ~  nR  +  MR. 
Then  R  is  the  integral  closure  of  R  in  K,  R  is  a  d  dimensional  regular  local  domain  with 
maximal  ideal  M,  M  O  R  =  M,  and  for  any  rf  €  K  with  F(r£)  =  0  and  any  ¥2,  .  .  .  ,  Fd 
in  R  with  (Fo,  F^,  .  .  .  ,  Fd)R  =  M  we  have  (rf,  F^,  .  .  .  ,  Fd)R  =  M,  and  hence  for  any 
rf  e  K  with  F(5J)  =  0  we  have  'rf  €  M  \  M2.  Moreover,  if  for  some  positive  integer 
D  <  N  -  1  we  have  FD  <£  M2  +  F0R  and  Ft  eJiD+2^  +  F^R  for  1  <  i  <  D  -  1, 
and  r?i,  .  .  .  ,  T]D  are  pairwise  distinct  elements  in  K  with  F(T]J)  =  Qfor  1  <  j  <  D,  then 
F(Y)  =  F(Y)  Y\I<J<D(Y  ~  nj)  where  F(Y)  is  a  polynomial  of  degree  N  -  D  in  Y  which 
is  Eisenstein  relative  to  (R,  M). 

In  proving  (1.10),  the  following  consequence  of  (1.11)  was  implicitly  used;  in  §2  we 
shall  explicitly  deduce  it  from  (1.11). 

Two  transitivity  lemma  1.12.  Assume  that  s  is  irreducible  in  R,  and  we  are  in  the  generic 
case  of  type  J*  with  Jb  c  J*  and  m  >  1.  [Note  that  by  (1.2)  we  know  that  then 
Gal(£*$,  K*)  <  GL(V[S])  &  GL(m,  qn)  and  hence  we  may  regard  Gal(£*$,  K*)  to  be 
acting  on  the  (m  —  1)  -dimensional  projective  space  P(m  —  1,  qn)  over  GF(qn)  (where  the 
action  is  not  faithful  unless  qn  =  2).]  Let  N  =  qmn  -  1  and  F(7)  =  Y~lE$i(Y)  = 
Z!o</<JV  Ftf1  with  Ft  e  R*  =  kq[{Xj  :  j  €  J*}].  Assume  that  the  localization  of  R*  at 
some  nonzero  prime  ideal  in  it  is  a  regular  local  domain  R  with  maximal  ideal  M  such  that 
F(7)  is  Eisenstein  relative  to  (R,  M).  Let  D  =  qn  —  1  and  assume  that  Frj  £  M2  -f  F$R 
and  FI  €  Mz>+2~/  +  F0Rfor  1  <  i  <  D  -  1.  Then  Gal(£^],  K*)  is  two  transitive  on 
the  (m  —  I)  -dimensional  projective  space  P(m  —  1,  qn)  over 


In  Theorem  I  of  [CKa],  Cameron-Kantor  proved  the  following: 

Cameron-Kantor's  two  transitivity  theorem  1.13.  If  m  >  2  and  G  <  FL(m,  q)  is  two 
transitive  on  the  projective  space  P(m  —  l,q),  then  either  SL(w,  q)  <  G  or  G  =  the 
alternating  group  Aj  inside  SL(4,  2). 

As  a  consequence  of  (1.6),  (1.7),  (1.12),  (1.13),  and  the  coefficient  computations  of 
§3,  but  without  using  (1.5)  or  (L8)  to  (1.10),  in  §4  we  shall  prove  the  following  theorem. 
With  an  eye  on  further  applications,  the  computations  of  §3  are  more  extensive  than  what 
we  need  here. 

Main  theorem  1.14.  Assume  that  s  is  irreducible  in  R,  and  n  <  m  with  GCD(m,  n)  =  1 
and  /J  c  J*.  Then  in  the  generic  case  of  type  J*  we  have  Gal(£^],  X"*)  =  GL(V[s])  « 
GL(m,  qn). 

In  §5  we  shall  make  some  motivational  and  philosophical  remarks. 


144  Shreeratn  S  Abhyankar  and  Pradipkwnar  H  Keskar 

2.  Proof  of  two  transitivity  lemma 

To  continue  with  the  discussion  of  (  1  .2),  for  a  moment  assume  that  ,v  is  irreducible  in  R 
s(Xm)  ^  0  and  m>  1.  Then  by  (1.2.3)  and  (1.2.4)  we  have  GaS(/:iv|,  K)  <  GL(V{* 
GL(m,  qn)  and  hence  we  may  regard  Gal(/:IiVl,  K  )  to  be  acting  on  the  (m  -  1  )-dimensi 
projective  space  P(m  —  1  ,  q"  )  over  GF(q!l  )  (where  the  action  is  not  faithful  unless  qn  = 
Let  N  =  qmn  -  1  and  F(Y)  =  Y'{E^(Y).  Then  F(Y)  €  K\Y\  is  of  K  -degree  A/. 
a  moment  assume  that  FjT)  is  irreducible  in  K[Y\  and  let  K  =  K(n)  where  //  is  a 
of  F(Y)  in  £2.  Then  [AT  :  tf|  =  A/  and  GaKE1*1,  A')  is  transitive  on  P(m  ~  1, 
Let  /?o  be  the  set  of  all  nonzero  members  of  R  of  /'-degree  less  than  /;.  Then,  ir 
notation  of  (1.2),  (o>(r)r))r€K(]  are  all  the  distinct  'nonzero  scalar  multiples1  of  rj  ir 
(R/s)-  vector  space  0tV',  and  clearly  /fy  is  the  set  of  all  oro  -f  a\  T  -f  •  •  •  +  uv~i  7V'~"! 
(a0,  ai,  .  .  •  ,  an~\)  €  GFU/)"  \  (((),  0,  .  .  u  ())}.  This  gives  us  /)  distinct  roots  of  F(Y) 
where  D  =  qn  -  1.  Therefore  F(K)  =  F*(X)  n/-eVK  ~  w{r)/^  where  ^*^)  6  ^ 
is  of  F-degree  N  -  D  =  qmn  -  qil  >  1  .  Now  (w(r  )//)re/?{)  is  the  inverse  image  of  a  f 
in  P(m  -  1,0")  under  the  natural  surjection  GF(£/")/;i  \  {0}  -*  P(m  -  1,  q")  obta 
by  jdentifying  V[s]  with  GF(^/7)/w  via  a  basis.  It  follows  that  if  F*(Y)  is  irredu< 
in  K  then  Gal(£[>vl,  K)  is  two  transitive  on  ptni  -  !,</").  It  is  also  clear  that  if  F(\ 
F(Y)Yl\<i<D(Y~~JH)  where/?],  .  ..,  /;/>  are  distinct  roots  of  F(K)  in  /C  and  F(Y)  €  / 
is  irreducible  then  we  must  have  F*(K)  =  F(Y).  Therefore  we  get  the  following: 

Projective  action  lemma  2.1.  ///  the  situation  of  (1.2)  assume  that  s  is  irreducible 
with  s(Xm)  ^  0  and  m  >  1.  Let  F(Y)  =  K""1  El'vl(y)  a«J  «ore  f/?af  r/z^/z  F(  X)  6  K[ 
of  Y  -degree  N  =  qmn  -  1.  Aww/?^  rta/  F(K)  is  irreducible  in  K\Y\  ami  let  K  =  i 
where  r\  is  a  root  of  F(Y)  in  S2.  Then  [K  :  K]  =  N  and  Gal(E|AJ,  K)  is  transits 
P(m-l,qn).  Moreover,  if  upon  letting  D  =  qn-\  wehave  F(Y)  =  F(Y)  Yli<j<i)(Y 
where  r)\,  .  .  .  ,  r\D  are  distinct  roots  of  F(Y)  in  K  and  F(Y)  €  K[Y\  is  irreducible 
Gal(£[5],  K)  is  two  transitive  on  P(m  ~  1  ,  qn}. 

Since  Eisenstein  polynomials  are  irreducible,  upon  taking  E  =  Efn  (]  with  F  =  F 
K  =  K*  =  Jf  in  (2.1),  by  (1.11)  we  get  (1.12). 

3.  Coefficient  computations 

Let  R*  =  GF($)[Xi  ,  .  .  .  ,  Xm].  Then  clearly  for  every  v  >  0  we  have 


mv 


E[[V]](Y)  =  Y"  .....  +        D^Y"  .....  -'  with  D,,,.  €  R*. 

/=! 

Also 


and  hence  for  every  integer  v  >  1  we  have 

/  mv—m 


w=l 


Modular  Galois  theory 


145 


m    mv~m 


v—\    w=l 


and  therefore,  for  any  positive  integer  /,  upon  letting 

the  set  of  all  pairs  of  integers  (u,  it;) 
Q(i)  =  \    with  1  <  v  <  m  and  1  <  w  <  mv  —  m 
such  that  v  +  w  =  i 


E 


we  get 


and 


By  induction  we  shall  show  that  for  every  v  >  0  we  have 

n  _  yv 

•*-/v,mv  —  ^m 


if 


and 


and 


if/  is  an  integer  with  1  <  /  <  m 

such  that  Xi  —  0  whenever  m  —I  <  i  <  m 

then  Dyj  =  0  whenever  mv  —  I  <  i  <  mv 

and  D          ,  -  Y      /  Y^1  v 

ana  ^ymy_/  —  Am-/  A^ 


if  j  is  an  integer  with  1  <  j  <  m 
such  that  Xt  =  0  whenever  1  <  i  <  j 
then  for  1  <  i  <  min(m,  2j  —  1)  we  have 

n    .  _  V^1  Y*W* 
^vv  —  2U.=0  A/ 

which  we  know  to  be  zero  if  1  <  /  <  j. 


mv—v—w 


(3.3) 


(3.4) 


(3.5) 


(3.6) 


(3.7) 


(3.8) 


By  (3.2),  this  is  obvious  for  v  =  1.  So  let  v  >  1  and  assume  true  for  v  —  1.  Then  clearly 
2(mv)  =  {(m,  mv  —  m)},  and  hence  by  (3.4)  and  the  v  —  1  version  of  (3.6)  we  get 


—  Y    Y 

—  -Am  .A 


V-l 


Likewise,  if  /  is  an  integer  with  1  <  /  <  m  such  that 
then,  by  (3.4)  we  get 


=  0  whenever  m  —  /  <  /  <  m, 

if  mv  —  /  <  i  <  mv 

,  _     if  my  —  /  =  / 


146  Shreeram  S  Abhyankar  and  Pradipkumar  H  Keskar 

and  hence  by  the  v  -  1  versions  of  (3.6)  and  (3.7)  we  get 

Dy  z  =  0     if     mv  —  I  <  i  <  mv 

and 


V  Y         y( 

—  ^-m-/  /  ^  A 


A.=0 


Similarly,  if  j  is  an  integer  with  1  <  ;  <  m  such  that  X/  =  0  whenever  1  <  /  <  7,  then 
for  all  z,  v,  u;  with  I  <i  <2j  -I  and  (v,  iy)  €  Q(0  we  have  either  V<JOTW<  j,  and 
hence  by  (3.5)  and  the  v  -  1  version  of  (3.8)  we  see  that  for  1  <  i  <  min(m,  2j  -  1)  we 
have 


4.  Proof  of  main  Theorem 

To  prove  the  Main  Theorem  1.14,  assume  that  s  is  irreducible  in  R  and  n  <  m  with 
GCD(m,  n)  =  1.  Also  assume  that  we  are  in  the  generic  case  of  type  7*  with  yj"  C  J*. 
In  view  of  (1.2.3)  and  (1.2.4),  after  identifying  V[s]  with  GF(<?")m  via  a  basis>  we  have 
Gal(£^  ,  K*)  <  GL(m,  qn)  and  we  may  regard  Gal(£*$,  £*)  as  acting  on  P(m  -  1  ,  ^n) 
(where  the  action  is  not  faithful  unless  qn  =  2).  We  want  to  show  that  Gal(£^],  K*)  = 


Let  N  =  <?mrt  -  1  and  F(7)  =  F'^^^F)  =  EQ<KN  Wl  with  f«y1'  €  R*  = 
^[{X;-  :  7  €  J*}].  Let  D  =  qn  -  1.  Note  that  5  =  5(7)  =  Eo<u<n^7u  with 
sv  €  GF(^)  and  sn  i=-  0.  Let  ^  be  an  algebraic  closure  of  kq  in  fl,  and  let  f  be  a  root 
of  s(T)  in  £3.  Since  $(7)  is  irreducible  in  R,  we  get  f  *n~1  j=  1  and  s7(f  )  ^  0  where 
5x(r)  is  the  T-derivative  ofs(T).  Let  R  be  the  localization  of  kq[Xn,  Xm]  at  the  maximal 
ideal  generated  by  Xn  and  Xm  -  f  .JThen  ^  is  two  dimensional  regular  local  domain  with 
maximal  ideal  M  =  (Xn,  XTO  -  ?)#; 

For  a  moment  suppose  that  ^  =  kq  and  jj  ==  J*,  and  let  us  write  ^t  for  /^^and  £^,^ 
for  E*  ^.  Now  by  (1.6)  and  (1.7)  we  see  that  F(Y)  is  Eisenstein  relative  to  (R,  M),  and 
the  determinantal  map  Gal(£™,  J^1")  ->  GF(^n)  \  {0}  is  surjective.  By  (1.2.1)  we  have 


By  taking  l  =  nm  (3.7)  we  see  that 
Fi  :  =  0  for  1  <  /  <  D  - 
and 


Modular  Galois  theory  147 

where 


0<v<« 

Since  f^~l  =  1,  we  get 


and  therefore 


(y-lHJ^-l)   _          v-1 

^  ' 


It  follows  that 


and  hence  by  (1.12)  we  conclude  that  Gal(£"^,  AT1")  is  two  transitive  on  P(m  -  1,  #").  If 
n  >  lthenby(1.13)weseethatSL(m,gn)  <  Gal(£^,  K  t)  and  hence,  because  the  deter- 
minantalmapGal(£^,  K*)  -+  GFG?")\{0}issurjective,  we  must  have  Gal  (£^3,  tf1")  = 
GL(m,  <?n).  If  /i  =  1  then  by  (1.3)  we  get  GaKE^,  #f)  =  GL(m,  <?").  Thus  in  both  the 
cases  we  have  Gal(E™,  K*)  =  GL(m,  qn). 

Now  let  us  return  to  the  case  when  the  field  kq  need  not  be  algebraically  closed.  Since 
kq  is  an  overfield  of  kq  and  E^q  is  obtained  from  E^q  by  putting  Xi  =  0  for  all  i  e 
7*\./J,  in  view  of  the  extension  principle  (cf.  p.  93  of  [Ab2])  and  the  specialization 
principle  (cf.  p.  1894  of  [AbL]),  see  that  Gal(£™,  #f)  <  Gal(£*[,^],  AT*).  Therefore 


5.  Concluding  remarks 

Let  us  end  with  some  remarks  on  motivation  and  philosophy. 

Remark  5.1  (Algebraic  fundamental  groups).  The  algebraic  fundamental  group 
of  the  affine  line  L&  over  a  field  k  is  defined  to  be  the  set  of  all  Galois  groups  of  finite 
unramified  Galois  coverings  of  the  affine  line  L&  over  k.  Similarly  we  define  7tA(Lk,t)  for 
Lk,t  =  Lk  punctured  at  t  points,  and  more  generally  we  define  TCA  (CgiW)  for  a  nonsingular 
projective  genus  g  curve  C  over  k  punctured  at  w  +  1  points.  Let  Q(p)  be  the  set  of  all 
quasi-/?  groups,  i.e.,  finite  groups  G  such  that  G  =  p(G)  where  p(G)  is  the  subgroup 
of  G  generated  by  all  of  its  ^-Sylow  subgroups,  and  more  generally  let  Qt(p)  be  the  set 
of  all  quasi-(p,  t)  groups,  i.e.,  those  G  for  which  G/p(G)  is  generated  by  t  generators. 
In  [Abl],  as  geometric  conjectures  it  was  predicted  that  if  k  is  an  algebraically  closed 
field  of  characteristic  p  then  JtA(Lk)  =  Q(p)>  and  more  generally  7tA(Lk,t)  =  Qt(p) 
and  7TA(CgjU;)  =  Q2g+w(p)-  In  1994,  these  were  settled  affirmatively  by  Raynaud  [Ray] 
and  Harbater  [Har].  For  higher  dimensional  versions  of  the  geometric  conjectures  see 
[Ab5].  Then,  mostly  inspired  by  Fried-Guralnick-Saxl  [FGS]  and  Guramick-Saxl  [GuS], 
we  turned  our  attention  to  coverings  defined  over  finite  fields.  In  [Ab6]  this  led  to  the 
arithmetical  question  asking  whether  ^(LoF^))  =  Q\(p)*  the  philosophy  behind  this 
being  that  dropping  from  an  algebraically  closed  field  to  a  finite  field  is  somewhat  like 
adding  a  branch  point.  In  particular  we  may  ask  whether  KA  (Lk,i)  contains  Q\  (p)  where 


148  Shreeram  S  Abhyankar  and  Pradipkumar  H  Keskar 

k  is  an  overfield  of  GF(#).  As  indicated  in  the  introduction,  in  doing  this  arithmetic; 
problem,  the  linear  groups  got  bloated  towards  their  semilinear  versions  and  the  attempt  t 
unbloat  them  led  us  to  generalized  iterations. 

Remark  5.2  (Division  points  and  Drinfeld  modules}.  The  generalized  iterations  memselve 
came  out  of  the  theory  of  Drinfeld  modules  as  developed  in  his  paper  [Dri].  This  work  c 
Drinfeld  seems  to  have  been  inspired  by  Serre's  work  [Sel]  on  division  points  of  ellipti 
curves  which  was  later  generalized  by  him  [Se2]  to  abelian  varieties.  In  turn,  our  descriptio 
of  the  module  E^J  in  (\2)  is  based  on  the  ideas  of  Drinfeld  modules.  For  a  discussion  c 
Drinfeld  modules  and  their  relationship  with  division  points  of  elliptic  curves  and  abelia 
varieties  see  Goss  [Gosj.  Very  briefly,  the  roots  of  the  separable  vectorial  g-polynomii 
E  of  ^-degree  2m  exhibited  in  (1.1)  form  a  1m  dimensional  GF(g)- vector-space  on  whic 
the  Galois  group  of  E  acts.  The  said  Galois  group  also  acts  on  the  roots  of  E^  discusse 
in  (1.2)  which  are  the  analogues  of  '^-division  points  of  £.'  Indeed,  we  have  used  th 
letter  E  to  remind  ourselves  of  elliptic  curves  in  case  of  m  =  1  and  more  generally  of  2t 
dimensional  abelian  varieties.  We  hope  that  the  present  descent  principle  can  somehow 
be  'lifted'  to  characteristic  zero.  Before  that  it  should  be  made  to  work  in  the  symplecti 
situation,  the  bloated  semilinear  equations  for  which  can  be  found  in  [Ab7].  Prior  to  tfu 
the  GL  work  of  this  paper  should  be  completed. 

Acknowledgement 

This  work  was  partly  supported  by  NSF  Grant  DMS  99-88166  and  NSA  grant  MDA  904 
97-1-0010. 


References 

[Abl]  Abhyankar  S  S,  Coverings  of  algebraic  curves,  Am.  J.  Math.  79  (1957)  825-856 

[Ab2]  Abhyankar  S  S,  Galois  theory  on  the  line  in  nonzero  characteristic,  Bull  Am.  Math.  Soi 

27(1992)68-133 

[Ab3]  Abhyankar  S  S,  Nice  equations  for  nice  groups,  Israel  J.  Math.  88  (1994)  1-24 
[AM]  Abhyankar  S  S,  Protective  polynomials,  Proc.  Am.  Math.  Soc.  125  (1997)  1643-1650 
[Ab5]  Abhyankar  S  S,  Local  fundamental  groups  of  algebraic  varieties.  Proc.  Am.  Math.  Soc.  12 

(1997)  1635-1641 

[Ab6]  Abhyankar  S  S,  Semilinear  transformations,  Proc.  Am.  Math.  Soc.  127  (1999)  251 1-252 
[Ab7]  Abhyankar  S  S,  Galois  theory  of  semilinear  transformations,  Proceedings  of  the  UF  Galoi 

Theory  Week  1 996  (ed.)  Helmut  Voelklein  et  al,  London  Math.  Soc.,  Lecture  Note  Serie 

256(1999)1-37 

[Ab8]  Abhyankar  S  S,  Desingularization  and  modular  Galois  theory  (to  appear) 
[Ab9]  Abhyankar  S  S,  Two  step  descent  in  modular  Galois  theory,  theorems  of  Burnside  an 

Cayley,  and  Hilbert's  thirteenth  problem  (to  appear) 
[AbL]  Abhyankar  S  S  and  Loomis  P  A,  Once  more  nice  equations  for  nice  groups,  Proc.  An 

Math.  Soc.  126  (1998)  1885-1896 
[AS1]  Abhyankar  S  S  and  Sundaram  G  S,  Galois  theory  of  Moore-  Carlitz-Drinfeld  modules,  C 

R.  Acad.  Scl  Paris  325  (1997)  349-353 
[AS2]  Abhyankar  S  S  and  Sundaram  G  S,  Galois  groups  of  generalized  iterates  of  generic  vectors 

polynomials  (to  appear) 
[Cam]  Cameron  P  J,  Finite  permutation  groups  and  finite  simple  groups,  Bull  London  Math.  So< 

13(1981)1-22 
[CKa]  Cameron  P  J  and  Kantor  W  M,  2-Transitive  and  antiflag  transitive  collineation  groups  c 

finite  projective  spaces,  J.  Algebra  60  (1979)  384-422 

[Car]  Carlitz  L,  A  class  of  polynomials,  Trans.  Am.  Math.  Soc.  43  (1938)  167-182 
[Dri]  Drinfeld  V  G,  Elliptic  Modules,  Math.  Sbornik  94  (1974)  594-627 


Modular  Galois  theory  149 

[FGS]  Fried  M  D,  Guralnick  R  M  and  Saxl  J,  Schur  covers  and  Carlitz's  conjecture,  Israel  J.  Math. 

82(1993)  157-225 
[GuS]  Guralnick  R  M  and  Saxl  J,  Monodromy  groups  of  polynomials,  Groups  of  Lie  Type  and 

their  Geometries  (eds)  W  M  Kantor  and  L  Di  Marino  (Cambridge  University  Press)  (1995) 

125-150 

[Gos]  Goss  D,  Basic  Structures  of  Function  Field  Arithmetic  (Springer- Verlag)  (1996) 
[Har]  Harbater  D,  Abhyankar's  conjecture  on  Galois  groups  over  curves,  Invent.  Math.  117  ( 1994) 

1-25 
[Hay]  Hayes  D  R,  Explicit  class  field  theory  for  rational  function  fields,  Trans.  Am.  Math.  Soc. 

189(1974)77-91 
[He  1  ]  Hering  C,  Transitive  linear  groups  and  linear  groups  which  contain  irreducible  subgroups 

of  prime  order,  Geometriae  Dedicata  2  (1974)  425-460 
[He2]  Hering  C,  Transitive  linear  groups  and  linear  groups  which  contain  irreducible  subgroups 

of  prime  order  II,  J.  Algebra  93  (1985)  151-164 

[Kal]  Kantor  W  M,  Linear  groups  containing  a  Singer  cycle,  /  Algebra  62  (1980)  232-234 
[Ka2J  Kantor  W  M,  Homogeneous  designs  and  geometric  lattices,  /.  Combinatorial  Theory  A38 

(1985)66-74 
[Lie]  Liebeck  M  W,  The  affme  permutation  groups  of  rank  three,  Proc.  London  Math.  Soc.  54 

(1987)477-516 
[Ray]  Raynaud  M,  Revetment  de  la  droit  affme  en  characteristic  p  >  0  et  conjecture  d'  Abhyankar, 

Invent.  Math.  116  (1994)  425-462 
[Sel]  Serre  J-P,  Proprietes  galoisiennes  des  points  d'ordre  fini  des  courbes  elliptiques,  Invent. 

Math.  15(1972)259-331 

[Se2]  Serre  J-P,  Resume  des  cours  et  travaux,  Annuaire  du  College  de  France  85-86  (1985) 
[Sin]  Singer  J,  A  theorem  in  finite  projective  geometry  and  some  applications  in  number  theory, 

Trans.  Am.  Math.  Soc.  43  (1938)  377-385 


Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  Ill,  No.  2,  May  2001,  pp.  151-161. 
Printed  in  India 


Obstructions  to  Clifford  system  extensions  of  algebras 


ANTONIO  M  CEGARRA  and  ANTONIO  R  GARZON 

Departamento  de  Algebra,  Universidad  de  Granada,  18071  Granada,  Spain 
E-mail:  acegarra@ugr.es;  agarzon@ugr.es 

MS  received  24  November  1999;  revised  13  September  2000 

Abstract.  In  this  paper  we  do  phrase  the  obstruction  for  realization  of  a  generalized 
group  character,  and  then  we  give  a  classification  of  Clifford  systems  in  terms  of 
suitable  low-dirnensional  cohomology  groups. 

Keywords.     Clifford  system;  character;  cohomology  groups;  obstructions. 

1.  Introduction 

The  problem  of  Clifford  system  extensions  resides  in  the  classification  and  the  construction 
of  the  manifold  of  all  Clifford  systems  over  a  commutative  ring  k,  S  =  0ae^  Sa,  the 
type  being  given  group  G  and  with  1 -component  Si  isomorphic  to  a  given  fc-algebra  R. 
Each  such  G -graded  Clifford  system  extension  realizes  a  generalized  collective  charac- 
ter of  G  in  R,  that  is  a  group  homomorphism  $  :  G  ->  Pic&(#)  of  G  into  the  group  of 
isomorphism  classes  of  invertible  left  R  <S>k  R° -modules,  and  this  leads  to  a  problem  of 
obstruction.  When  a  generalized  collective  character  is  specified,  it  is  possible  that  no  Clif- 
ford system  extensions  realizing  the  specified  homomorphism  can  exist.  The  main  result 
in  this  paper  is  to  obtain  a  necessary  and  sufficient  condition  for  the  existence  of  such  a 
Clifford  system  extension,  formulated  in  terms  of  a  certain  3-dimensional  group  cohomol- 
ogy class  T(^>),  referred  to  here  as  the  Teichmiiller  obstruction  of  <l>.  The  construction 
of  T(<l>)  is  closely  analogous  to  a  construction  by  Kanzaki  [9],  for  a  description  of  the 
Chase-Harrison-Rosenberg  seven  term  exact  sequence  [2]  about  the  Brauer  group.  In  the 
case  where  a  generalized  collective  character  <1>  has  an  extension,  the  manifold  of  such 
strongly  graded  extensions  is  shown  as  a  principal  and  homogeneous  space  under  a  2nd 
cohomology  group. 

This  paper  has  been  strongly  influenced  by  the  work  on  the  classification  of  crossed- 
product  rings  by  Hacque  in  [7,8],  where  he  makes  a  systematic  analysis  of  the  important 
phenomenon  bound  to  the  existence  of  obstructions.  Clifford  systems,  also  called  strongly 
graded  algebras,  are  a  direct  generalization  of  crossed  product  algebras  and  they  were 
introduced  and  applied  by  Dade  in  several  important  papers  [3,  4],  where  he  develops 
Clifford's  theory  axiomatically,  and  which  can  be  referred  to  for  general  background. 

In  §2,  we  state  a  minimum  of  needed  notation  and  terminology.  Section  3  contains 
the  main  results  of  the  paper,  namely  the  construction  of  the  Teichmiiller  obstruction  map 
and  the  obstruction  theorems.  We  conclude  in  §4  by  exhibiting  a  non-realizable  collective 
character. 

2.  Clifford  system  extensions  and  generalized  collective  characters 

Throughout  the  paper  k  is  a  commutative  ring  with  identity  and  G  is  a  group. 


152  Antonio  M  Cegarra  and  Antonio  R  Garzon 

A  G-gmded  Clifford  system  over  k  S  is  a  ^-algebra  with  identity,  also  denoted  by  , 
together  with  a  family  of  &-submodules  Sff,  a  €  G,  such  that  5  =  ©ff€£  Sff  and  SaST 
Scr  for  ail  cr,  r  €  G,  where  the  product  5,j5r  consists  of  all  finite  sums  of  ring  produc 
xy  of  elements  x  G  Sa  and  y  €  5T.  Note  that  the  1  -component  S\  is  a  &-subalgebra  of 
and  each  or  -component  5a,  cr  €  G,  is  a  two-sided  Si-submodule  of  5. 

By  a  Clifford  system  extension  of  a  ^-algebra  R  we  mean  a  Clifford  system  /:-algebra 
whose  1  -component  Si  is  isomorphic  to  R.  More  precisely,  we  have  the  following: 

DEFINITION  2.1 

Let  R  be  a  ^-algebra  and  G  a  group.  A  G  -graded  Clifford  system  extension  of  R  is  a  pa 
(5,  y'),  where  S  =  0ae^  ^  *s  a  G~graded  Clifford  system  fc-algebra  and  j  :  R  c^  5  is 
^-algebra  embedding  with  j(R)  —  S\. 

If  (5,  7),  (S',  /)  are  two  G-graded  Clifford  system  extensions  of  R,  by  a  morphism  b< 
tweenthem  /  :  (S,  j)  -»  (S',  /),  we  mean  a  grade-preserving  ^-algebra  homomorphis] 
/  :  S  ~>  S7  that  respects  the  embeddings  of  R,  that  is,  such  that  /  j  =  /. 

The  most  striking  example  is  the  group  algebra  #[G],  but  also  crossed  products  of 
and  G  yield  examples  of  G-graded  Clifford  system  extensions  of  a  ^-algebra  R. 

From  ([4],  Corollary  2.10)  it  follows  that  any  Clifford  system  extension  rnorphisi 
/  :  (5,  /)  ->  (S",  /)  is  necessarily  an  isomorphism.  Therefore  the  existence  of  a  mo 
phism  is  an  equivalence  relation  between  G-graded  Clifford  system  extensions  of  R  an. 
in  this  case,  we  usually  say  that  the  extensions  are  equivalent.  Then 


denotes  the  set  of  equivalence  classes  of  G-graded  Clifford  system  extensions  of  the  I 
algebra  R. 

If  (5,  7)  is  a  G-graded  Clifford  system  extension  of  R,  then  each  Saj  a  e  G,  is  a 
invertible  R  ®^  jR°-module  and,  for  every  cr,  r  e  G,  the  canonical  morphism  Sff  ®R  Sr  - 
San  *a  ®  XT  *~*  ^axi->  is  an  R  ®^  J?°-isomorphism.  Hence,  there  is  a  canonical  map 

X  :  Cliff*(G,  R)  —  >  Homc/?(G,  Pic^/?)),  (: 

where  HomGp(G,  Pic^(/?))  is  the  set  of  group  homomorphisms  of  G  into  Pick(R),  tit 
group  of  isomorphism  classes  of  invertible  R  %  ^°-modules,  which  carries  the  class  < 
a  G-graded  Clifford  system  extension  (S,  j)  to  the  group  homomorphism  X[S,  j]  :  G  ~ 
),  given  by 


We  have  the  Baer  notion  of  Kollectivcharakter  in  mind,  and  we  define  a  genera 
ized  collective  character  of  the  group  G  in  the  ^-algebra  R  as  a  group  homomorphisi 
<t>  :  G  ->  Pick(R).  Let  us  recall  the  exact  group  sequence  ([1],  Chapter  II,  (5.4)), 

1  ->  InAut(/Z)  —  *  Autk(R)  -^  Pick(R),  (<• 

in  which  .8  maps  a  ^-algebra  automorphism  of  J?,  a  €  Autfc(/0,  to  the  class  of  the  invertib] 
/?  ®jt  ^°-module  #a,  which  is  the  same  left  R  -module  as  R  with  right  action  given  b 

x-y=  xct(y),  x,  y  €  R.  Then,  there  is  a  canonical  embedding  Out*(jR)  ^  Pic^(/Z),  c 
the  group  of  outer  automorphisms  of  the  A:-algebra  R,  Out^(R)  =  Aut^(^)/InAut(/?),  int 
the  Picard  group  Pick(R).  A  group  homomorphism  *  :  G  ->  Out^(^)  has  been  called 


Clifford  system  extensions  153 

collective  character  (cf.  Hacque  [7,  8]);  so  that  collective  characters  of  G  in  R  are  those 
generalized  ones  factoring  through  the  embedding  Out^(R)  *-*  Pic^(R).  Of  course,  by 
character  we  understand  a  group  homomorphism  G  —  >•  Aut/t(jR). 

Hence  Honi^G,  Pic^(R))  is  the  set  of  generalized  collective  characters  of  G  in  /?,  and 
the  map  x  associates  with  each  equivalence  class  of  G  -graded  Clifford  system  extensions 
of  R  a  generalized  collective  character.  We  refer  to  a  generalized  collective  character 
<£>  :  G  -+  Picfc  (R)  as  realizable  if  it  is  in  the  image  of  /  ,  that  is,  if  it  is  induced  as  explained 
above  from  a  G  -graded  Clifford  system  extension  of  R.  The  map  x  produces  a  partitioning 
of  the  set  of  equivalence  classes  of  G  -graded  Clifford  system  extensions  of  R, 

Cliff  ^(G,  R)  =  IJ  Cliff*  (G,  tf;  O),  (5) 

<D 

where,  for  any  generalized  collective  character  <f>  e  Hom£/7(G,  Pic^  (/?)),  we  denote  by 
Cliff  £(G,  /?;<!>)  =  x  -1  (3>)  the  fiber  of  x  over  <1>  .  Thus  a  generalized  collective  character 
O  is  realizable  if  the  set  Cliff  *(G,  /?;  <J>)  is  not  empty.  We  refer  to  Cliff^G,  /?;  <t>)  as  the 
set  of  equivalence  classes  of  realizations  of  the  generalized  collective  character  O. 

3.  The  Teichmiiller  cocycle  and  the  obstruction  theorems 

If  R  is  a  Jk-algebra,  let  C(R)  =  {r  e  R\rx  =  jcr,  x  e  R}  denote  its  center.  Then  C(R)  is  a 
fc-algebra  whose  group  of  units  we  denote  by  C  (/?)*. 

We  will  often  use  the  following  elementary  fact,  which  is  a  consequence  of  ([1],  Chapter 
II,  (3.5)). 

Lemma  3.1.  If  P,  Q  are  invertible  R  ®£  Ra  -modules,  then  for  any  two  R  ®£  R°  -isomor- 
phisms a,  /3  :  P  —  >  Q,  //x^re  exwf5  0  unique  u  e  C(/?)*  swc/z  f/z«f  p  =  ua  =  an  (i.e., 
fi(x\=ua(x)  =  a(ux)  for  all  x  e  P). 


Proof.  Given  a  :  P  ->  Q,  an  R  <8>&  /?°  -isomorphism,  the  map  C(R)*   - 

(P,  Q),  u  H>  wa,  is  bijective  since  it  can  be  obtained  as  the  composite  map  of  the  canonical 

group  isomorphism  C(R)*  =  Aut/j^  /?•=(/?),  the  group  isomorphism  —  <8>#  F 

(/?)  =  Aut^(g,A/jo(P)  and  the  bijection  induced  by  a,  a^  :  AutR®kR°(P)  = 

C/5,  fi).  a 

If  P  is  any  invertible  R  <g>£  /?°-module  and  u  G  C(/?)*,  since  x  H>  xw  is  an  /?  ®#  /?°- 
automorphism  of  P,  there  exists  a  unique  element  ctp(u)  e  C(R)*  such  that  ap(u)x  =  xu 
for  all  x  e  P.  Clearly  ap  :  C(R)*  ->  C(R)*  is  an  automorphism  and 

Pick(R)  -1+  Aut(C(/?)*),     p([P])  =  aP  (6) 

is  a  group  homomorphism  (note  that  p  is  the  restriction  to  C(R)*  of  Bass'  homomorphism 
h  :  Pick(R)  ->  Autjk(C(/?))  ([1],  Chap.  II,  (5.4)).  Hence  C(/f)*  is  a  Pic^(^ 
By  composition  with  the  homomorphism  (6)  we  have  for  any  group  G  a  map 


HomG/?(G,  Kcfc(*))  ->  HomG^(G,  Aut(C(/Z)*))  (7) 


that  to  each  generalized  collective  character  of  G  in  the  ^-algebra  jR,  <l>  :  G  — 
associates  a  character  <l>*  —  pO  :  G  ->  Aut(C(/?)*)  from  group  G  in  the  abelian  group 
C(J?)*.  Of  course,  the  set  of  characters  HomGp(G,  Aut(C(/?)*))  is  the  set  of  G-module 
structures  on  C(J?)*.  Hence  every  generalized  collective  character  4>  :  G  ->  Pic^(jR),  of 


154  Antonio  M  Cegarra  and  Antonio  R  Garzon 

G  in  R,  determines  a  G-module  structure  on  C(/?)*  for  which  the  corresponding  G-action 
of  an  element  a  e  G  on  an  element  M  e  C  (R)*  is  given  by  au  =  ap(w)forany  P  e  <£>(<r). 
In  particular, 

xu  =  aux  (8) 

for  any  a  e  G,  u  e  C(/Z)*,  *  e  P  and  P  €  <&(<r).  We  will  denote  by  #  J(G,  C(R)*), 
7?  >  0,  the  nth  cohomology  group  of  G  with  coefficients  in  this  G-module. 

We  will  now  show  how  every  generalized  collective  character  $  :  G  ->  Pic^P)  has  a 
cohomology  class  T($)  €  #|>(G,  C  (/?)*)  canonically  associated  with  it,  whose  construc- 
tion has  several  precedents:  the  Teichmuller  cocycle  homomorphism  /f°(G,  Br(R))  -> 
#3(G,  /?*)  [10,6],  defined  when  R/k  is  a  field  Galois  extension  with  group  G;  the 
Eilenberg-Mac  Lane  obstruction  defined  by  a  G  -kernel,  defined  in  [5]  for  the  study  of 
group  extensions  with  a  non-abelian  kernel;  the  description  by  Kanzaki  [9]  of  the  homo- 
morphism Hl(G,  PicR(R))  -^  H3(G,  7?*),  in  the  Chase-Harrison-Rosenberg  seven  term 
exact  sequence  [2],  about  the  Brauer  group  relative  to  a  Galois  extension  of  commuta- 
tive rings  R/k;  the  Teichmuller  obstruction  associated  to  a  collective  character  <J>  :  G  -» 
Out(jR),  by  Hacque  in  [7,8]  for  the  study  of  obstructions  to  the  existence  of  crossed  product 
rings. 

Let  <f>  :  G  ->  Pic^(^)  be  a  generalized  collective  character  of  a  group  G  in  a  fc-algebra 
R.  In  each  isomorphism  class  *(or)  €  Pic^R),  choose  an  invertible  R  ®*  #°-module 
Pa  e  *(or);  in  particular,  select  PI  =  R.  Since  *  is  a  homomorphism,  the  modules 
Pff  ®R  ^1  ^^  Pa*  Inust  be  ^  ®fc  ^°-isomorphic  for  each  pair  a,  r  e  G.  Then  we  can 
select  R  <S>£  /^-isomorphisms 

r*,r  :  ^  ®J?  Pr  ^  Pert  (9) 


with  ro->i(jc(g)r)=^r  and  TiiCr(r  ®x)  =  rjc,  r  € 
For  any  three  elements  a,  r,  y  e  G,  the  diagram 


(10) 


-  ^  p 

I  <7,T7 


need  not  be  commutative  but,  by  Lemma  3.1,  there  exists  a  unique  element  T*T     e  C  (R)* 
such  that 

iWdVr  ®  PX)  -  r^x(rff,ry(pa  ®  rTty)).  (ii) 

Clearly  r^r  y  =  T^y  =  7^^  =  1  so  that  the  choices  of  Pa  and  ra>T  determine  a 
normalized  3-dimensional  cochain  of  G  with  coefficients  in  C(J?)*. 

Lemma  3.2.  TTze  coc/wm  r  =  T®  :  G3  ->  C(R)*  is  a  3-cocyde  ofG  with  coefficients  in 
the  G-module  C(R)*. 

Proof.  We  must  prove  the  identity 

7ir,r,x^r,Tx,5  ^^r.y.S  =  ^or^^a^-yl  (12) 

for  any  (a,  r,  y,  5)  €  G4.  To  see  this,  we  compute  the  isomorphism 

J  =  (I 


Clifford  system  extensions  155 

in  two  ways.  On  one  hand,  for  all  x  e  Pa  ,  y  e  PT  ,  z  e  Py  and  t  e  P&,  we  have 

/(*  ®  y  ®  z  ®  *  )    =    rffrj(rffr(ras(x  ®  y)  ® 


and  on  the  other  hand 

y  ®  z  ®  o    =    r^y^r^ysorv^c*  ®  y)  ®  ry,5(z  0  0) 


and  comparing  the  two  expressions  together  with  Lemma  3.1  gives  (12).  D 

We  now  observe  the  effect  of  different  choices  of  P0  and  T0^  in  the  construction  of  the 
3-cocycle  T®  for  a  given  generalized  collective  character  O  :  G  -> 


Lemma  3.3.  (i)  If  the  choice  ofT  in  (9)  is  changed,  then  T®  is  changed  to  a  cohomologous 
cocycle.  By  suitably  changing  F,  T®  may  be  changed  to  any  cohomologous  cocycle. 

(ii)  If  the  choice  of  the  invertible  R  ®&  RQ  -modules  P  is  changed,  then  a  suitable  new 
selection  ofV  leaves  cocycle  T®  unaltered. 

Proof,  (i)  By  Lemma  3.1,  any  other  choice  of  ra>r  in  (9)  has  the  form  T^T  =  hffiTrfftT, 
where  h  :  G2  ->>  C(R)*  is  a  normalized  2-cochain  of  G  in  C(R)*. 

For  any  a,  T,  y  €  G  we  have  the  following  expressions  for  the  isomorphism  J  =  F^y 
(r^T  ®  Py)  from  Pa  ®R  Pr  ®R  PY  onto  Pary: 


=      W  Vt/a,r,yra,ry  (x  0  rr>y(>;  0  z)) 
and 


rTfy  (y  ®  z)) 

(^}     T^yh^y  °hI,YYa^Y(x®Vr,Y( 
and  comparing  the  two  expressions  together  with  Lemma  3.1  yield 


an  identity  that  asserts  that  the  3-cocycles  T  and  Tf  are  cohomologous. 

(ii)  If  Pfa  e  *(or),  CT  €  G,  is  another  selection  of  invertible  R  <S)k  /?°-modules,  then  we 
can  select  R  0^  /^-isomorphisms  <pa  :  P'a  -*  P0  and  choose  Tr0  T  :  P'a  ®R  PfT  —  >•  P^T, 
the  isomorphism  making  the  following  diagram  commutative: 


156  Antonio  M  Cegarra  and  Antonio  R  Garzon 


T,-y  1 


(14) 

"9 


1  <r,T~f 

for  each  cr9  r  e  G.  Thus  we  have 


for  all  ;c  e  />£,  y  G  P^  and  z  €  P£. 

Hence  T'ar^  (F^r  (x  ®y)®z)  =  Ta^yYf^Y(x  ®  r^y(y  ®  z))  and  the  3-cocycle  T  is 
unchanged.  D 


These  lemmas  show  that  each  generalized  collective  character  <f>  :  G  ->  Pic^C/^)  de- 
termines in  invariant  fashion  a  3-dimensional  cohomology  class  T(&)  =  [T®]  e  //| 
(G,  C(R)*)  .  We  refer  to  the  map  O  h^  r(O)  as  the  Teichmuller  obstruction  map  (see  [8] 
for  background). 

Next  we  prove  the  main  objective  of  this  paper. 

Theorem  3.4.  A  generalized  collective  character  <$  :  G  ->  Pic^(,R)  w  realizable  if  and 
only  if  its  Teichmuller  obstruction  r(O)  e  ff^(G,  C(J?)*)  vanishes. 

Proof.  Suppose  first  that  (5  =  ©(T€G  Sff  ,  ;)  is  a  realization  of  <D.  Then,  in  the  construction 
of  the  Teichmuller  3-cocycle  T®  of  G  with  coefficients  in  the  G-module  C(/?)*,  one  can 
take  just  the  invertible  R  ®#  j^°-modules  5^,  a  e  G,  a  ^  1,  and  the  canonical  R  (%  R°- 
isomorphisms  rfftT  :  Sa  ®R  ST  ->  Sffr,  T^r(x  ®  j)  =  ^  r^i^  ®  r)  =  ^7  (r)  and  rij<r 
(r  ®  x)  =  7  (r)%  for  each  a,  r  e  G.  Since  multiplication  in  the  ^-algebra  5  is  associative 
r<rr,x(ra,r  ®  5y)  =  F^y^  ®  TT?y)  for  all  a,  r,  /  e  G,  and  then  T^y  =  1  in  (11). 
Therefore,  T($>)  =  [T®]  is  the  zero  cohomology  class. 

Conversely,  suppose  that  the  generalized  collective  character  O  has  a  vanishing  coho- 
mology class  /(<£).  Select  any  invertible  R  ®£  /^-modules  P0  e  O(cr),  a  €  G,  with 
PI  =  /?.  By  Lemma  3.3(i),  there  is  a  choice  of  R  ®£  ^""-isomorphisms  ra>T  :  Pa(8)^ 
A:  -*  FOT  with  Fii(y  .and  r^i  the  canonical  ones,  such  that  the  Teichmuller  3-cocycle 
T®  is  identically  1.  This  means  that  (10)  is  commutative  for  any  or,  r,  x  €  G.  Hence, 
the  family  (Pff,  ra,T)  gives  rise  to  a  generalized  crossed  product  algebra  in  the  sense  of 
Kanzaki  [9]  A  =  0a€G  Pa,  where  the  product  of  elements  x  e  Pa  and  y  e  Pr  is  defined 
by  xy  =  rajT(jc  ®  y),  which  is  a  G  -graded  Clifford  system  over  fc,  extension  of  #  by  the 
canonical  injection  j  :  R  =  PI  ^  A.  Since  /[A  j](^)  =  [/Vl  =  ^(cO,  *  is  realized, 
that  is,  Cliff  *(G,  /?;  <E>)  ^  0.  '  D 

Now,  to  complete  the  classification  of  G  -graded  Clifford  system  extensions  of  a  fc-algebra 
R,  we  have  the  following  result. 


Clifford  system  extensions  1  57 


Theorem  3.5.  If  a  generalized  collective  character  O  :  G  -*  PiCk(R)  is  realizable, 
then  the  set  of  isomorphism  classes  of  realizations  0/<f>,  Cliff  #(G,  R',  4>),  w  a  principal 
homogeneous  space  under  the  abelian  group  //^(G,  C(R)*).  In  particular,  there  is  a 
(non-canonical)  bijection 

Cliff*(G,  /?;  *)  =  #|(G,  C(*)*). 

Proof.  We  will  describe  an  action 

#|(G,  C(*)*)  x  Cliffy,  *;  *)  —  >  Cliffft(G,  *;  *)  (15) 

below. 

Let  /z  :  G2  ->  COR)*  be  a  normalized  2-cocycle  representative  of  an  element  [/z]  €  #<| 
(G,  C(/?)*)  and  (5  =  0a€G  Sa,  j  :  R  =  Si)  be  a  G-graded  Clifford  system  extension 
of  J?,  representative  of  an  element  [5,  j]  €  Cliff£(G,  R\  O).  A  new  G-graded  Clifford 
system  extension  of  R,  (hS,  j)  is  defined  by  considering  the  fc-algebra  hS  which  is  the 
same  G-graded  fc-algebra  as  5  =  ®ff€Q  Sa,  where  the  product  of  elements  x  e  Sa  and 
y  e  Sr  is  now  defined  by 

x*y  =  j(hfftT)xy. 
Since  for  any  x  £  Sa,y  €  ST  and  z  e  SY  we  have 


/0\ 

a  =  j  (har,y)j  (h 


the  multiplication  is  associative  and  so  hS  is  a  /:-algebra.    Furthermore,  Sff  *  ST  =  j 
(ha,r)SaST  =  7  (^O-,T)^OT  =  ^(rr»  §ince  ^<T,T  is  invertible  for  all  a,  r  €  G. 

Therefore  (^5,  7)  is  actually  a  G-graded  Clifford  system  over  k  extension  of  R,  clearly 
representing  an  element  [^5,  j]  €  Cliff&(G,  R;  <l>),  which  we  maintain  depends  only  on 
[h]  and  [5,  j].  To  see  this,  let  us  suppose  that  h'  is  another  representative  of  [h]  and 
(Sf,  j')  is  another  representative  of  [S,  j].  Then,  there  must  exist  a  1-cochain  ifr  :  G  -> 
C(R)*  such  that  hfa  Tff^T^o-  =  ^OT^T*  ^,  r  e  G,  and  a  grade-preserving  isomorphism 
/  :  5  ->  5'  such  that  //  =  /,  from  which  we  build  the  grade-preserving  ^-isomorphism 
*f:hS-+  h's'9  +f(x)  =  f(j(1rff)x)  if  ;c  €  Sff.  For  each  *  €  5a  and  y  €  5T,  we  have 


sothat^/  :  (^5,  j)  ~>  (/z/5/,  7  0  is  actually  an  isomorphism  of  Clifford  system  extensions 
of  J?,  that  is,  [*S,  7]  =  [^'S7,  /]. 

Therefore,  ([A],  [5,  7])  H>  [A5,  7]  is  a  well-defined  action  of  the  abelian  group  H^ 
(G,  C(JR)*)onCliff^(G,  R;  <E>),  which  furthermore  is  a  principal  one.  In  fact,  if  we  suppose 
that  [^5,  7]  =  [5,  7],  there  must  exist  a  grade  preserving  fc-algebra  isomorphism  /  : 
^5  ->  5  such  that  fj  =  j1.  For  each  a  €  G,  the  restriction  f/$a  :  Sa  -»  5^  is 
a  R  ®£  /?°-isomorphism,  and,  by  Lemma  3.1,  there  exists  a  unique  V/v  e  C(J?)*  such 
that  f(x)  =  7(^cr)-^  for  all  ^  €  5ff.  Thus  ^  :  G  ->  C(J?)*  is  a  1-cochain.  Since 
/(z  *  j)  =  f(x)f(y)9  for  any  x  €  S^,  ?  €  5T,  a,  r  6  G,  we  have 


U  fffa)xy-  Therefore,  since  5a5T  =  5aT,  Lemma  3.1  implies  that 


^  T  =  ^  aT/rT,  that  is,  /i  =  8(VO  represents  the  zero  class  in  H^(G,  COR)*). 


158  Antonio  M  Cegarra  and  Antonio  R  Garzon 

Finally,  we  observe  that  action  (15)  is  transitive.  Let  (S,  j),(S',  /)  be  any  two  G- 
graded  Clifford  system  extensions  of  R  representing  elements  in  Cliff ^(G,  R-,  <l>).  Since 
Sff9S'a  €  O(jc)  for  any  a  €  G,  there  must  exist  R  ®&  R° -isomorphisms  fa  :  Sa  ->  S£, 
a  e  G,  with  /i  =  /y""1.  For  each  pair  cr,  r  €  G,  the  square 


ST 


where  the  horizontal  arrows  represent  the  canonical  isomorphisms  x  ®  y  h->  jry,  need 
not  be  commutative.  But,  by  Lemma  3.1,  there  exists  a  unique  hal  e  C(R)*  such  that 
fffl(xy)  =  j'(hfftr)fff(x)My)  for  all  x  e  Sa,y  e  ST.  Thus  h  :  G  -+  C(R)*  is  a 
normalized  2-cochain.  For  any  x  e  5a,  y  e  ST  and  z  e  Sy,  we  have 


and  analogously, 


=      /(VryflrAr,y)/crW 

Lemma  3.1  implies  that  har,yh^T  =  h^ryffhTtY,  that  is,  A  is  a  2-cocycle  of  G  on  C  (/?)*. 
Clearly  /  =  0aeG  /^  establishes  a  G-graded  Clifford  system  extension  isomorphism 
(5,  7)  ->  (^iS17,  /),  and  so  action  (15)  is  transitive.  D 

To  end  this  section,  we  shall  focus  on  that  class  of  rings  known  as  crossed-product  group 
^-algebras.  According  to  ([4],  §5)  an  extension  of  a  k-algebra  R  by  a  group  G  in  the 
sense  of  Hacque  [8],  is  the  same  as  G-graded  Clifford  system  extension  of  R  satisfying 
the  condition  that  in  any  component  there  is  at  least  one  unit.  As  in  Hacque's  paper  [8], 
let  Ext^(G,  R)  denote  the  set  of  isomorphism  classes  of  extensions  of  a  ^-algebra  R  by 
a  group  G.  Then  Ext^(G,  R)  c  Cliff  ^(G,  R),  and  we  shall  characterize  this  subset  of 
Clifffc(G,  R)  by  means  of  collective  characters  as  in  the  following  proposition,  where  we 

take  into  account  the  canonical  group  embedding  Out^(7?)  <-*  PiCk(R)  induced  by  the 
group  exact  sequence  (4),  whose  image  is  ([1],  Chap.  II,  (5.3)) 


Img((5)  =  {[P]  €  Pic£(#)|P  =  R  as  left  tf-module}. 

PROPOSITION  3.6 

For  any  k-algebra  R  and  group  G  there  is  a  cartesian  square 
Ext*(G,  R)>  -  >  Cliff*(G,  R) 

I*  (16) 


HomGp(G,  Out*(*))> >  HomG,(G;  Pic 

that  is,  Extfc(G,  R)  =  %~~l  QIomGp)(G ,  Outjt(,R))  is-  the  set  of  classes  of  those  G-graded 
Clifford  system  extensions  of  R  which  realize  collective  characters  (in  the  sense  of[l,  8]). 


Clifford  system  extensions  159 

Proof.  Let  (5  =  ®O^Q  Sa,  j  )  be  a  G  -graded  Clifford  system  extension  of  R  such  that  for 
any  a  €  G,  there  exists  ua  e  S*  Pi  Sff9  that  is,  an  extension  of  R  by  G.  Right  multiplication 
by  ua  is  an  isomorphism  of  left  ^-modules  R  ->  Rua  =  Sa,  a  €  G,  and  therefore  the 
generalized  collective  character  realized  by  [5,  7],  X[S,/]  :  G  -*  Pic&(/?),  X[S,j](&)  = 
[S0]  factors  through  Outfc(#).  Conversely,  suppose  (S  =  0^^^,  j)  is  a  G-graded 
Clifford  system  extension  of  R  such  that  X[S,  j]  =  ^  f°r  some  <f>  :  G  ->  Out^(j?).  Then, 
if  we  choose  any  ^-automorphism  /(a)  e  <$(a)  for  each  o  e  G,  there  must  exist  an 
R  ®k  /^-isomorphism  cp0  :  R/(a)  =  Sa.  If  uff  =  ^(1),  then  Sa  =  /?«<j  =  WCT^.  From 
SffSa-i  =  /?!$  =  Sff-iSff,  it  follows  thatuaRua-i  =  RI$  =  ua-\Rua.  Then  there  exist 
a,b  e  R  such  that  1  =  uaaua-\  —  u0~\bua  so  that  WQ.  e  5*  fl  Sa  and  therefore  (5,  7) 
represents  an  extension  of  R  by  G.  D 

From  the  general  results  about  Clifford  system  extensions  of  algebras,  we  deduce  the 
following  group  cohomology  classification  of  extensions  of  an  algebra  by  a  group,  which 
was  proved  by  Hacque  in  [8]. 

COROLLARY  3.7 

Let  G  be  a  group  and  R  be  a  k-algebra. 

(i)  Each  collective  character  of  G  in  R,  $  :  G  -»  Outk(R)  determines  in  an  invariant 
fashion  a  three-dimensional  cohomology  class  r(4>)  e  //^(G,  C  (/?)*)  <?/  G  wif/z 
coefficients  in  the  G  -module  (via  O)  o/a//  umto  m  ?/z^  center  of  R. 
(ii)  T/z^r^  w  ^2  canonical  partition  of  the  set  of  equivalence  classes  of  extensions  ofRbyG, 


Ext*(G,  /?)  =       Ext£(G,  Ri  *), 

<E> 

where,  for  any  collective  character  <2>  :  G  -»  Outjt(jR),  Ext&(G,  /^;  <J>)  w  fA^  set  of 

equivalence  classes  of  those  extensions  realizing  <£>. 
(iii)  A  collective  character  <£  :  G  —  >  Out^(/?)  w  realizable,  that  is,  Ext^(G,  /?;  4>)  ^  0  £/ 

an^  on/y  i//^  obstruction  vanishes. 
(iv)  Tff/ze  obstruction  of  a  collective  character  <l>  :  G  ~>  Out^(/^)  vanishes,  then  Ext/, 

(G,  /?;  $)  w  «  principal  homogeneous  space  under  H^(G,  C(jR)*).   /«  particular, 

there  is  a  bijection 

Ext*(G,  /J;  *)  =  F2(G,  COR)*).  (17) 

4.  An  obstructed  collective  character 

It  is  very  easy  to  find  unobstructed  generalized  collective  characters.  Of  course  any  Clifford 
system  yields  one  of  them.  In  this  example  we  shall  exhibit  a  non-realizable  collective 
character,  that  is,  a  group  homomorphism  <I>  :  G  -+  Pic]c(R),  for  particular  group  G  and 
fc-algebra  jR,  such  that  there  is  no  G-graded  Clifford  system  extension  of  R,  (S  =  0a€£ 
Sa,  j  :  R  =  Si)  such  that  <&(<r)  =  [Sa],  o  e  G. 

For  example  consider  G  =  €2  =  (t]t2  =  1  ),  the  cyclic  group  of  order  two,  k  =  FS, 
the  Galois  field  with  five  elements  and  R  =  Fs[Z>io],  the  group  Fs-algebra  of  the  dihedral 
group  DIG  =  (r,  s;  r10  =  1  =  s2,  srs  =  r~l  }. 

Let  /?  :  FS[DIO]  —  >  ^S[D\Q]  be  the  algebra  automorphism  defined  by  fl(r)  =  r1  and 
f}(s)  =  r5s.  Since  ^62(r)  =  r"1  =  srs  and  /82(.s)  =  5r5(r5)7  =  s,  the  automorphism  ^2 


160  Antonio  M  Cegarra  and  Antonio  R  Garzon 


is  simply  conjugation  by  s.  Therefore,  the  equations  3>(1)  =  1  and  <£(r)  =  [ft]  determine 
a  homomorphism 

<D  :  C2  —  >  Out  (F5[Dio])  c  Pic  (F5[Diol)  (18) 

of  the  cyclic  group  C?  into  the  Picard  group  of  Fs[Z)io],  that  is,  a  collective  character  of 

C2inF5[Z>ioL 

PROPOSITION  4.1 

The  Teichmuller  obstruction  T($>)  e  #|(C2,  C(F5[£>i0])*)  is  non-zero. 

Proof.  First,  let  us  observe  that  in  this  case,  a  (normalized)  n-cochain  h  :  €2  x  •  •  •  x 
C2  —  >  C(F5[Z>io])*  is  determined  by  a  single  constant  A(r,  .  .  .  ,  f)  =  h  €  C(Fs[Z>io])*, 
whose  coboundary  is  given  by  Sh  =  f$(h)h~~l  if  n  is  even  or  Sh  =  ft(h)h  if  n  is  odd. 
Since  we  easily  see  that  the  Teichmuller  3-cocycle  is  T®  =  r5,  the  proof  of  the  proposition 
amounts  to  checking  that  there  is  no  unit  h  in  the  center  of  FsfDio]  such  that  ft(h)  =  r5h. 
The  center  of  Fs[Dio]  can  be  described  as  the  8-dimensional  space  over  FS  generated  by 
the  elements 


y,      eg  =  (c2  -f-  04 
with  multiplication  given  by 

2  __         i^  s\               ___         I  i 

Cf\  — —  c3  ~i   -^     ^2^3  *~~  ^2  ~i   ^4  C^C4  —• *-  C3  ~t~  ^5 

C'jCn  —  2cs           C9Cs  *~~  2^7  C    ~~~  /"*c  —I—  2 

C3C7  =  2<77  C3Cg  =  2cg               C^  =  C5  +  2 

C4C7  =  2cg  C4Cg  =  2C7               C$  =  C3  -h  2                C5C6  =  i 

=  2C7           C5Cg  =  2cg  c^  =  1 


Let  CQ  be  the  F5-subalgebra  generated  by  c2;  that  is,  the  span  of  ci, . . . ,  05  and  note 
that  the  minimal  polynomial  of  c2  is  '(t  +  2)3(r  -  2)3.  Then  C(F5[Di0])  =  Co  0  F5c7© 
F5cg  with  multiplication  given  by  c^  =  C7Cg  =  c|  =  0  and  c2c7  =  2cg,  c2cg  =  2c7  and 
we  see  that  there  is  a  homomorphism  <p  :  C(Fs[Dio])  ->  F5  mapping  c2  to  -2  and  cj,  eg 
to  0.  Hence,  <p(c3)  =  2,  <p(c4)  =  -2,  <p(c5)  =  2  and  <p(c6)  =  -1.  Since  /3(c2)  =  c4, 
^3(c7)  =  eg  and  y3(cg)  =  C7,  this  homomorphism  <p  satisfies  that  <p(fi(h))  =  ft(h)  for  all 
h  e  C  (F5[Z>io]).  Therefore,  if  h  €  C(F5[Di0])  is  such  that  0(h)  =  r5h,  then  comparing 
the  image  of  each  side  under  <p  yields  <p(h)  =  —<p(h),  whence  <p(h)  =  0,  and  it  follows 
that  h  is  not  invertible.  D 

Remark  4.2.  Proposition  4.1  is  an  effect  of  inseparability.  If  we  consider  the  Galois  field 
F3  instead  of  F5,  the  resulting  collective  character  (18),  *  :  C2  -+  Pic(F3[Di0])  defined 
similarly  by  *  (t)  =  [ft],  where  ft  is  the  corresponding  algebra  automorphism  determined 
by  ft(r)  =  r1  and  fi(s)  =  r5^,  is  unobstructed.  In  this  case  the  Teichmuller  cocycle  T®  = 
r5  =  3  (h),  is  the  coboundary  of  the  order  4  element  h  =  c44-C5  +  c74-2cg  e  C  (F3[Z>i0])*, 
and  therefore  T(9)  =  0. 


Clifford  system  extensions  161 

Acknowledgement 

The  authors  wish  to  thank  the  referee  for  his  careful  observations.  Proposition  4. 1  was 
proved  in  the  present  improved  form  by  the  referee.  This  work  is  supported  by  DOES: 
PB97-0897. 

References 

[1]  Bass  H,  Algebraic  K-theory  (Benjamin)  (1968) 

[2]  Chase  S  U,  Harrison  D  K  and  Rosenberg  A,  Galois  theory  and  cohomology  of  commutative 

rings,  Memoirs  Am.  Math.  Soc.  52  (1965) 

[3]  Bade  E  C,  Compounding  Clifford's  theory,  Am.  Math.  91  (1970)  236-290 
[4]  Dade  E  C,  Group  graded  rings  and  modules,  Math.  Z.  174  (1980)  241-262 
[5]  Eilenberg  S  and  Mac  Lane  S,  Cohomology  theory  in  abstract  group.  II  Group  extensions  with 

a  non-abelian  kernel,  Ann.  Math.  48  (1946)  326-341 
[6]  Eilenberg  S  and  Mac  Lane  S,  Cohomology  and  Galois  theory  I,  normality  of  algebras  and 

Teichmuller's  cocycle,  Trans.  Am.  Math.  Soc.  64  (1948)  1-20 

[7]  Hacque  M,  Cohomologies  des  anneaux-groupes,  Comm.  Algebra  18  (1991)  3933-3997 
[8]  Hacque  M,  Produits  croises  mixtes:  Extensions  des  groupes  et  extensions  d'anneaux,  Comm. 

Algebra  19  (1991)  3933-3997 
[9]  Kanzaki  T,  On  generalized  crossed  product  and  Brauer  group,  Osaka  J.  Math.  995  (1968) 

175-188 
[10]  Teichmiiller  O,  Uber  die  sogenannte  nichtkommutative  Galoissche  theorie  und  die  relation 

^M.^Mv^U*  =  ^.M.VJT^M.V.JT.  Deutsche  Math.  5  (1940)  138-149 


Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  1 11,  No.  2,  May  2001,  pp.  163-172. 
Printed  in  India 


The  raoltiplicatioe  map  for  global  sections  of  line  bundles 
and  rank  1  torsion  free  sheaves  on  corves 


E  BALLICO 

Department  of  Mathematics,  Universita  di  Trento,  38050  Povo  (TN),  Italy 
E-mail:  ballico@science.unitn.it 

MS  received  21  June  2000;  revised  1 1  September  2000 

Abstract.  Let  X  be  an  integral  projective  curve  and  L  e  Pica(X),  M  €  Pic^(X) 
with  hl  (X,  L)  =  h]  (X,  M)  =  0  and  L,  M  general.  Here  we  study  the  rank  of  the 
multiplication  map  HL,M  :  HQ(X,  L)®H°(X,  M)  -»  HQ(X,  L(g>M).  We  also  study 
the  same  problem  when  L  and  M  are  rank  1  torsion  free  sheaves  on  X.  Most  of  our 
results  are  for  X  with  only  nodes  as  singularities. 

Keywords.  Singular  projective  curve;  rank  1  torsion  free  sheaf;  nodal  curve; 
cuspidal  curve;  line  bundle;  special  divisor. 

1.  Introduction 

Let  X  be  a  smooth  projective  curve  of  genus  g  >  0  and  L ,  M  €  Pic(X)  with  L ,  M  spanned. 
Call  hL  :  X  ->  P(#°(X,  L))  and  hM  :  X  -+  P(#°(X,  M))  the  associated  morphisms. 
Denote  with /XL, M  :  H°(X,  L)<8>#°(X,  M)  ->  #°(X,  L  ®  M)  the  multiplication  map  and 
iL,M  :  P(H°(X,'  L))  xP(#°(X,  M))  -»  P(#°(X,  L)(g>#°(X,  M))  the Segre embedding. 
Let  hLM  :  X  -*  P(ff  °(X,  L))  x  P(#  °(X,  M))  be  the  morphism  induced  by  hL  and  /ZM 
on  the  two  factors.  Call  /L,M  "  X  -^  P(/:/0(X,  L  0  M))  the  morphism  obtained  from 
hi,M  and  the  multiplication  map  ML,M-  The  surjectivity  of  ML,M  means  that  /L,M(X)  is 
linearly  normal  in  its  linear  span  and  dim(Ker(/u-L,M))  is  the  codimension  of  its  linear  span. 
For  any  L,  M  the  surjectivity  of  H>L,M  has  several  important  geometric  consequences  (see 
e.g.  [7])  and  very  good  criteria  for  the  surjectivity  of  /ZL,M  are  known  (see  [10],  Th.  4.a.l, 
and  [7],  p.  514). 
In  §2  we  will  give  a  proof  the  following  result,  proved  also  in  [3]. 

Theorem  1.1.  Fix  integers  m,  n  and  g  with  m  >  1,  n  >  1  and  g  >  0.  Let  X  be  a  general 
smooth  projective  curve  of  genus  g.  Take  a  general  pair  (L ,  M)  €  Pic^+ m  (X)  x  Pic5"1""  (X). 
Then  the  multiplication  map  \JLLM  '  #°(X,  L)  (g)  #°(X,  M)  ->  H°(X,  L  ®  M)  /^ 
maximal  rank,  i.e.  it  is  injective  ifg>  mn  and  it  is  surjective  ifg<  mn. 

Remark  1.2.  In  the  set-up  of  1.1  since  deg(L)  >  g,  deg(M)  >  g  and  both  L  and  M  are 
general,  we  have  hl  (X,  L)  =  hl  (X,  M)  =  0.  Hence  by  Riemann-Roch  we  have  /*°(X,  L) 
=  m  +  1  and  /i°(X,  M)  =  n  +  1.  We  explain  the  numerology  in  the  statement  of  1.1  with 
the  following  example.  Fix  positive  integers  m  and  n.  Let  C  be  a  smooth  projective  curve 
of  genus  mn  and  A  G  Picm+mrt(C),  B  €  Pic"+mw(C)  with  /^(C,  A)  =  ft^C,  £)  =  0. 
Wehave/i°(C,  A)  =  m  +  1,  A°(C,  B)  =  n  +  1,  deg(A®£)  =  n-fm  +  2mn,  fc!( 
=  0  and/i°(C,  A).fc°(C,  B)  =  (m  +  l)(n  +  1)  =  n  +  m  +  1.+  mn  =  fc°(C,  A  ( 
At  the  end  of  §2  we  will  prove  the  following  result. 


164  E  Ballico 

Theorem  1.3.  Fix  integers  m,  n,  g  and  q  with  g>q>Q,m>l,n>l  and  g  >  3.  Let 
n  :  Y  -*  C  be  a  birational  morphism  with  Y  general  curve  of  genus  q  and  C  general  nodal 
curve  with  g  —  q  nodes  and  Y  as  normalization,  i.e.  assume  that  n l  (Sing(C))  is  formed  by 
2g-2q  general  points  ofY.  Take  a  general  pair  (L,  M)  e  Picg+m(C)  x  Pic8+n(C).  Then 
the  multiplication  map  [AL,M  '  #°(C,  L)  ®  #°(C,  M)  -»  #°(C,  L  (g)  M)  has  maximal 
rank,  i.e.  it  is  injective  if  g  >  mn  and  it  is  surjective  ifg  <mn. 

In  §3  we  will  use  the  classical  Brill-Noether  theory  of  special  divisors  to  study  the 
multiplication  map  for  line  bundles  on  nodal  or  cuspidal  curves.  In  §4  we  will  use  1.3  to 
study  some  problems  related  to  the  multiplication  map  for  rank  1  torsion  free  sheaves  on 
nodal  curves. 

2.  Proofs  of  1.1  and  1.3 

We  work  over  an  algebraically  closed  field  K  with  char(K)  =  0;  for  the  case  char(K)  >  0, 
see  Remark  3.4.  For  all  positive  integers  m  and  n  set  ]\(m,n)  :=  Pm  xPn.  Call  n\  (m,ri)  : 
n(w»  n)  -+  Pm  and  ni(m,  n)  :  [~[(m>  n)  -*  Pn  (or  just  n\  and  112)  the  projections.  We 
have  Pic(H(m,  n))  =  Ze2  and  we  will  take  7Ti*(0pm  (1))  and  jr2*(0p«  (1))  as  generators 
ofPic(Y[(m,  n)).  Sometimes  we  will  write  f]  instead  of  Y[(m,  n).  Set  O  :=  On  and  call 
0(1, 0)  and  0(0,  1)  the  two  choosen  generators  of  Pied"])-  Every  one-dimensional  cycle 
T  of  H  has  a  bidegree  (a,  b)  with  a  :=  7\0(1,  0)  and  b  :=  T.0(0, 1).  If  T  is  effective 
and  irreducible  we  have  a  =  deg(jri  |r)deg(jri(r))  and  b  =  deg(7r2|7T)deg(7r2(r)).  The 
tangent  bundle,  T  Y[(m,  n),  of  H(m,  n)  is  isomorphic  to  7ii*(rPm)  ©  7r2*(TPn).  Notice 
that  rPm(-l)  and  TPn(-l)  are  spanned  (e.g.  by  the  Euler  sequence  of  TPS ,  s  =  m  or  n). 
Hence  for  every  integral  curve  X  C  I~I  °f  tvPe  («»&)> tne  vector  bundle  T  Y[(m,n)\X  is 
the  direct  sum  of  a  rank  m  vector  bundle  which  is  the  quotient  of  m  4- 1  copies  of  Ox  ( 1 , 0) 
(and  hence  the  quotient  of  line  bundles  of  degree  a)  and  a  rank  n  vector  bundle  which  is  a 
quotient  of  n  4- 1  copies  of  Ox  (0,  1)  (and  hence  a  quotient  of  72  -f- 1  line  bundles  of  degree 
b).  For  any  locally  complete  intersection  curve  X  C  H(m>  n)> let  NX]/J[(m,n)  be  its  normal 
bundle.  If  X  is  smooth,  then  the  normal  bundle  Nx/Y[(m,n)  of  X  in  Y[(m'  n) is  a  quotient 
of  T  fl(m>  n)\X.  To  prove  Theorem  1.1  we  introduce  the  following  statement: 

ff(m,n),m  >  l,n  >  1:  There  exists  a  smooth  connected  curve  X[/n,  n]  C  ]\(m,ri) 
such  that  pa(X[m,  n])  =  mn,  X[m,  n]  has  bidegree  (mn  +  m,mn  +  n),  the  embedding  of 
X[m,  n]  in  H(m,  n)  is  induced  by  a  pair  of  line  bundles  (L,  M)  with  ft1  (X[m,  n],  L)  = 
/^(Xtm,  n],  M)  =  0,  X[m,  n]  spans  p™+m+»  and  ft^Xfrn,  n],  ^[m)/l]/n(m,n))  =  0. 

Since/i^Xtm.nLL)  =  A^Xtm,  /i],  M)  =  A^Xt/n./i],  L  ®  M)  =  0,  the  condition 
that  X[m ,  n]  spans  pmn+m+n  in  the  statement  of  H  (m ,  n)  is  equivalent  to  the  condition  that 
the  two  maps  X[m,  n]  ->  Pm  and  X[m,  ?i]  ->  Pn  induced  the  inclusion  of  X[m,  n]  into 
H(^,  n)  are  given  by  a  complete  linear  system  (i.e.  by  Riemann-Roch,  that  they  are  non- 
degenerate)  and  that  the  multiplication  map  fMLtM  :  HQ(X[m,  n],  L)®HQ(X[m,  n],  M)  -^ 
HQ(X[m,  n],  L  (g)  M)  is  bijective. 

Remark  2.2.  H(l,  1)  is  true  because  a  smooth  quadric  surface  H(l>  1)  C  P3  contains  a 
smooth  non-degenerate  elliptic  curve  of  bidegree  (2, 2)  and  such  curve  has  as  normal  bundle 
a  degree  4  line  bundle. 

PROPOSITION  2.2 

Fix  an  integer  m  >  1.  IfH(m,  m)  is  true,  then  H(m  +  1,  m  +  1)  is  true. 


Rank  1  torsion  free  sheaves  165 

Proof.  See  P™2+2™  as  a  codimension  2m  +  3  linear  subspace,  A,  of  p>«2+4m+3  Take 
a  solution  X[m,m]  c  f](m»m)  f°r  H(m9m)  and  see  ]~l(m'  m)  as  a  linear  section  of 
Yl(m  +  1,  m  +  1)  c  P™2+4™+3.  Fix  5  C  X[m,  m]  with  card(S)  =  2m  -f  2  and  5  spanning 
a  linear  subspace  (S)  of  p™2+2m  with  dim((S»  =  2m  + 1 .  Let  C  be  a  smooth  rational  curve 
and  consider  the  pair  (R,  R)  e  Pic(C)  xPic(C)  withdeg(/?)  =  2m +  2.  The  multiplication 
map  M^  :  ff°(C,  J?)  0  #°(C,  fl)  ->  //°(C,  T?®2)  is  surjective  and  tf®2  embeds  C  into 
a  (4m  +  4) -dimensional  projective  space  W  as  a  rational  normal  curve;  call  D  C  W  its 
image.  Hence  D  may  be  seen  both  as  a  smooth  rational  curve  of  degree  4m  H-  4  in  W 
and  a  curve  of  bidegree  (2m  -f  2,  2m  +  2)  in  f](r  >  0  f°r  anv  r  >  2m  4-  1 .  We  may  take 
W  C  Pm  +4m+3  in  such  a  way  that  W  n  A  contains  5;  here  we  use  that  A  has  codimension 
2m  +  3  =  dim(W)  -  (2m  +  1)  in  P™2+4w+3  and  that  card(S)  <  dim(W).  The  group 
Aut((5))  acts  transitively  on  the  set  of  ordered  (2m  +  2)-ples  of  points  in  linear  general  in 
(S) .  Any  such  (2m +2) -pie  is  contained  in  a  codimension  2m +3  linear  section  of  a  rational 
normal  curve  of  W.  Hence  we  may  assume  that  D  D  A  =  5.  Set  Y  :=  X[m,  m]  U  D.  Y 
has  bidegree  ((m  +  l)(m  -I-  2),  (m  +  l)(m  -f-  2)),  the  same  bidegree  of  X[m  +  1,  m  -f-  1]. 

Claim.  We  may  find  such  D  with  D  C  Yl(m  +  1,  m  +  1),  i.e.  with  F  C  fl(m  +  1,  w  +  1) 
andF  fl  A  =  X[m,m]. 

Proof  of  the  Claim.  First  we  will  check  that  Pic(7)  is  an  extension  of  Pic(X[m,  m])  x 
Pic(D)  =  Pic(X[m,  m])  x  Z  by  a  multiplicative  group  isomorphic  to  (K*)e(2m+1).  More 
precisely,  every  E  e  Pic(7)  is  uniquely  determined  by  E\X[m,  m],  E\D  and  by  the  gluing 
data  at  each  of  the  2m  +  2  points  of  5;  since  D  =  P1,  E\D  is  uniquely  determined 
by  the  integer  deg(£|Z));  each  of  these  gluing  data  is  uniquely  determined  by  a  non- 
zero scalar  (and  vice  versa,  each  non-zero  scalar  induces  a  gluing  datum  at  one  point  of 
5);  however,  since  for  any  E'  e  Pic(X[m,m])  and  E"  e  Pic(D)  we  have  Aut(£")  = 
Aut(£'//)  =  Aut(£")  =  K*,  we  may  multiply  all  these  gluing  data  by  a  common  non- 
zero scalar  and  obtain  an  isomorphic  line  bundle  on  Y.  Hence  Pic(F)  is  an  extension 
of  Pic(X[m,m])  x  Pic(D)  by  (K*)®^2m+1>.  Take  any  L'  e  Pic(F),  M'  €  Pic(7)  with 
Z/|X[m;  m]  =  L,  M'|X[m,  m]  =  M  and  deg(Z/|D)  =  deg(M;|D)  =  2m  +  1.  Consider 
the  Mayer- Vietoris  exact  sequence  for  Z/, 

0  ->  L'  ->  Z/|X[m,  m]  ®  Lf\D  ->  L'|S  ->  0  (1) 

and  the  corresponding  Mayer-Vietoris  exact  sequence  for  M' '.  Since  card(S)  =  2m  +  2  and 
deg(L'\D)  =  deg(M/|Z))  =  2m  +  1,  the  restriction  maps  H°(D,  Lf\D)  ->  //°(5,  Lf\S) 
and  #°(Z),  M'\D)  ->•  //°(5,  M'15)  are  surjective.  Hence  by  the  Mayer-Vietoris  exact  se- 
quences we  obtain  /z°  (7,  L')  =  m  +  1,  /z°(F,  MO  =  m  +  1,  hl(Yt  L'}  =  Oand/zl(7,  Mx)  = 
0.  Similarly,  we  obtain  that  L'  and  M7  are  spanned  and  (for  general  gluing  data)  induce  an 
embedding  of  Y  into  f~[(m  +  l,m-|-l),  proving  the  Claim. 

The  variety  H(m> m)  is  me  complete  intersection  of  two  Cartier  divisors  of  Yl(m  + 
l,m  +  1),  one  of  type  (1,0)  and  one  of  type  (0,  1).  Hence  NX[m,m\/ ri(m+i, w+i)  = 
^X[m,ni]/n(w,m)  ®  £  0  M.  Thus  /z1  (X[m,  m],  ^m5m]/n(m+1,m+1))  =  0.  By  con- 
struction D  fi  X[m,  m]  =  5  and  D  intersects  quasi-transversally  X[m,  m].  Hence  F 
is  a  connected  nodal  curve  with  pa(Y)  =  m2  +  2m  H-  1.  Since  A  n  W  =  (S)  and 
dim({S'»  +  dim(W)  =  codim(A),  Y  spans  pm2+4™+3.  Hence  by  semicontinuity  it  is 
sufficient  to  prove  that  Y  is  smoothable  and  that  hl(Y,  A/V/[](m+i,m-i-i))  =  0.  Since 
D  has  bidegree  (2m  +  1,  2m  +  1)  in  Yl(m  H-  1,  m  +  1),  its  normal  bundle  is  a  quo- 
tient of  a  direct  sum  of  line  bundles  of  degree  2m  -f-  1.  Since  every  vector  bundle  on 
D  =  P1  is  a  direct  sum  of  line  bundles,  we  obtain  that  every  line  bundle  appearing  in 


166  EBallico 

a  decomposition  of  ND/Y[(m+\,  m+i)  nas  degree  at  least  2m  4-  1.  By  [11],  Cor.  3.2  and 
Prop.  3.3,  or  [13],  NY/H(m+i,  m+\)\X[m,  m]  (resp.  #y/r](m+i,  m+\)\D)  is  obtained  from 
tf*[m,m]/no«+i,  m+1>  (resp-  ^/nfa+i.  "H-n)  making  2™  +  2  positive  elementary  trans- 
formations. Hence  hl(X[m,  m],  NY/i\(m+i,  m+i)I^IX  m])  =  0  and  every  line  bundle 
appearing  in  a  decomposition  of  NY/Y[(m+i,  m+i)l^  has  degree  at  least  2m  -f  1.  The  last 
remark  implies  the  surjectivity  of  the  restriction  map  p  :  /f°(Z),  Afy/  r](m+i,  m+i)|£>)  ~* 
#°(.S,  Ny/[](m-t-i,  m+i)|5).  By  the  Mayer-Vietoris  exact  sequence 

0  ~>  Afy/non+l,  m+i)  -»  Wy/nc/n+i,  m+i)|X[m,  m]  0  %/r](m+i,  m+i)lD 

*  0,  (2) 


we  obtain  /z1  (7,  A^y/j-j^+i^  m+i))  =0.  Furthermore,  as  in  [11],  Th.  4.1,  or  [13]  we  obtain 
also  that  Y  is  smoothable.  Notice  that  we  may  apply  the  semicontinuity  theorem  for  the 
dimension  of  the  kernel  of  the  multiplication  map  for  a  flat  family  of  pairs  of  non-special 
line  bundles  on  a  flat  family  of  curves,  because  the  non-speciality  condition  implies  that 
the  corresponding  cohomology  groups  have  constant  dimension.  By  semicontinuity  we 
obtain  the  result  for  a  general  triple  (Z,  L",  M")  with  Z  of  genus  (m  +  I)2  and  (L",  M"} 
a  general  pair  of  line  bundles  on  Z  with  degree  (m-f-l)2-|-m-hl. 

PROPOSITION  2.3 

Fix  integers  m,  n  with  n  >  m  >  1.  Assume  that  H(m,n)  is  true.  Then  H(m,  n  +  1)  is  true. 

Proof.  We  will  show  how  to  modify  the  proof  of  2.2.  Notice  that  pa(X[m,  n  -f  1])  = 
pa(X[m,  n])  4-  m.  We  start  with  (X[m,  n],  L,  M)  satisfying  H(m,  n).  Hence  L,  M  € 
Pic(X[w,  n]),  deg(L)  =  pa(X[m,  n])  -f-  m  =  mn  -f-  n  and  deg(M)  =  mn  +  n.  We  take 
S  C  X[m,  n]  C  A  :=  (Il(m,  n))  with  card(S)  =  m  -f  1  and  dim((S»  =  m.  Now  D  is 
a  smooth  rational  curve  and  it  is  embedded  into  Yl(x,y),x  >  m,y  >  m  +  1,  by  a  pair 
(R\,  #2)  with  deg(jRi)  =  m  and  deg(7?2)  =  m  4-  1,  i.e.  of  bidegree  (m,  m  +  1).  Hence 
degCKi<8>#2)  =2m+l.  SetF  :=X[m,n]UD.  Since  A°(D,  /?/)  =  deg(^/)  +  l  >  card  (5) 
for  i  =  1,  2,  every  part  of  the  proof  of  2.2  works  in  our  new  set-up,  proving  2.3. 

Proof  of  1.  1.  (i)  Here  we  will  cover  the  case  0  <  g  <  mn,  i.e.  when  we  need  to  prove 
that  for  a  general  triple  (X,  L,  M)  the  multiplication  map  \J(,LM  is  surjective.  Since  the 
case  g  <  I  is  well-known  and  trivial,  we  assume  g  >  2  and  hence  n  >  2.  Since  H(m,ri) 
is  true,  we  know  the  case  g  =  mn.  Hence  we  may  assume  2  <  g  <  mn.  We  start  with 
X[19  1]  satisfying  H(l,  1)  and  then  we  follow  the  proofs  of  2.2  and  2.3  made  to  obtain 
a  proof  of  H(m,  n).  However,  at  each  step  of  the  proof  we  take  D  intersecting  the  other 
curve  in  a  subset,  Sf,  of  S.  For  instance  if  n  >  m  and  mn  —  m  —  I  <  g  <  mn,  we  take 
card(S')  =  g  -  mn  +  n.  Call  Yf  the  curve  X[m,  n  -  1]  U  D  with  D  n  X[m,  n  -  1]  =  Sf  . 
The  proofs  of  2.2  and  2.3  and  semicontinuity  proves  1.1  for  this  triple  (m,  n,  g). 

(ii)  Now  we  assume  g  >  mn.  By  induction  on  g  for  a  fixed  pair  (m,  n)  and  the  case 
g'  =  mn  (the  bijective  case)  proved  in  part  (i)  we  may  assume  the  result  for  the  triple 
(m,  n,g  -  1).  Let  (C,  A,  B)  a  general  triple  satisfying  the  statement  of  1.1  for  the  triple 
(m,  n,  g  -  1).  Fix  two  general  points  {P,  Q]  of  C  and  let  7  be  the  nodal  curve  CUD 
with  D  =  P1  and  C  n  D  =  {P,  g}.  By  semistable  reduction  F  is  the  flat  limit  of 
a  flat  family  of  smooth  connected  curves  of  genus  g.  Take  any  L,  M  G  Pic(F)  with 
L\C  =  A,  Af|C  =  B  and  deg(L|D)  =  deg(M  |Z>)  =  1.  We  saw.in  the  proof  of  2.2  that 
the  set  of  all  such  L  (resp.  M)  is  not  empty  and  parametrized  by  an  extension  of  Pic°(C) 
by  K*.  Since  the  restriction  maps  H°(D,  L\D)  ~>  0{/>,0}  and  H°(D,  M\D)  ->  0{p,£) 


Rank  1  torsion  free  sheaves  167 

are  surjective,  as  in  the  proof  of  2.2  a  Mayer-Vietoris  exact  sequence  similar  to  (1)  shows 
that  /z°(F,  L)  =  m  +  1,  /z°(F,  M)  =  m  +  1  and  /z1  (F,  L)  =  /z1  (F,  M)  =  0.  Furthermore, 
the  same  exact  sequence  induces  an  isomorphism  of  //°(C,  A)  (resp.  //°(C,  5))  with 
H°(F,  L)  (resp.  #°(F,  M))  and  a  surjection  of  #°(C,  A  ®  5)  onto  #°(F,  L  ®  M ).  Hence 
the  injectivity  of  H,A,B  implies  the  injectivity  of  ^LM-  By  semicontinuity  we  conclude  as 
in  the  last  part  of  the  proof  of  2.2. 

Proof  of '1.3.  Look  again  to  the  proof  of  1 .1  and  in  particular  to  the  proof  of  2.2.  Now  we 
take  as  X[l,  1]  a  rational  curve  with  an  ordinary  node  as  only  singularity.  As  in  the  proof 
of  2.2  we  obtain  the  result  in  the  case  m  =  n  =  1.  Now  we  consider  the  inductive  step  in 
the  proofs  of  2.2  and  2.3.  Just  to  fix  the  notation  we  assume  the  case  (m,  m)  and  prove  the 
case  (m  +  1 ,  m  +  1).  Now  X[m,m]  is  the  general  rational  curve  with  mn  ordinary  nodes  as 
only  singularities.  Set  Y  :=  X[m,  m]  U  D.  We  need  to  deform  Y  inside  Y[(m  +  1,  wi  +  1) 
to  an  irreducible  rational  curve  with  only  nodes  as  singularities.  Hence  it  is  sufficient  to 
prove  that  we  may  smooth  exactly  one  node  (any  node  we  chose)  in  Y  n  D  keeping  singular 
the  other  singular  points  of  Y  Pi  D  and  without  smoothing  the  other  points  and  keeping 
singular  the  singular  points  of  X[m,  n].  If  instead  of  X[mt  n]  we  would  have  a  smooth 
curve,  this  would  be  the  notion  of  strong  smoothability  considered  in  [11],  §1.  The  part 
concerning  the  nodes  in  X[m,  m]  fl  D  is  easy  because  card(X[m,  m]  fl  D)  =  2m  +  2  and 
every  line  bundle  appearing  in  a  decomposition  of  Ny/Y[(m+i,  m+i)l^  nas  degree  at  least 
2m  +  1.  Hence  hl(D,  (A^y/pj(m+i,  m+i)l^)("~^))  =  0  and  we  may  apply  the  proof  of 
[11],  Th.  4.1.  We  know  that  h[(Y,  Ny/r^/n+i./w-i-i))  =  0  an<^  hence  that  Y  is  a  smooth 
point  of  Hilb(H(^  +  1,  m  +  1)).  Furthermore,  by  induction  on  m  we  may  assume  that 
each  subset  of  the  set  of  all  nodes  of  X[m,  m]  may  be  smoothing  independently,  i.e.  that 
for  every  subset  F  of  Sing(X[m,  m])  the  set  of  curves  inf|(m  +  l,/n  +  l)  near  X[m,  m]  in 
which  we  smooth  exactly  the  nodes  in  Sing(X[m,  m])\F  has,  near  7,  codimension  card(F) 
in  Hilb(n(^  +  1,  m  +  1)).  The  same  assertion  for  Y  follows  from  this,  card(S)  =  2m  + 
2,  that  every  line  bundle  appearing  in  a  decomposition  of  Wy/]-](m+i,m+i)l^  has  degree 
at  least  2m  +  1  and  a  Mayer-Vietoris  exact  sequence  as  in  the  proof  of  [11],  Th.  4.1. 
Hence  we  obtain  the  case  q  =  0  of  1.3.  If  q  >  0  we  just  smooth  q  nodes  and  apply 
semicontinuity. 


3.  Line  bundles  on  singular  curves 

For  any  triple  g,  r,  d  of  integers,  let  p(g,  r,  d)  :—  g  —  (r  +  l)(g  +  r  —  d)  be  the  so-called 
Brill-Noether  number  associated  to  g,  r  and  d.  For  any  smooth  projective  curve  X,  set 
Wrd(X)  :=  {L  €  Pic(X)  :  h°(X,  L)  >  r  +  1}.  On  a  general  smooth  curve  X  of  genus 
g  >  2  we  have  Wrd(X))  ^  0  if  and  only  if  p(g,  r,  d)  >  0;  if  p(g,  r,  d)  >  0,  then  WJ(X) 
is  non-empty,  smooth  outside  Wd*l(X)  and  of  pure  dimension  p(g,r,d)\  Wd(X)  is 
irreducible  if  p(g,  r,  d)  >  0  ([1],  chs  V  and  VII,  and  in  particular  the  references  [9]  and 
for  the  smoothness  and  irreducibility  in  arbitrary  characteristic).  If  p(g,  r,  d)  >  0  this 
implies  that  a  general  L  e  WJ(X)  has  no  base  points  and  hQ(X,  L)  =  r  +  1;  here  and 
in  the  statements  of  3.1,  3.2  and  3.5  if  p(g,  r,  d)  =  0  (i.e.  if  WJ(X)  is  finite)  the  word 
'general  L  €  WJ(X)'  means  'every  L  €  W^(X)';  if  C  is  singular  (i.e.  q  ^  g)  in  the 
statement  of  3.1, 3.2  and  3.3  the  word  'general'  means  only  'general  in  a  smooth  component 
with  the  expected  dimension  p(g,  x  —  1,  a)  and  p(g,  y  —  1,  by  because  we  do  not  claim 
any  irreducibility  result  for  the  schemes  WJ(C)  when  C  is  a  singular  curve.  In  the  smooth 
case  (q  =  g)  when  p(g,  x  —  1,  b)  =  0  to  have  'for  all  L  G  Wd(Xy  we  need  to  use  [6] 
and  hence  we  need  to  assume  char(K)  =  0. 


168  E  Ballico 

Theorem  3.1.  Fix  integers  g  and  q  with  g  >  q  >  0  and  g  >  3.    Let  n  :  Y  -*  C 

be  a  birational  morphism  with  Y  general  curve  of  genus  q  and  C  general  nodal  curve 
with  g  —  q  nodes  and  Y  as  normalization,  i.e.  assume  that  jr^CSingCC))  is  formed  by 
2g  -  2q  general  points  of  Y.  Fix  integers  a,b,x  and  y  with  2  <  x  <  g  —  2,2  <  y  < 
g+x-a-  l,p(g,*-  1,#)  >  0,  Q<a<2g-2andg  +  y-x-l  <b<g  +  y-~l. 
Let  L  e  W%~l(C)  and  M  e  W^~  (X)  be  general  elements.  Then  the  multiplication  map 
VLM  :  #°(C,  L)  ®  H°(C,  M)  -*  #°(C,  L  <g>  M)  is  injective. 

Proof.  By  [9],  Prop.  1.2,  there  is  a  nodal  curve  D  with  pa(D)  =  g  and  exactly  g  ordinary 
nodes  such  that  for  every  L  €  W%~l(D)  with  /z°(D,  L)  =  jc  the  multiplication  map 
ML,wD<g)L*  :  #°(A  £)  ®  #°(A  ^D  ®  L*)  ->  //°(£>,  &>£>)  is  injective;  we  will  only  use 
that  this  is  true  just  for  one  L  €  W£~l(D)  with  /z°(Z),  L)  =  AT.  By  semicontinuity  for  a 
general  nodal  curve,  C,  with  p^  (C)  =  g  and  with  exactly  g  —  q  nodes  as  only  singularities 
there  is  L  €  Pic(C)  with  deg(C)  =aJh®(C,L)  —  x  and  such  that  the  multiplication  map 
VL,O>C®L*  :  #°(C,  L)  <g>  /f  °(C,  o;c  (8)  L*)  ->  #  °(C,  o>c)  is  injective.  By  Riemann-Roch 
wehave/z!(C,  L)  =  g+x  —  a  —  1.  By  assumption  we  have  g—  a  —  1  <  y  <  g-Kx—a  —  1  and 
fe  >  g+v  —  jc  —  1.  LetZ)  (resp.  ZX)  be  the  union  of  g+x  —  a  —  1  —  y  (resp.  fc>  —  g  —  v+x  +  1) 
general  points  of  C.  Set  fl  :=  coc  ®  L*(-D)  and  A  :=  ^(D7).  Hence  deg(J?)  = 
^  +  ^-^  +  1  <^  =  deg(A).  Since  Z)  is  general,  we  have  A°(C,  /?)  =  j.  Adding 
Z)  as  a  base  locus  we  may  see  the  vector  space  //°(C,  R)  as  a  subspace  of  //°(C,  &>c  ® 
L*).  Thus  the  multiplication  map  fjLLtR  :  HQ(C,  L)  ®  H°(C,  /?)  ->  //°(C,  L  ®  /?)  is 
injective.  By  Rieman-Roch  we  have  hl(C,  R)  =  x.  Thus  hl(C,  R)  >  deg(D').  Hence 
/z°(C,  A)  =  A°(C,  /?)  by  the  generality  of  D7,  i.e.  A  e  Wyb~l(C)  and  the  complete 
linear  system  associated  to  A  has  Dr  in  its  base  locus.  Thus  the  multiplication  map  /z^,  A  '• 
#°(C,  L)®#°(C,  A)  ->•  /f°(C,  L0  A)  is  injective.  Hence  by  semicontinuity  for  general 
M  e  W/'^C)  the  multiplication  map  /XL,M  :  H°(C,  L)  ®  /f°(C,  M)  ->  H°(C,  L  ®  M) 
is  injective.  Quoting  [5]  instead  of  [9]  we  have  the  following  result. 

Theorem  3.2.  Fix  integers  g  and  q  with  g  >  q  >  0  and  g  >  3.   Le?  TT  :  Y  ->  C  Z?e 

fl  birational  morphism  with  Y  general  curve  of  genus  q  and  C  general  cuspidal  curve 
with  g  -  q  nodes  and  Y  as  normalization,  i.e.  assume  that  Tr^SingCC))  is  formed  by 
g  -  q  general  points  of  Y.  Fix  integers  a,  b,  x  and  y  with  2  <  x  <  g  ~  2,  2  <  y  < 
g+x-a-l,p(g,x-l,a)  >  0,  0  <  a  <  2g-2andg  +  y-x-l  <b  <  g  +  y-l.  Let 
L  e  W%~l(C)  and  M  €  W%~  (X)  be  general  elements.  Then  the  multiplication  map 
,  L)  (g)  H°(C,  M)  -^  H°(C,  L  (g)  M)  w  in> 


Remark  3.3.  Theorem  3.2  is  true  with  the  same  proof  for  every  rational  cuspidal  curve,  not 
just  the  general  one  ([5]). 

4.  Rank  1  torsion  free  sheaves 

Let  C  be  an  integral  protective  curve  and  F  and  G  rank  1  torsion  free  sheaves  on  C.  The 
sheaf  F  (g)  G  may  have  torsion,  but  the  sheaf  F  ®  G/Tors(F  ®  G)  is  a  rank  1  torsion 
free  sheaf.  Call  £F,G  :  #°(C,  F)  ®  H°(C,  G)  ->  #°(C,  F  (g)  G/Tors(F  (g)  G))  the 
composition  of  the  multiplication  map  IJLF,G  •  H°(C,  F)  (g)  HQ(C,  G)  ->-  H°(C,  F  (g)  G) 
with  the  map  //°(C,  F  ®  G)  -»  /f°(C,  F  ®  G/Tors(F  (g)  G))  induced  by  the  quotient  map 
F  <g>  G  ->  F  (g>  G/Tors(F  (g)  G).  We  believe  that  the  linear  map  PF,G  is  more  significant 
and  has  better  behaviour  than  the  plain  multiplication  map  /ZF,G-  In  this  section  we  study 
^d  MF,G  in  the  case  of  nodal  curves.  The  general  set-up  works  for  curves  with  only 


Rank  I  torsion  free  sheaves  169 

ordinary  nodes  and  ordinary  cusps  as  singularities  (see  (4.1)).  The  restriction  to  nodal 
curves  come  from  the  use  of  1.3.  In  many  interesting  cases  the  map  $/7  G  is  induced  from  a 
multiplication  map  for  line  bundles  on  a  partial  normalization  of  C  (see  (4.2)).  Here  is  the 
general  set-up.  Let/  :  Y  ->•  C  be  a  birational  morphism  between  integral  projective  curves. 
Set<5  :=  pa(Y)  —  pa(C).  We  have  <5  >  Oand<5  =  0  if  and  only  if  /  is  an  isomorphism.  For 
every  rank  1  torsion  free  sheaf  A  on  7  the  coherent  sheaf  /*  (A)  is  a  rank  1  torsion  free  sheaf 
onC.  If  A  =  f*(B)  for  some  rank  1  torsion  free  sheaf  B  on  C,  then/*  (A)  =  #® /*(0y) 
(projection  formula)  and  hence  deg(/*(A))  =  deg(A)  +  <5.  By  the  very  definition  of 
the  direct  image  functor  we  have  /i°(C,  /*(A))  =  /i°(F,  A).  Since  /  is  finite,  we  have 
hl(C,  /*(A))  =  hl(Y,  A).  It  is  easy  to  check  that  for  every  rank  1  torsion  free  sheaf  B 
on  C  the  natural  map  fB*  :  HQ(C,  B)  -»  HQ(Y,  /*(B)/Tors(/*(£)))  is  injective.  Let 
L,  M  be  rank  1  torsion  free  sheaves  on  Y.  Since  #°(7,  L)  =  /f  °(C,  /*(£)),  //°(F,  M)  = 
#°(C,  /*(M)),  the  multiplication  map  /ZL,M  :  H°(Y,  L)  <g>  HQ(Y,  M)  ->  //°(r,  L  0  M) 
induces  a  morphism  £L,M  :  #°(^  L)  ®H°(Y,  M)  ->  /f°(y,  L  ®  M/Tors(L  ®  M)), 
a  morphism  0LfAft/  :  /f°(C,  /*(L))  ®  H°(C,  /*(M))  ->  /f°(C,  /*(L  ®  M))  and  a 
morphism  <xLtMtf  :  #°(C,  /*(L))  ®  #°(C,  /*(M))  -»  #°(C,  /*(L  ®  M)/Tors(/*(L  ® 
M))).  A  section  of  a  torsion  free  sheaf  on  a  reduced  curve  is  uniquely  determined  by  its 
restriction  to  a  Zariski  open  dense  subset  of  the  curve.  Hence  if  \JLL,M  is  injective,  then 
«L,M,/  is  injective. 

(4. 1 )  Let  R  be  the  local  ring  either  of  an  ordinary  node  (i.e.  of  an  A  i  singularity),  P,  of  an 
irreducible  curve  or  of  an  ordinary  cusp  (i.e.  of  an  A2  singularity).  Let  m  be  the  maximal 
ideal  of  R.  If  R  is  an  ordinary  node  will  say  that  a  coherent  sheaf  on  Spec(/?)  is  torsion 
free  near  P  if  its  completion  has  no  nonzero  element  killed  by  an  element  of  R  which  is 
not  a  zero-divisor  of  R',  this  is  the  definition  used  in  [4],  With  this  convention  every  finitely 
generated  torsion  free  R  -module  M  (up  to  a  completion)  is  of  the  form  R®a  ©  m®b  for 
some  integers  >  0,  b  >  0,  a  +b  >  0,  with  a+b  =  rank(M)  ([4],  Th.  2.4.2  and  Remark  1 
after  that,  or  [14],  Prop.  2  at  p.  162).  The  same  is  true  if  R  is  an  ordinary  cusp.  We  will 
need  only  the  case  rank(M)  =  1;  hence  either  M  =  R  or  M  =  m.  It  is  easy  to  check  that 
m  contains  a  rank  1  submodule  M  with  M  =  R  and  m/M  ==  K;  obviously  m  is  contained 
in  the  rank  1  free  module  R  and  jR/m  =  K. 

For  any  coherent  sheaf  F  on  an  integral  projective  curve  X  with  pure  rank  r  the  degree 
deg(F)  of  F  is  defined  by  the  Riemann-Roch  formula  deg(F)  :=  x(F)  4-  r(g  —  1).  If  F 
is  a  torsion  free  sheaf  on  X,  set  Sing(F)  :=  {P  e  X  :  F  is  not  locally  free  at  P]  =  {P  € 
Sing(X)  :  F  is  not  locally  free  at  X}. 

(4.2)  Let  C  be  an  integral  projective  curve  whose  only  singularities  are  ordinary  nodes 
and  ordinary  cusps.   Let  F  be  a  rank  1  torsion  free  sheaf  on  C.   Set  S  :=  Sing(F)  = 
{P  G  C  :  F  is  not  locally  free  at  P}.  Hence  by  4.1  for  every  P  e  Sing(F)  near  P  the 
sheaf  F  is  formally  equivalent  to  the  maximal  ideal  of  OC,P-    Set  5  :=  card(S).    Let 
TT  :  Y  — >  C  be  the  partial  normalization  of  C  in  which  we  normalize  only  the  points 
of  S.  We  have  pa(C)  =  pa(Y)  +  8.   Set  L  :=  7T*(F)/Tors(jr*(F)).  By  4.1  we  have 
L  €  Pic(C),  F  =  ;r*(L)  and  deg(F)  =  deg(L)  +  S.  Let  M(C;  x,  S)  be  the  set  of  all  rank  1 
torsion  free  sheaves,  G,  on  C  with  deg(G)  =  x  and  Sing(G)  =  S.  Now  we  will  use  the 
following  observation. 

Remark  4.3.  Let  C  be  an  integral  projective  curve  whose  only  singularities  are  ordinary 
nodes  and  ordinary  cusps.  Fix  5  C  Sing(C)andlet7r  :  Y  -»  C  be  the  partial  normalization 
of  C  in  which  we  normalize  only  the  points  of  S.  Pic°(F)  is  a  g -dimensional  algebraic 
group,  q  :—  pa(C)  —  card(S)  =  pa(Y),  which  is  an  extension  of  an  abelian  variety  of 


170  EBallico 

dimension  pa(C)  —  card(Sing(C))  by  a  connected  affine  group  G;  G  is  the  product  of 
some  copies  of  the  additive  group  (the  number  of  copies  being  the  number  of  cusps  of  7, 
i.e.  of  the  cusps  in  Sing(C)\S)  and  some  copies  of  the  multiplicative  group  (the  number 
of  copies  being  the  number  of  nodes  of  Y).  In  particular  Pic°(F)  is  an  irreducible  q- 
dimensional  variety.  Hence  for  every  integer  .x  the  set  M(C\  x,  S)  has  a  natural  structure  of 
g -dimensional  irreducible  algebraic  variety.  Hence  we  are  allowed  to  consider  the  general 
element  of  M(C;  *,  S). 

Take  another  rank  1  torsion  free  sheaf  G  with  5  =  Sing(G).  Set  M  :=  jr*(G)/Tors(7T* 
(G)).  HenceG  =  ;r*(M)anddeg(G)  =  deg(M)+<5.  By(4.1)wehaveF(g>G/Tors(F<g>G) 
=  jr*(L  ®  M).  Since  #°(C,  F)  ~  HQ(Y,  L),  #°(C,  G)  ~  HQ(Y,  M)  and  H°(Y,  L  ®  M) 
=  #°(C,  n*(L  <8>  M)),  the  linear  maps  ^L,M  and  ULM^  nave  kernel  and  cokernel  with 
the  same  dimension.  In  particular  AIL.M  is  surjective  (resp.  injective)  if  and  only  ifai,Mj 
is  surjective  (resp.  injective).  Hence  by  Theorem  1.3  for  the  integer  q  :=  g  —  8  we  obtain 
the  following  result. 

PROPOSITION  4.4 

Let  C  be  an  integral  projective  curve  whose  only  singularities  are  ordinary  nodes.  Fix  a 
set  S  C  Sing(C)  and  set  g  :=  pa  (C)  and  8  :=  card(S).  Let  n  :  Y  ->  C  be  the  partial 
normalization  ofC  in  which  we  normalize  only  the  points  ofS.  Fix  integers  a,  b  with  a  >  g 
and  b  >  g.  Then  for  general  element  jr*(L)  6  M(C\  a,  S)  and  n*(M)  €  M  (C;  b,  S)  the 
map  C£L,M,TT  has  maximal  rank. 

Remark  4.5.  Let  C  be  an  integral  projective  curve  whose  only  singularities  are  ordinary 
nodes  or  ordinary  cusps.  Set  g  :=  pa(C).  Fix  S  c  Sing(C)  and  set  s  :=  card(S).  Let 
TT  :  Y  ~>  C  be  the  partial  normalization  of  C  in  which  we  normalize  the  set  S.  For  every 
L  e  Pic(C)  we  have  jr^L)  €  M(C;  x,  S)  with*  =  deg(L)+s  =  deg(L)+/?fl(C)- />fl(F) 
and  /i°(F,  L)  =  /i°(C,  jr*(L)),  /z!(F,  L)  =  /^(C,  jr*(L)).  Hence  taking  a  general  L  e 
Pic*~5(7)  we  obtain  that  for  every  integer  x  >  g  —  1  a  general  F  €  M  (C;  jc,  5)  has 
/i^C,  F)  =  0,  i.e.  A°(C,  F)  -  deg(F)  +  1  -  g. 

(4.6)  Let  C  be  an  integral  projective  curve  whose  only  singularities  are  ordinary  nodes 
and  ordinary  cusps.  Let  F  and  G  be  rank  1  torsion  free  sheaves  on  C  with  Sing(F)  n 
Sing(G)  =  0.  This  condition  is  equivalent  to  the  torsion  freeness  of  F  0  G.  We  have 
deg(F  ®  G)  =  deg(F)  4-  deg(G)  and  Sing(F  ®  G)  =  Sing(F)  U  Sing(G).  Since  F  0  G 
has  no  torsion,  here  we  will  consider  the  usual  multiplication  map  /XF,G  •  For  the  injectivity 
of  /^F,G  it  is  usually  not  restrictive  to  assume  F  spanned  (otherwise  we  reduce  to  the  study 
of  the  subsheaf  F'  of  F  spanned  by  #°(C,  F),  although  Smg(Fx)  ^  Sing(F)  in  general), 
Usually  we  will  consider  a  range  in  which  F  (g)  G  is  spanned  and  hence  to  obtain  the 
surjectivity  of  IJLF ,G  it  is  necessary  to  assume  that  F  and  G  are  spanned. 

(4.7)  Let  C  be  an  integral  projective  curve  whose  only  singularities  are  ordinary  nodes  and 
ordinary  cusps.  Let  F  be  a  rank  1  spanned  torsion  free  sheaf  on  C  and  n  :  Y  ->  C  the 
partial  normalization  of  C  in  which  we  normalize  exactly  the  points  of  Sing(F).  Set  L  := 
;r*(F)/Tors(7r*(F)).  By  4.1  we  have  L  G  Pic(C),  F  =  ;r*(L)  and  deg(F)  =  deg(L)  +  8 
and  /*°(F,  L)  =  /z°(C,  F).  Since  F  is  spanned,  jr*(L)  is  spanned  and  hence  L  is  spanned. 

Remark  4.8.  Let  U  be  a  quasi-projective  one-dimensional  scheme  with  a  unique  singular 
point,  P,  which  is  either  an  ordinary  node  or  an  ordinary  cusp.  Let  F  and  G  be  rank  1 
*orsion  free  sheaves  on  U  such  that  F  is  not  locally  free  at  P,  while  G  is  locally  free  at 


Rank  1  torsion  free  sheaves  171 

P,  i.e.  with  G  6  Pic(U).  Let  K/>  be  the  skyscraper  sheaf  on  U  supported  by  P  and  with 
length  /.  By  the  last  part  of  (4.1)  there  exist  rank  1  torsion  free  sheaves  F7,  F"  ',  G1  ',  G"  on 
U  with  F7  C  F  c  F",  G7  C  G  C  G",  F/F7  £  F"/F  =  G/G'  =  G77/G7  =  KP  and 
such  that  F'  and  F"  are  locally  free,  while  G'  and  G"  are  not  locally  free  at  P. 

Remark  4.9.  Let  C  be  an  integral  projective  curve  whose  only  singularities  are  ordi- 
nary nodes  and  ordinary  cusps.  Take  S  c  Sing(C)  and  any  spanned  R  e  Pic(C)  with 
hl(C,R)  =  0.  By  4.8  we  obtain  the  existence  of  F  e  M(C;  jc,  S),x  =  deg(/?)  +  card(S), 
such  that  /?  is  a  subsheaf  of  F  and  F/R  =  O$.  Since  Os  is  a  skyscraper  sheaf  and 
hl(C,  R)  =  0,  we  obtain  A1  (C\F)  =  0.  Hence  A°(C,  F)  =  A°(C,  fl)  +  card(S).  Since  R 
is  spanned,  this  implies  the  spannedness  of  F. 

PROPOSITION  4.  10 

Fix  non-negative  integers  g,  q,  s,  s1  ',  a,  &  vwY/z  g  >  s  +  s7  4-  <?,  <3  >:  g  +  s,b  >  g  -{-  s  and 
(a-\-l  —  g  —  s)(b  +  l  —  g  —  sf)  >  a  -\-b-\-  1  —  g  —  s  —  s'.  Let  C  be  a  general  integral  nodal 
curve  with  pa(C)  =  g  and  normalization  of  genus  q.  Fix  S  c  Sing(C)  and  Sf  C  Sing(C) 
with  card(S)  =  s,  card(S7)  =  s'  and  S  H  S7  =  0.  Then  for  a  general  F  e  M(C;  a,  5)  and 
a  general  G  e  M  (C;  fr,  S7)  the  multiplication  map  fJip,G  is  surjective. 

Proof.  By  Remark  4.9  for  general  F  and  G  we  have  hl  (C,  F)  =  ft1  (C,  G)  =  hl  (C,  F  (g) 
G)  =  0.  Take  general  L  e  Pic(C)  and  M  e  Pic(C).  By  1.3  and  the  assumptions  on 
g,  a,  b,  s  and  sf  the  linear  map  HL,M  is  surjective.  Take  as  Ff  (resp.  G1)  any  element  of 
M(C\  a,  S)  (resp.  M  (C;  6,  5')  containing  L  (resp.  M)  and  with  F'/£  =  Os  (resp.  Gr/M  = 
Os)  (Remark  4.8).  By  Remark  4.9  we  have  hl(C,  F')  =  A1^,  G;)  =  0,  7z°(C,  F;)  = 
A°(C,  L)  +  J,  A°(C,  GO  =  A°(C,  M)  +  j'  and  both  F1  and  G'  are  spanned.  See  L  0  M 
as  a  subsheaf  of  F'  (g)  G7  with  F'  (g)  G'/£  ®  Af  =  Osus'-  Since  both  ^x  and  C1'  ^e 
spanned  Im^f^G')  spans  F'^G'.  Hence  dim  (Im^/r/^O)  >  dim^m^z^M))^^-!-^7  = 
a  +  b  —  s  —  sf  +  1  —  g  +  s  -i-  s'  =  A°(C,  F7  0  GO-  Hence  AI/T',G'  is  surjective  and  we 
conclude  by  semicontinuity. 

PROPOSITION4.il 

Fix  non-negative  integers  g,  q,  s,  sf  ,  a,  b  with  g  >  s  +  s'  +  q,  a  >  g,  b  >  g-f£  and 
(a  -f  /  —  g  +  s)(b  +  1  —  g  +  s')  <a-f-i>  +  /  —  g  +  5'  +  5/.  L€f  C  be  a  general  integral  nodal 
curve  with  pa(C)  =  g  and  normalization  of  genus  q.  Fix  S  C  Sing(C)  and  Sf  C  Sing(C) 
wiYA  card(5')  =  j,  card(S')  =  s'  and  S  H  5X  =  0.  Then  for  a  general  F  e  M(C;  a,  S) 
a  general  G  €  M(C\  b,  S')  the  multiplication  map  IAF,G  is  infective. 


Proof.  Since  (<*+/-#  +  j)(i  +  /  -  g  -f  .y7')  <^  +  Z?  +  /-(?-l-54-^/,  Theorem  1.3 
shows  that  for  a  general  L  e  Pica+s(C)  and  a  general  M  e  Pic^"1"5  (C)  the  multiplication 
map  HL,M  is  injective.  By  Remark  4.9  there  is  F7  e  Af  (C;  a,  5)  and  G7  e  M(C\  b,  57) 
with  F7  C  L,  G7  C  M,  L/F7  =  O,s  and  M/G7  =  O$'  .  Since  F7  is  a  subsheaf  of  L  and  G7 
is  a  subsheaf  of  M  the  map  i^pf,Gf  is  injective.  Since  h^(C,  Ff  ®  G7)  =  0,  we  conclude 
using  semicontinuity. 

There  is  a  geometrically  important  case  in  which  iterations  of  the  multiplication  maps  do 
occur.  Let  X  be  a  smooth  projective  curve  of  genus  q  and  L  e  Pick(X)  with  h°(X,  L)  =  2 
and  L  spanned.  The  ordered  sequence  of  integers  {fe°(X,  L®r  )}r>o  uniquely  determines 
the  so-called  scrollar  invariants  of  the  pencil  L  (see  e.g.  [12],  §2).  If  2k  <  q  and  X  is  a 
general  &-gonal  curve  of  genus  q  we  have  A°(X,  L®')  =  f  +  lifO<f<  [#/(&  -  1)], 


172  E  Ballico 

while  fe°(X,  L®0  =  M  +  1  -  q  (i.e.  A^X,  L®r)  =  0)  if  t  >  [q/(k  -  1)]  ([2]).  Fix  an 
integer  with  2  <  a  <  g.  The  equalities  A°(X,  L®r)  =  r  4-1  if  0  <  t  <  a  are  equivalent  to 
the  surjectivity  of  all  multiplication  maps  ^L®b  L  with  1  <  b  <  a.  On  singular  curve  when 
L  is  not  locally  free  the  sheaf  L  <g>  L  has  always  torsion  and  hence  it  is  more  interesting  to 
consider  the  associated  map  <XL,LJ  and  its  iterations. 

PROPOSITION  4.  12 

Fix  integers  g,  q  and  k  with  g  >  q  >  2k  >  4.  Let  C  be  an  integral  projective  curve 
with  pa(C)  =  g  and  whose  only  singularities  are  ordinary  nodes  and  ordinary  cusps  and 
f  :  X  ->  C  its  normalization.  Assume  that  X  is  a  general  k-gonal  curve  of  genus  q  and 
call  L  its  degree  k  pencil.  For  every  integer  t  >  1  set  Ft  :=  /(L®r).  For  every  integer 
t  <  [q/(k  -  1)]  we  have  fc°(C,  Fr)  =  t  +  1  and  for  every  integer  a  <  [q/(k  -  1)]  the  map 
otFa,F\,f  is  surjective. 

Proposition  4.  12  follows  at  once  from  the  next  observation  which  also  explain  the  mean- 
ing of  the  sheaves  involved  in  the  statement  of  4.  12. 


Remark  4.13.  By  4.2  each  Ft  is  a  rank  1  torsion  free  sheaf  on  X  with  deg(Fr)  = 
andSing(Fr)  =  Sing(C).  Since  A°(C,  Ft)  =  /z°(X,  L0r),  the  first  assertion  of  4.  12  follows 
from  [2].  If  t  >  2  we  have  Ft  =  Ft-i  ®  Fi/Tors(Fr_i  ®  FI)  =  Ff  r/Tors(Ffr)  (4.1  and 
induction  on  t).  Hence  we  obtain  the  last  assertion  of  4.12  from  the  first  assertion  of  4.12. 

Acknowledgements 

The  author  would  like  to  thank  the  referee  for  several  useful  remarks.  This  research  was 
partially  supported  by  MURST,  Italy. 

References 

[1]  Arbarello  E,  Cornalba  M,  Griffiths  P  and  Harris  J,  Geometry  of  Algebraic  Curves,  I  (Springer- 

Verlag)  (1985) 
[2]  Ballico  E,  A  remark  on  linear  series  on  general  fc-gonal  curves,  Boll.  U.M.I.  3-A  (1989) 

195-197 

[3]  Ballico  E,  Line  bundles  on  projective  curves:  the  multiplication  map,  preprint 
[4]  Cook  Ph  R,  Local  and  global  aspects  of  the  module  theory  of  singular  curves,  Ph.D.  thesis 

(Liverpool)  (1983) 
[5]  Eisenbud  D  and  Harris  J,  Divisors  on  general  curves  and  cuspidal  rational  curves,  Invent 

Math.  74  (1983)  371-418 
[6]  Eisenbud  D  and  Harris  J,  Irreducibility  and  monodromy  of  some  families  of  linear  series,  Ann. 

EC.  Norm.  Sup.  20  (1987)  65-87 
[7]  Eisenbud  D,  Koh  J  and  Stillman  M,  Determinantal  equations  for  curves  of  high  degree,  Am. 

J.Math.  110(1988)513-539 
[8]  Fulton  W  and  Lazarfeld  R,  On  the  connectedness  of  degeneracy  loci  and  special  divisors, 

ActaMath.  146  (1981)  271-283 

[9]  Gieseker  D,  Stable  curves  and  special  divisors,  Invent.  Math.  66  (1982)  251-275 
[10]  Green  M,  Koszul  cohomology  and  the  geometry  of  projective  varieties,  J.  Diff.  Geom.  19 

(1984)  125-171 
[111  Hartshome  R  and  Hirschowitz  A,  Smoothing  algebraic  space  curves,  in:  Algebraic  Geometry, 

Sitges  1983,  Lect.  Notes  in  Math.  1124  (Springer-Verlag)  (1985)  98-131 
r!2]  Schreyer  F-O,  Syzygies  of  canonical  curves  and  special  linear  series,  Math.  Ann.  275  (1986) 

105-137 

Li3]  Seraesi  E,  On  the  existence  of  certain  families  of  curves,  Invent.  Math.  75  (1984)  25-57 
[14]  Seshadri  C  S,  Fibres  vectoriels  sur  les  courbes  algebriques,  Asterisque  96  (1982) 


Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  Ill,  No.  2,  May  2001,  pp.  173-178. 
Printed  in  India 


Boundedness  results  for  periodic  points  on  algebraic  varieties 


NAJMUDDIN  FAKHRUDDIN 

School  of  Mathematics,  Tata  Institute  of  Fundamental  Research,  Homi  Bhabha  Road, 
Mumbai  400  005,  India 
E-mail:  naf@math.tifr.res.in 

MS  received  27  July  2000;  revised  19  December  2000 

Abstract.     We  give  some  conditions  under  which  the  periods  of  a  self  map  of  an 
algebraic  variety  are  bounded. 

Keyword.     Periodic  points. 

Let  X  be  an  algebraic  variety  over  a  field  K  and  let  /  :  X  ->  X  be  a  morphism.  A  point 
P  in  X(K)  is /-periodic  if  fl(P)  =  P  for  some  n  >  0,  and  the  smallest  such  n  is  called 
the  period  of  P.  We  shall  prove  that  if  X  and  /  satisfy  certain  hypotheses,  then  the  set  of 
possible  periods  is  finite. 

Our  results  may  be  viewed  as  an  analogue  of  the  finiteness  of  the  torsion  of  abelian 
varieties  over  finitely  generated  fields.  It  is  then  natural  to  ask  for  an  analogue  of  the 
full  Mordell-Weil  theorem.  We  believe  that  the  following  conjecture  is  the  appropriate 
generalization. 

Conjecture  1.  Let  X  be  a  proper  algebraic  variety  over  a  finitely  generated  field  K  of 
characteristic  zero  and  /  :  X  ->  X  a  morphism.  Suppose  there  exists  a  subset  5  of  X(K) 
which  is  Zariski  dense  in  X  and  such  that  /  induces  a  bijection  of  5  onto  itself.  Then  /  is 
an  automorphism. 

This  can  be  easily  checked  for  X  =  Pn  or  X  an  abelian  variety  using  heights  and  the 
Mordell-Weil  theorem  respectively. 

1.  Finitely  generated  fields 

Theorem  1.  Let  X  be  a  proper  variety  over  afield  K  which  is  finitely  generated  over  the 
prime  field  and  let  f  :  X  -»  X  be  a  morphism. 

(i)  Ifchar(K)  =  0  then  the  set  of  periods  of  all  f-periodic  points  in  X(K)  is  finite. 

(ii)  If  char  (K)  =  p  ^  0  then  the  prime  to  p  parts  of  the  set  of  periods  is  finite  i.e.  there 
exists  n  >  0  such  that  all  the  fn -periodic  points  in  X(K)  have  periods  which  are 
powers  of  p. 

Many  special  cases  of  this  result  have  been  known  for  a  long  time,  the  first  such  being 
the  theorem  of  Northcott  ([5],  Theorem  3),  proving  the  finiteness  of  the  number  of  periodic 
points  in  certain  cases.  We  refer  the  reader  to  [4]  for  a  more  detailed  list  of  references. 

Remark.  We  do  not  know  whether  the  periods  can  really  be  unbounded  if  char(J^)  >  0. 


174  Najmuddin  Fakhruddin 

The  theorem  is  obvious  if  AT  is  a  finite  field  and  we  will  reduce  the  general  case  to  this 
one  by  a  specialization  argument.  A  little  thought  shows,  that  the  following  proposition 
suffices  to  prove  the  theorem. 

PROPOSITION  1 

Let  R  be  a  discrete  valuation  ring  with  quotient  field  K  and  residue  field  k.  Let  X  be  a 
proper  scheme  of  finite  type  over  Spec(/?)  and  f  :  X  — >  X  an  R-morphism.  Assume  that 
the  conclusions  of  the  theorem  hold  for  f  restricted  to  the  special  fibre,  and  that  for  each 
n  >  0  there  are  only  finitely  many  roots  of  unity  contained  in  all  extensions  ofk  of  degree 
<  n.  Then  the  same  holds  for  f  restricted  to  the  generic  fibre,  except  possibly  in  the  case 
char(^)  =  0  and  char(fc)  =  p  >  0,  when  the  result  holds  modulo  powers  of  p. 

Proof.  Let  p  =  1  if  char(fc)  =  0 .  The  hypotheses  imply  that  by  replacing  /  with  a  suitable 
power  we  may  assume  that  all  the  /-periodic  points  in  X(k)  have  period  a  power  of  p.  Let 
P  be  a  /-periodic  point  in  X(K).  By  replacing  /  by  f?" ,  for  some  n  which  may  depend  on 
P,  we  may  assume  that  the  specialization  of  P  in  X(k)  (which  exists  since  X  is  proper)  is 
a  fixed  point  of  /  restricted  to  the  special  fibre.  Let  Z  be  the  Zariski  closure  (with  reduced 
scheme  structure)  of  the  /-orbit  of  P.  Z  is  finite  over  Spec(P)  with  a  unique  closed  point, 
hence  is  equal  to  Spec(A)  where  A  is  a  finite,  local  P-algebra  (with  rank  equal  to  the  period 
of  P)  which,  since  A  is  reduced,  is  torsion  free  as  an  P-module. 

The  key  observation  of  the  proof  is  that  /  restricted  to  Z  induces  an  automorphism  of 
finite  order  of  A  (which  we  also  denote  by  /):  Since  /  preserves  the  orbit  of  P  and  Z  is 
reduced,  it  follows  that  /  induces  a  map  from  Z  to  itself,  hence  an  endomorphism  of  A. 
/"  is  the  identity  on  the  orbit  of  P  for  some  n  >  0,  hence  /"  is  the  identity  on  A  ®/?  K. 
Since  A  is  torsion  free,  it  follows  that  fn  is  the  identity  on  A  as  well. 

Let  m  be  the  maximal  ideal  of  A.  Since  Z  is  a  closed  subscheme  of  X,  it  follows  that  the 
dimension  of  m/m2  is  bounded  independently  of  P.  By  the  hypothesis  on  roots  of  unity, 
we  may  replace  /  by  some  power,  independently  of  P,  so  that  the  endomorphism  of  m/m2 
induced  by  /  is  the  identity.  Thus,  /  is  a  unipotent  map  with  respect  to  the  (exhaustive) 
filtration  of  A  induced  by  powers  of  m.  This  implies  that  the  order  of  /,  hence  the  period 
of  P,  is  a  power  of  p.  D 

Remarks.  (1)  For  any  explicitly  given  example,  the  proof  furnishes  an  effective  method  for 
computing  a  bound  for  the  periods.  (2)  In  the  non-proper  case  one  can  prove  the  following 
result  by  the  methods  of  this  paper:  Let  S  be  a  flat,  separated,  integral  scheme  of  finite 
type  over  Z,  let  X  be  a  separated  scheme  of  finite  type  over  S  and  let  /  :  X  ->  X  be  an 
S-morphism.  If  one  defines  the  notion  of  /-periodic  points  and  periods  for  elements  of 
X(S)  in  the  obvious  way,  then  the  set  of  periods  is  again  bounded.  (3)  One  may  also  ask 
whether  Theorem  1  itself  holds  without  the  assumption  of  properness,  for  example  when 
X  is  arbitrary  but  /  is  finite.  The  results  of  Flynn-Poonen-Schaefer  [1]  may  be  viewed  as 
some  positive  evidence,  however,  aside  from  this  we  do  not  have  many  other  examples.  If 
true,  this  would  imply  the  uniform  boundedness  of  torsion  of  abelian  varieties  and  other 
similar  conjectures. 

In  general  the  set  of  periodic  points  is  of  course  not  finite.  However,  one  can  often 
use  some  geometric  arguments  to  deduce  finiteness  of  the  number  of  periodic  points  from 
Theorem  1  as  in  the  following: 

Lemma  1.  Let  X  be  a  proper  variety  over  a  finitely  generated  field  K  of  characteristic  zero 
and  f  :  X  -»  X  a  morphism.  Suppose  that  there  does  not  exist  any  positive  dimensional 


Periodic  points  on  algebraic  varieties  175 

subvariety  Y  of  X  such  that  f  induces  an  automorphism  of  finite  order  of  Y.  Then  the 
number  of  f -periodic  points  in  X(K)  is  finite. 

Proof.  Theorem  1  implies  that  /  induces  an  automorphism  of  finite  order  on  the  closure 

of  the  set  of  /-periodic  points  in  X  (K).  D 

The  following  gives  a  useful  method  for  checking  the  hypothesis  of  the  previous  lemma. 

-Lemma  2.  Let  X  be  a  projective  variety  over  afield  K  and  f  :  X  -»  X  a  morphism. 
Suppose  there  exists  a  line  bundle  C  on  X  such  that  f*(L)  ®  L~l  is  ample.  Then  there  is 
no  positive  dimensional  subvariety  YofX  such  that  f  induces  an  automorphism  of  finite 
order  of  Y. 

Proof.  By  replacing  X  by  fn(X)  for  some  large  n,  we  may  assume  that  /  is  a  finite 
morphism.  Suppose  there  exists  a  K  as  above  and  assume  that  fm\Y  is  the  identity  of  Y. 
Then 


fm(L)  > 


By  assumption  /*(L)  (g)  L  l  is  ample  so  fm(L)  0  L  *,  being  a  tensor  product  of  am- 
ple bundles,  is  also  ample.  But  fm(L)  0  L~l\y  is  trivial,  so  it  follows  that  Y  must  be 
0-dimensional.  D 

In  case  C  is  also  ample,  finiteness  can  also  be  proved  using  heights,  see  for  example 
[2].  One  advantage  of  our  method  is  that  it  applies  also  when  /  is  an  automorphism,  in 
which  case  an  ample  C  as  above  can  never  exist.  Using  this,  one  can  for  example  extend 
the  finiteness  results  of  Silverman  [6]  to  apply  to  all  automorphisms  of  infinite  order  of 
projective  algebraic  surfaces  X  with  Hl(X,  O%)  =  0  =  HQ(X,  TX)  and  the  Picard  rank 
p=2. 

The  following  proposition  gives  a  simple  class  of  examples  for  which  boundedness  of 
the  periods  holds  for  non-proper  varieties. 

PROPOSITION  2 

Let  K  be  a  finitely  generated  extension  ofQ  and  let  G  be  a  linear  algebraic  group  over  K. 
Then  there  exists  an  integer  M(G)  such  that  for  all  varieties  X  over  K  with  a  G  action  and 
for  all  g  in  G(K),  the  set  of  periods  of  g -periodic  points  in  X  is  bounded  above  by  M(G). 

Proof.  The  G  orbit  of  any  x  €  X  (K)  is  isomorphic  to  G/H ,  where  H  is  a  closed  subgroup 
of  G,  hence  we  may  restrict  ourselves  to  the  case  where  X  =  G/H.  By  Weil  restriction 
of  scalars  we  may  assume  that  K  —  Q(t\ ,  ^, . . . ,  tr)  for  some  r  >  0,  and  since  any  linear 
algebraic  group  can  be  embedded  in  GLn,  we  may  also  assume  that  G  =  GLn  for  some 
integer  n . 

Assume  K  =  Q.  Let  GZ  =  GLn^z  and  let  HZ  be  the  Zariski  closure  of  H  in  GZ-  We 
may  also  form  the  quotient  Gz/Hz  on  which  there  is  a  natural  action  of  GZ  extending  the 
action  of  G  on  G/H.  Let  x  be  a  ^-periodic  point  of  G/H  with  period  equal  to  /.  There 
exists  a  finite  set  of  primes  S  such  that  g  extends  to  an  element  of  Gz(R)  and  x  extends  to 
an  element  of  GZ///Z(#),  where  R  is  the  ring  of  S-integers.  For  a  prime  number  p  not  in 
S,  let  Gp,  Hp,  Gp/Hp,  gp,  xp  denote  the  reductions  mod  p  of  the  corresponding  objects 
defined  above  and  let  lp  denote  the  gp -period  of  xp.  It  is  clear  that  lp  divides  the  order  of 


176  Najmuddin  Fakhruddin 

Gp(fp)  =  GLn(¥p)  for  all  p  £  S  and  for  p  »  0,  lp  =  /.  By  Lemma  1  below,  it  follows 
that  /  <  M  for  some  constant  M  independent  of  H,  g  and  x. 

Now  let  K  =  Q(/i,  *2,  •  •  • » */•)  with  r  >  0.  We  repeat  the  arguments  of  the  above 
paragraph,  replacing  Z  with  Q[r i ,  ti, . . . ,  rr ].  Since  the  rational  points  are  dense  in  Spec(Q 
[fj. ,  *2,  •  •  • » *r])»  ^  follows  that  the  same  constant  M  bounds  the  periods.  D 

Lemma  3.  For  each  positive  integer  n,  there  exists  an  integer  Mn  such  that  if  I  is  any  integer 
which  divides  \GLn(¥p)\for  all  p  »  Q,  then  I  <  Mn. 

Proof.  Let  TV  be  an  integer  such  that  for  all  a  >  N,  (Z/aZ)*  contains  an  element  of  order 
greater  than  n.  Let  q  be  a  prime  and  assume  that  qb  divides  \GLn(¥p)\  =  pn(n~^l2(pn  - 
l)(pn~l  -  1)  -  -  -  (p  -  1)  for  all  p  »  0.  If  #c  >  #,  then  by  Dirichlet's  theorem  on 
primes  in  arithmetic  progressions  there  exist  infinitely  many  primes  p  such  that  the  order 
of  p  mod  qc  is  greater  than  n.  This  bounds  the  powers  of  q  that  can  divide  each  of 
pn  -  1,  pn~l  —  1, ...,/?-  1  and  hence  bounds  b.  It  is  clear  that  if  q  >  N  then  b  =  0,  so 
we  obtain  a  bound  for  /  by  multiplying  together  the  bounds  for  each  prime  q  <  TV.  O 

Remark.  Note  that  Proposition  2  applies  to  all  automorphisms  of  affine  space,  where  in 
many  cases  finiteness  of  the  number  of  periodic  points  is  also  known;  see  for  example  the 
paper  [3]  of  Marcello. 

2.  /7-Adic  fields 

Proposition  1  shows  that  one  also  has  boundedness  of  periods  for  p-adic  fields,  up  to  powers 
of  /?,  as  long  as  the  variety  and  the  morphism  extend  to  the  ring  of  integers.  We  now  show 
that  in  fact  we  can  bound  the  extra  powers  of  p. 

Theorem  2.  Let  O  be  the  ring  of  integers  in  K,  a  finite  extension  ofQp,  and  let  X  be  a 
proper  scheme  of  finite  type  over  Spec(O).  Then  there  exist  a  constant  M  >  0  such  that 
for  any  O-morphism  f  :  X  -^  X,  the  periods  of  the  f -periodic  points  ofX(K)  are  all 
less  than  M. 

If  X  is  any  variety  over  a  finite  field  k  then  it  is  clear  that  a  statement  analogous  to  the 
theorem  holds  for  X(k):  since  this  is  a  finite  set  the  periods  are  bounded  above  by  \X(k)\, 
and  hence  are  bounded  independently  of  the  morphism.  To  bound  the  powers  of  p  that 
occur,  one  sees  from  the  proof  of  Proposition  1  that  it  is  enough  to  prove  the  following: 

PROPOSITION  3 

Let  O  be  the  ring  of  integers  in  K,  a  discrete  valuation  ring  of  characteristic  zero  with 
residue  field  k  of  characteristic  p.  Let  (A,  rri)  be  a  local  sub-O-algebra  of  OP"  of  rank  pn 
which  is  preserved  by  the  automorphism  a  given  by  cyclic  permutation  of  the  coordinates. 
Furthermore,  assume  that  or  acts  trivially  on  m/m2.  Then  n  <  r  =  v(p)  if  p  >  2  and 
n<rifp  =  2,  where  v  is  the  normalized  valuation  on  K. 

Proof.  Assume  that  n  >  r  if  p  >  2  and  n  >  r  if  p  =  2.  Since  a  acts  trivially  on  m/m2,  it 
follows  that  a?*  acts  trivially  on  m/mt+2  for  all  t  >  0.  Thus,  by  replacing  A  by  a  quotient 
algebra  corresponding  to  the  Zariski  closure  in  Spec(A)  of  the  aPn~~l  orbit  of  any  O  valued 
point,  we  obtain  a  local  rank  p  subalgebra  of  OP  which  is  stable  under  (the  new)  a  and 
such  that  a  acts  trivially  on  m/mr+l  (m/mr+2  if  p  =  2). 


Periodic  points  on  algebraic  varieties 


111 


For  a  in  A  we  denote  by  v(a)  the  minimum  of  the  valuations  of  the  coordinates.  Let 

U(m)  =  {a  6  m\v(ai)  ^  v(dj)  for  some/,  j] 
and  let 

P(m)  =  {a<E  U(m)\v(a)  <  v(fe)forallfc  €  £/(m)}. 

Suppose  v(a)  =  1  for  a  €  P(m).  Since  P(m)  C  t/(ra),  it  follows  that  v(crs(a)  -  a)  =  I 
for  some  s,  which  in  turn  implies  that  as(a)  ~  a  £  m2.  This  is  a  contradiction,  hence 
v(fl)  >  1  for  all  a  €  P(m).  Also,  one  easily  sees  that  any  element  of  m  can  be  written  as 
a  =  x  +  b  with  :c  €  n  •  O  and  b  e  U(m)  U  {0},  where  n  in  O  is  a  uniformizing  parameter. 
Now  let  a  e  P(m)  and  consider  a  (a)  —  a.  Letting  r  =  r+lif/?>2  and  f  =  r  +  2  if 
p  =  2,  we  see  that 


7  =  1 


with  xij  €  n  -  O  and  fyj  e  U(m)  U  {0}.  Expanding  the  products  and  using  the  fact  that 
v(fefj)  >  1,  we  see  that 


a  (a)  -a  = 


mod  TT  v^ 


withx,  z^67r?~1  -(9  and^/jfc  G  P(m).  Further,  using  the  fact  that  a  is  in  t/(m),  one  sees  that 
=  v(cr(a)  —  a)  =  v(a).  Thus,  we  get 


0  = 


-  a  = 


-  a)  =  p  •  x 


Now  the  <4's  are  also  in  P(m\  so  we  have 

o-(4)  -dk  =  wk  mod  jr1 
with  u;^  in  TT?  •  O.  This  implies  that 


mod  7rv<fl>+r.  (1) 


<-  (4)  =  /»  •  4 


Substituting  this  in  eq.  (1)  (using  that  the  z^'s  are  in  n  •  O)  we  get 

=  0  mod  »"<">•". 


We  have  v(jc)  =  v(c)  =  v(^)  =  ^(^),  v(zjk)  >  t  —  1  and  v(p)  =  r.  By  the  choice  of 
t  it  follows  that  the  only  term  in  the  above  equation  with  valuation  less  than  or  equal  to 
v(a)  +  r  is  p  •  x.  This  is  a  contradiction  since  t  >  r.  D 

Remark.  The  assumption  of  properness  is  used  only  to  guarantee  the  existence  of  spe- 
cializations. If  we  consider  an  arbitrary  separated  scheme  X  of  finite  type  over  Spec(O), 
then  we  obtain  boundedness  of  the  periods  for  the  set  of  periodic  points  in  X(O).  One 
can  also  construct  examples  for  which  the  set  of  periods  of  the  periodic  points  in  X(K)  is 
unbounded. 


178  Najmuddin  Fakhruddin 

Acknowledgements 

The  author  would  like  to  thank  Bjorn  Poonen  and  Ramesh  Sreekantan  for  some  interesting 
conversations  and  helpful  correspondence. 

References 

[1]     Flynn  E  V,  Poonen  B  and  Schaefer  E  F,  Cycles  of  quadratic  polynomials  and  rational  points 

on  a  genus-2  curve,  Duke  Math.  J.  90  (1997)  435-463 

[2]     Kawaguchi  S,  Some  remarks  on  rational  periodic  points,  Math.  Res.  Lett  6  (1999)  495-509 
[3]     Marcello  S,  Sur  les  proprietes  arithmetiques  des  iteres  d'automorphismes  reguliers, 

C.R.  Acad.  ScL  Paris  331  (2000)  1 1-16 
[4]     Morton  P  and  Silverman  J  H,  Rational  periodic  points  of  rational  functions,  Internat.  Math. 

Res.  Notices  (1994)97-110 

[5]     Northcott  D  G,  Periodic  points  on  an  algebraic  variety,  Ann.  Math.  51  (1950)  167-177 
[6]     Silverman  J  H,  Rational  points  on  K3  surfaces:  a  new  canonical  height,  Invent.  Math.  105 

(1991)347-373 


>roc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  1 11,  No.  2,  May  2001,  pp.  179-201. 
Printed  in  India 


Spectra  of  Anderson  type  models  with  decaying  randomness 


M  KRISHNA  and  K  B  SINHA* 

Institute  of  Mathematical  Sciences,  Taramani,  Chennai  600  113,  India 
Indian  Statistical  Institute,  7  SJS  Sansanwal  Marg,  New  Delhi  110016,  India 
*Present  address:  Indian  Statistical  Institute,  203,  BT  Road,  Kolkata  700  035,  India 
E-mail:  Jmshna@imsc.ernet.in;  kbs@isical.ac.in 

MS  received  1 1  May  2000;  revised  17  November  2000 

Abstract  In  this  paper  we  consider  some  Anderson  type  models,  with  free  parts 
having  long  range  tails  and  with  the  random  perturbations  decaying  at  different  rates 
in  different  directions  and  prove  that  there  is  a.c.  spectrum  in  the  model  which  is 
pure.  In  addition,  we  show  that  there  is  pure  point  spectrum  outside  some  interval. 
Our  models  include  potentials  decaying  in  all  directions  in  which  case  absence  of 
singular  continuous  spectrum  is  also  shown. 

Keywords.  Anderson  model;  absolutely  continuous  spectrum;  mobility  edge; 
decaying  randomness. 

L.  Introduction 

fhere  have  been  but  few  models  in  higher  dimensional  random  operators  of  the  Anderson 
nodel  type  in  which  presence  of  absolutely  continuous  spectrum  is  exhibited.  We  present 
lere  one  family  of  models  with  such  behaviour. 

The  results  here  extend  those  of  Krishna  [10]  and  part  of  those  in  Kirsch-Krishna- 
Dbermeit  [9],  Krishna-Obermeit  [12]  while  making  use  of  wave  operators  to  show  the 
existence  of  absolutely  continuous  spectrum,  the  results  of  Jaksic-Last  [14]  to  show  its 
>urity  and  those  of  Aizenman  [1]  for  exhibiting  pure  point  spectrum. 

The  new  results  in  this  paper  allow  for  long  range  free  parts,  have  models  with  com- 
>act  spectrum  (in  dimensions  2  and  more)  which  contains  both  absolutely  continuous  and 
lense  pure  point  spectrum.  Our  models  include  the  independent  randomness  on  a  surface 
considered  by  Jaksic-Molchanov  [15,  16]  and  Jaksic-Last  [14,  13],  while  allowing  for  the 
•andomness  to  extend  into  the  bulk  of  the  material. 

The  literature  on  the  scattering  theoretic  and  commutator  methods  for  discrete  Laplacian 
ncludes  those  of  Boutet  de  Monvel-Sahbani  [4,  5]  who  study  deterministic  operators  on 
he  lattice. 

The  scattering  theoretic  method  that  we  use  is  applicable  even  when  the  free  operator  is 
lot  the  discrete  Laplacian  but  has  long  range  off  diagonal  parts.  We  impose  conditions  on 
he  free  part  in  terms  of  the  structure  it  has  in  its  spectral  representation. 

I.  Main  results 

Fhe  models  we  consider  in  this  paper  are  related  to  the  discrete  Laplacian  (Aw)(n)  = 
£|i|=i  u(n  +  0  on  ^2(^v)-  We  denote  by  Tv  the  v  dimensional  torus  Rv/27rZv  and  a  the 
invariant  probability  measure  on  it.  We  use  the  coordinate  chart  {#  :  #  =  (9\9 . . . ,  #v)>  0  < 
3i  <  2n]  and  the  representation  a  =  nr=i(d#//2jr)  on  the  torus  for  calculations 


1  80  M  Krishna  and  K  B  Sinha 

below  without  further  explanation.  Then  A  is  unitarily  equivalent  to  multiplication  by 
22^=1  cos(0/)  acting  on  L2(Tv,o-),  written  in  the  above  coordinates.  We  consider  a 
bounded  self  adjoint  operator  HQ  which  commutes  with  A  and  which  is  given  by,  on 
L2(TU,  dcr),  an  operator  of  multiplication  by  a  function  /z(#)  there  with  h  satisfying  the 
assumptions  below. 

Hypothesis  2.1.  Let  h  be  a  real  valued  C3H"3(TU)  function  satisfying 

1.  h  is  separable,  i.e.  h(&)  =  £^=1  ^y(0/)- 

2.  The  sets 


are  finite  for  each  j  =  1  ,  .  .  .  ,  v.  Let 

C(hj)  =  T  x  .  .  .  x  T  x  C(hj)  x  T  .  .  .  x  T, 
where  the  set  C(/zy)  occurs  in  the  y'th  position.  We  denote  by 

C  =  U]f=1C(A7-) 
and  note  that  this  is  a  closed  set  of  measure  zero  in  "P. 

We  consider  random  perturbations  of  bounded  self  adjoint  operators  coming  from  func- 
tions as  in  the  above  hypothesis.  We  assume  the  following  on  the  distribution  of  the 
randomness. 

Hypothesis  2.2.  Let  //,  be  a  positive  probability  measure  on  R  satisfying: 

1.  \L  has  finite  variance  a2  =  f  x2d/z(x). 

2.  /x  is  absolutely  continuous. 

Finally  we  consider  some  sequences  of  numbers  an  indexed  by  the  lattice  Zv  or  Z++1  = 
Z+  x  Zv  and  assume  the  following  on  them. 

Hypothesis  2.3.   (1)  an  is  a  bounded  sequence  of  non-negative  numbers  indexed  by  Zv 

which  is  non-zero  on  an  infinite  subset  of  Zv  . 

(2)Letg(J?)  =0nX{n€Zy:|n/|>*,  vi<i<v).  Then  £  e  L1^!,  oo)). 

(10  an  is  a  bounded  sequence  of  non-negative  numbers  which  are  non-zero  on  an  infinite 

subset  of  Zl{.+1. 

(20 


Remark  1.  In  the  case  of  Zv  our  hypothesis  on  the  sequence  an  allows  for  the  following 
type  of  sequences 

•  an  =  (l  +  |n|)a,    a  <  -1. 

•  an  =  (1  -h  |«f|)a,  for  some  z,   a  <  —1. 

•  ««  =  riLid  +  |n,-|)°"  ,    a,-  <  0  with  E       «/  <  -1- 


Spectra  of  random  operators  181 

Therefore  in  the  theorems,  on  the  existence  of  absolutely  continuous  spectrum,  we  can 
allow  the  potentials  to  be  stationary  along  all  but  one  direction  in  dimensions  v  >  2. 
2.  In  the  case  of  Z++1  ,  we  can  allow  the  sequence  to  be  of  the  type 

©  an  =  0,    n\  >  N  and  an  =  1,    for  n\  <  N,  for  some  Q  <  N  <  oo. 


•  ««  =  [DUO  +  I"/  1)"1'.    «/  <  0  with   ELi  «/  <  -1- 

Thus  allowing  for  models  with  randomness  on  a  the  boundary  of  a  half  space. 

For  the  purposes  of  determining  the  spectra  of  the  models  we  are  going  to  consider  here 
in  this  paper  we  recall  a  definition  given  in  Kirsch-Krishna-Obermeit  [9],  namely, 

DEFINITION  2.4 

Let  an  be  a  non-negative  sequence,  indexed  by  Zy  or  Z^+1  .  Let  /x  be  a  positive  probability 
measure  on  IR.  Then  the  a-supp(/x)  is  defined  as 

1.  In  the  case  of  Zu, 

a-supp(/z)  =  x  :  V<(anl(x  -  *,  *  +  e))  =  oo,    V  6  >  0  1  . 


2.  InthecaseofZ^1, 

a-supp(^)  = 


lj.(a-L(x-€,x  +  €))  =  oo9    Ve>0 

nzkl^1 


Remark.  \.  In  the  sums  occurring  in  the  above  definition  we  set  fJi(an  1  (x  —  e,  x  +  e))  =  0, 
for  those  n  for  which  an  =  0.  This  notation  is  to  allow  for  sequences  an  that  are  everywhere 
zero  except  on  an  axis  for  example. 

2.  We  note  that  when  an  is  a  constant  sequence  an  =  X  ^  0, 

a-supp(/x)  =  A  •  supp(/x). 

3.  When  an  converge  to  zero  as  \n\  goes  to  oo,  the  a-supp(/x)  is  trivial  if  fju  has  compact 
support.  It  could  be  trivial  even  for  some  class  of  JJL  of  infinite  support  depending  upon  the 
sequence  an . 

4.  If  an  is  bounded  below  by  a  positive  number  on  an  infinite  subset  along  the  directions 
of  the  axes  in  Zu  (respectively  ZlJ.+1),  then  the  a-supp(/x)  could  be  non-trivial  even  for 
compactly  supported  //,. 

We  consider  the  operator  (for  u  G  £2(Z+)), 

i(n  +  l)+u(n  -  1),    n  >  0, 


Below  we  use  either  A+  or  its  extension  by  A+  (8)  /  to  €2(Z!j_+1)  by  the  same  symbol,  the 
correct  operator  is  understood  from  the  context.  Given  a  real  valued  continuous  function  on 
the  torus  Tv ,  we  consider  the  bounded  self  adjoint  operators  HQ  on  t2  (Zv )  which  is  unitarily 


1  82  M  Krishna  and  K  B  Sinha 

equivalent  to  multiplication  by  h  on  £2(Ty,  a).  We  also  denote  the  extension  /  <g)  HQ  of 
HQ  to  -£2(Z++1)  by  the  symbol  HQ  and  L2(Ty,  a)  as  simply  L2(Ty)  in  the  sequel. 
We  then  consider  the  random  operators 

on  £2(Zy), 


#«!  =  #0+  +  V",    yw  =        ^^w(/2)Pn,//o+  =  A+  +  //o,    on  *(Z;+),        (1) 

«€/ 

where  Pn  is  the  orthogonal  projection  onto  the  one  dimensional  subspace  generated  by  &n 
when  [Sn]  is  the  standard  basis  for  12(I)  (/  =  Zu  or  Z++1).  {(f(n)}  are  independent  and 
identically  distributed  real  valued  random  variables  with  distribution  //.  The  operator  HQ 
is  some  bounded  self  adjoint  operator  to  be  specified  in  the  theorems  later. 

Then  our  main  theorems  are  the  following.  First  we  state  a  general  theorem  on  the 
spectrum  of  HQ  in  such  models.  For  this  we  consider  the  operator  HQ  to  denote  a  bounded 
self  adjoint  operator  on  £2(Zy)  coming  from  a  function  h  satisfying  the  Hypothesis  2.1  and 
A+  defined  as  before. 

Theorem  2.5.  Let  HQ  and  HQ+  be  the  operators  defined  as  in  eq.  (  1  ),  coming  from  functions 
h  satisfying  the  hypothesis  2.1(1)(2).  Let 

V 

sup    hi(6),    £_  =  y     inf     hj(6). 


Then,  the  spectra  of  both  HQ  and  HQ+  are  purely  absolutely  continuous  and 
=  [£_,£+],    and  a(HQ+)  =  [-2  +  JEL,  2  +  £+]. 


Part  of  the  essential  spectra  of  the  operators  /zw  and  H®  are  determined  via  Weyl  se- 
quences constructed  from  rank  one  perturbations  of  the  free  operators  HQ  and  //Q+  respec- 
tively. The  proof  of  this  theorem  is  done  essentially  on  the  line  of  the  proof  of  Theorerr 
2.4  in  [9]. 

Theorem  2.6.  Let  the  indexing  set  I  be  Zy  or  Z^+1  and  consider  the  operator  HQ  coming 
from  a  function  h  satisfying  the  conditions  of  hypothesis  2.1(1)  in  the  case  of  I  =  Zv  am 
consider  the  associated  HQ+  in  the  case  of  I  =  Z++1.  Suppose  q^(n),  n  e  I  are  Li.c 
random  variables  with  the  distribution  JJL  satisfying  the  hypothesis  2.2(1).  Let  an  be  c 
sequence  indexed  by  I  satisfying  the  hypothesis  2.3(1)  (or  (I7)  as  the  case  may  be).  Assum* 
also  that  0  6  a-supp()a),  then 

a(HQ  +  A.Po)  C  cress(#w)  almost  every  a> 

A.  e  a-supp(/x) 

and 

[J        <r(/fo+  +  APo)  C  creSs(Hp  almost  every  to. 

k  e  a-supp(/i) 

Remark  1.  When  JJL  has  compact  support  and  an  goes  to  zero  at  infinity,  or  when  /z  ha 
infinite  support  but  an  has  appropriate  decay  at  infinity,  there  is  no  essential  spectrur 
outside  that  of  HQ  for  H^  almost  every  a).  So  the  point  of  this  theorem  is  to  show  that  ther 
is  essential  spectrum  outside  that  of  HQ  based  on  the  properties  of  the  pairs  ({an},  /z). 


Spectra  of  random  ope  rators  183 

2.  In  Kirsch-Krishna-Obermeit  [9]  some  examples  of  random  potentials  which  have 
essential  spectrum  outside  a  (Ho)  even  when  an  goes  to  zero  at  oo  were  given.  The  examples 
presented  there  had  a-supp(/z)  as  a  half  axis  or  the  whole  axis,  this  is  because  of  the  decay 
of  the  sequences  an  .  Here  however,  since  we  allow  for  an  to  be  constant  along  some 
directions,  our  examples  include  cases  where  the  spectra  of  H*0  are  compact  with  some 
essential  spectrum  outside  CT(HQ). 

We  let  E±  be  as  in  Theorem  2.5..  We  also  set  Ji^.n  to  be  the  cyclic  subspace  generated 
by  &n  and  H">. 

Theorem  2.7.  Consider  a  bounded  self  adjoint  operator  HQ  coming  from  a  function  h 
satisfying  the  conditions  of  hypothesis  2.1(1),  (2).  Suppose  q^  are  Lid  random  variables 
with  the  distribution  /JL  satisfying  the  hypothesis  2.2(1). 

1.  Let  I  =  Zy  andan  be  a  sequence  satisfying  the  hypothesis  2.3(1),  (2).  Then, 
aac(ffa})  D  [£-,  £+]  almost  every  a>. 

Further  when  p.  satisfies  the  hypothesis  2.3(2),  an  =£  0  on  Zy,  7ia>>w,  Ha),m  not  mutually 
orthogonal  for  any  n,  m  in  Zy  for  almost  all  a)  and  E±  as  in  theorem  2.5.,  we  also  have 


C  R\(E_,  £+)  almost  every  co. 
2.  Let  I  =  Z^+   and  an  be  a  sequence  satisfying  the  hypothesis  2.3(10,  (2')-  Then, 
<?ac(H+)  D  [-2  +  £_,  2  +  £+]  almost  every  CD. 

Further  when  IJL  satisfies  the  hypothesis  2.3(2),  an^Qona  subset  ofl^1  that  contains 
the  surface  {(0,  h)  :  n  €  Zy},  the  subspaces  7iw,«,  Ww,m  are  not  mutually  orthogonal 
almost  every  CD  form,  n  in  {(0,  k)  :  k  e  Zv},  we  also  have 

OjC/O  c  R  \  (~2,  +£_,  2  H-  £+)  a/matf  every  o>. 

Remark  1.  When  /u,  is  absolutely  continuous  the  theorem  says  that  the  spectrum  of  H® 
in  (£_,  £+)  (respectively  in  (-2  +  E_,  2  +  £+)  for  the  Z+"1"1  case)  is  purely  absolutely 
continuous,  this  is  a  consequence  of  a  remarkable  theorem  of  Jaksic-Last  [14]  who  showed 
that  in  such  models  with  independent  randomness,  with  the  randomness  non-zero  a.e.  on  a 
sufficiently  big  set  (H$  can  be  any  bounded  self  adjoint  operator  in  their  theorem,  provided 
the  set  of  points  where  the  randomness  lives  gives  a  cyclic  family  for  the  operators  H®), 
whenever  there  is  an  interval  of  a.c.  spectrum  it  is  pure  almost  every  a>  .  Their  proof  is  based 
on  considering  spectral  measures  associated  with  rank  one  perturbations  and  comparing 
the  spectral  measures  of  different  vectors  (which  give  rise  to  the  rank  one  perturbations). 

2.  Our  theorem  extends  the  models  of  surface  randomness  considered  by  Jaksic-Last 
[13],  to  allow  for  thick  surfaces  where  the  randomness  is  located  in  a  strip  beyond  the 
surface  into  the  bulk  of  the  material.  Such  models  (which  are  obtained  by  taking  an  = 
0,  n\  >  N,  an  =  1,  n\  <  TV  for  some  finite  N)  have  purely  absolutely  continuous 
spectrum  in  (—  2v  —  2,  2v  -f  2).  The  purity  of  the  a.c.  spectrum  is  again  a  consequence  of 
a  theorem  of  Jaksic-Last  [14]. 

Finally  we  have  the  following  theorem  on  the  purity  of  a  part  of  the  pure  point  spectrum. 
We  denote 


e+  =  sup<r(#o+),    £-  =  inf<r  (#b+)  and  e$  =  max(|e_|,  \e+\).  (2) 


1  84  Af  Krishna  and  K  B  Sinha 

Theorem  2.8.  Consider  a  bounded  self  adjoint  operator  HQ  coming  from  a  function  h 
satisfying  the  conditions  of  'hypothesis  2.1.  Let  I  be  the  indexing  set  and  suppose  q^ty),  n  € 
/  are  Lid  random  variables  with  the  distribution  fj,  satisfying  the  hypothesis  2.2(1),  (2). 
Assume  further  that  the  density  f(x)  =  d/x(jc)/dx  isbounded.  Seta\  =  f  djj,(x)\x\.  Then, 

1.  Letl  —  iy  and  let  an  be  a  sequence  satisfying  the  hypothesis  2.3(1),  (2).  Then  there  is 
a  critical  energy  E(JJL)  >  EQ  depending  upon  the  measure  /z  such  that 

(TcCH^)  C  (-£(M)>  £(M))  almost  every  a). 


2.  Let  I  =  Z++1  and  let  an  be  a  sequence  satisfying  the  hypothesis  2.3(10,  (20-   Then 
there  is  a  critical  energy  e(ii)  >  e$  such  that 

+)  C  (—e(iJi),e(fji))  almost  every  a>. 


Remark  1.  The  E(^)  and  efa),  while  finite  may  fall  outside  the  spectra  of  the  operators 
H®  and  H+,  for  some  pairs  (an,  JJL)  when  /z  is  of  compact  support,  so  for  such  pairs  this 
theorem  is  vacuous.  However  since  the  numbers  E(fjC)  (respectively  e(^))  depend  only 
on  the  operators  HQ  (respectively  HQ+)  and  the  measure  \JL  we  can  still  choose  sequences 
an  and  JJL  of  large  support  such  that  the  theorem  is  non-trivial  for  such  cases.  Of  course 
for  fji  of  infinite  support,  the  theorem  says  that  there  is  always  a  region  where  pure  point 
spectrum  is  present. 

2.  Since  we  allow  for  potentials  with  an  not  vanishing  at  oo  in  all  directions,  we  could  not 
make  use  of  the  technique  of  Aizenman-Molchanov  [3],  for  exhibiting  pure  point  spectrum. 

3.  When  //,  has  compact  support,  comparing  the  smallness  of  a  moment  near  the  edges  of 
support  one  exhibits  pure  point  spectrum  there  by  using  the  Lemma  5.  1  proved  by  Aizenman 
[1],  comparing  the  decay  rate  in  energy  of  the  sums  of  low  powers  of  the  integral  kernels  of 
the  free  operators  with  some  uniform  bounds  of  low  moments  of  the  measure  JJL  weighted 
with  singular  but  integrable  factors  occurring  to  the  same  power. 

As  in  Kirsch-Krishna-Obermeit  [9],  Jaksic-Last  [14]  we  also  have  examples  of  cases 
when  there  is  pure  a.c.  spectrum  in  an  interval  and  pure  point  spectrum  outside.  The  part 
about  a.c.  spectrum  follows  as  a  corollary  of  theorem  2.6.,  while  the  pure  point  part  is 
proven  as  in  [9]  (following  the  proof  of  their  theorem  2.3,  where  A  can  be  replaced  by 
any  bounded  self  adjoint  operator  on  l2(Zd)  and  work  through  the  details,  as  is  done  in 
Krishna-Obermeit  [12],  Lemma  2.1).  Further  when  HQ  =  A,  the  Jaksic-Last  condition 
on  the  mutual  non-orthogonality  of  the  subspaces  H^n,  H^^m  is  valid  since  given  any 
n,  m  we  can  find  a  k  so  that  («5n,  A^<5m>  >  0  (reason,  take  k  =  \n  -  m\  =  £^=1  |n,-  —  m,-  1, 
then 


with  Ti  denoting  the  bilateral  shift  in  the  zth  direction  and  c  a  strictly  positive  constant 
coming  from  the  multinomial  expansion).  We  see  that  we  can  add  any  operator  diagonal 
in  the  basis  {Bn}  to  A  without  altering  the  conclusion. 

COROLLARY  2.9 

Let  an  be  a  sequence  as  in  Hypothesis  2.  3  and  JJL  as  in  Hypothesis  2.  2.  Let  HO  =  A.  Assume 
further  that  an  ^  0,  n  €  Zv  goes  to  zero  at  oo  and  a-supp(/z)  =  R.  Then  we  have,  for 
almost  all  co, 


Spectra  of  random  operators  185 

)  =  [-2v,2v]. 

3.  asc(H^)  =  0. 

The  h  given  in  the  corollary  below  is  a  smooth  27rZy  periodic  function,  so  it  satisfies 
the  conditions  of  the  Hypothesis  2.1.  It  is  also  not  hard  to  verify  that,  because  of  the  term 
S/=i  cos(#/)  occurring  in  its  expression,  the  cyclic  subspaces  generated  by  the  associated 
HQ  on  any  pair  of  [8n,  8m]  are  mutually  non-orthogonal. 

COROLLARY  2. 10 

Let  an  be  a  sequence  as  in  Hypothesis  2.3  and  IJL  as  in  Hypothesis  2.2.  Let  HQ  be  a  bounded 
self  adjoint  operator  coming  from  the  function  h  given  by  h($)  =  £^=i  ]QkLi  cos(£0/)- 
Assume  that  an  /  0,  n  e  Zv  goes  to  zero  at  oo  and  a-supp(^)  =  R.  Then  we  have,  for 
almost  all  CD, 

1.  aac(H<»)  =  [£-,£+]. 

2. 

3. 


3.  Proofs 

In  this  section  we  present  the  proofs  of  the  theorems  stated  in  the  previous  section. 

Proof  of  Theorem  2.5.  The  statement  about  the  spectrum  of  HQ  follows  from  the  Hypothesis 
2.1(1)  on  the  function  h.  Each  of  the  functions  hi  is  a  real  valued  continuous  2n  periodic 
function,  hence  has  compact  range.  By  the  intermediate  value  theorem,  we  see  that  the 
range  of  (0,  2n)  under  A/  is  also  an  interval.  Since  the  spectrum  of  77o  is  the  algebraic 
sum  of  the  intervals  7,-,  -  if  HQJ  denotes  the  operator  associated  with  hj  on  €2(T),  then 

7/o  =  T/oi  ®  7  +  7  ®  77o2  ®  1  H h  7  <8>  77ov  hence  this  fact  -  the  statement  follows. 

We  note  that  -£2(Z+)  is  unitarily  equivalent  to  the  Hardy  space  H2(T)  of  functions  on  T 
whose  negative  Fourier  coefficients  vanish.  Under  this  unitary  transformation,  the  operator 
A+  is  unitarily  equivalent  to  the  operator  of  multiplication  by  the  function  2  cos(#)  acting  on 
H2(T),  which  can  be  seen  by  the  definitions  of  A+,  IH]2(T)  and  the  unitary  isomorphism  U 
that  takes  H2(I)  to  12(I+)  (explicitly  this  is  2n(Uf)(n)  =  fj*"  d<9  e~in ef(0)).  Therefore 
the  spectrum  of  A+  is  [—2,  2]  and  is  purely  absolutely  continuous  (there  are  no  eigenvalues). 
Therefore  the  spectrum  of  7/o+  is  also  purely  a.c.  and  equals  cr  (A+)  +  [£_,  £+],  with  E± 
as  above.  Hence  the  theorem  follows. 

Proof  of  Theorem  2.6.  We  prove  the  theorem  for  the  case  H®  the  proof  for  the  case  77+ 
proceeds  along  essentially  the  same  lines  and  we  give  a  sketch  of  the  proof  for  that  case. 
We  consider  any  A,  €  a-supp  (//,),  which  means  that  we  have 

/i(a~1(A.-6,A,H-6))  =  oo,   V&  G  Z+,&  ^  0,    and  all  €  >  0. 

•v+l 

We  consider  the  distance  function  \n\  =  max|n;|,  i  =  1, . . . ,  v  on  Zu.  We  consider  the 
events,  with  €  >  0,  m  e  £ZV, 

Ak,m,€  =  {&  :  cimq^(m}  €&-€,  A.  +  €),    \an  q°*(n)\  <  €,  VO  <  \n  -  m\  <  *  -  1} 


1  86 

and 


M  Krishna  and  K  B  Sinha 


(/i)|  <  6,  VO  <  |/z  -  m\  <  k  -  1}, 


where  the  index  w  in  the  definition  of  the  above  sets  varies  in  Zy.  Then  each  of  the  events 
Ak,m,€  are  mutually  independent  for  fixed  k  and  €  as  m  varies  in  &Zy,  since  the  random 
variable  defining  them  live  in  disjoint  regions  in  Zy.  Similarly  B^m^  is  a  collection  of 
mutually  independent  events  for  fixed  k  and  e  as  m  varies  in  &Zy  .  Further  these  events  have 
a  positive  probability  of  occurrence,  the  probability  having  a  lower  bound  given  by 


and 

Prob(£^,<)  >  Gu(-c  6,  c  O)(*~1)W+l, 
where  we  have  taken  c  =  inf^g^v  a~l  >  0.  The  definition  of  c  implies  that 

(-c  €,  c  6)  C  ^(-e,  €),  V  m  e  Zy. 
Therefore  the  assumption  that  A.  €  a-supp(^t)  implies  that  Vk  e  Z+  \  {0}, 


and  similarly 


]T  Prob(ftfm.€)  =  oo,  V&  €  Z+  \  {0}. 


Then  Borel-Cantelli  lemma  implies  that  for  all  6  >  0,  (setting  R€  =  (X  —  6,  A.  +  6)  and 
S€  =  (-6,  6)  and  A*(m)  =  [n  €  Zv  :  0  <  \n  -  m|  <  k  -  1}),  the  events 

^  :  ^^(^)  €  *«•  fl«  ^W(n)  e  S€,  Vn  e  Ak(m)  \  {m}} 


me/cZy 

#/=oo 

have  full  measure.  Therefore  the  event 


=  n 


has  full  measure,  being  a  countable  intersection  of  sets  of  full  measure.  Similarly  the  sets 

{a>:anq°>(n)eS€,  Vrc 


#/=oo 


have  full  measure.  Therefore  the  events 


UeZ+\{0} 


have  full  measure. 
We  take 


Spectra  of  random  operators  1  87 

and  note  that  it  has  full  measure.  We  use  this  set  for  further  analysis.  We  denote  H(k)  — 
HQ  -h  IPo-  Then  suppose  E  e  a(H  (X)),  then  there  is  a  Weyl  sequence  ^/  of  compact 
support,  V/  €  t2(T)  such  that  ||  V/||  =  1  and 

y. 

Suppose  the  support  of  V/7  is  contained  in  a  cube  of  side  r(/),  centered  at  0.  Denote  by 
AjkCx)  a  cube  of  side  /:  centered  at  x  in  Zy.  We  denote  Vw(n)  =  anq^(n),  for  ease  of 
writing.  We  then  find  cubes  Ar(/)(a/)  centered  at  the  points  a/  such  that 

,    V*  e 


Now  consider  $/(*)  =  V"/U  ~  #/)•  Then  by  the  translation  invariance  of  //o  we  have  for 
any  a>  €  £2o> 


,      +  0, 

Li 

Clearly  since  fy  is  just  a  translate  of  i/r/,  ||0/  1|  =  1  for  each  /.  We  now  have  to  show  that  the 
sequence  <pi  goes  to  zero  weakly.  This  is  ensured  by  taking  successively  a^  large  so  that 

UjlJsuppO^/)  H  Ar(*)(ajk)  =  0,    and  supp(^)  C  Ar(*)(ote). 

This  is  always  possible  for  each  a)  in  QQ  by  its  definition,  thus  showing  that  the  point  E  is 
in  the  spectrum  of  H  ®  ,  concluding  the  proof  of  the  theorem. 

Proof  of  Theorem  2.7.  We  first  consider  the  part  (1)  of  the  theorem  and  address  the  proof 
of  (2)  later.  The  set  C  below  is  as  in  Hypothesis  2.1.  We  consider  the  set 

V  =  {(j>  e  £2(ZU)  :  supp(0)  C  Ty  \  C  and  0  smooth},  (4) 

where  we  denote  by  <p  the  function  in  -£2(TV)  obtained  by  taking  the  Fourier  series  of  (/). 
Since  the  set  C  is  of  measure  zero,  such  functions  form  a  dense  subset  of  £2(ZV).  We  also 
note  that  the  set  C  is  closed  in  Tu,  thus  its  complement  is  open  (in  fact  it  is  a  finite  union  of 
open  rectangles)  and  each  0  in  V  has  compact  support  in  Tv  \  C. 

We  first  consider  the  case  when  JJL  has  compact  support.  The  general  case  is  addressed 
at  the  end  of  the  proof. 

If  we  show  that  the  sequence  W(t,cx))  =  QltHO)Q~ltHo  is  strongly  Cauchy  for  any  a>9  then 
standard  scattering  theory  implies  that  o-flC(J!/6>)  D  crac(Ho)  for  that  co.  We  will  show  below 
this  Cauchy  property  for  a  set  co  of  full  measure. 

To  this  end  we  consider  the  quantity 

E{||(W(f,a))-W(r,a)))0||},    0eD  (5) 

and  show  that  this  quantity  goes  to  zero  as  t  and  r  go  to  +00.  Then  the  integrand  being 
uniformly  bounded  by  an  integrable  function  ||0||  and  since  0  comes  from  a  dense  set, 
Lebesgue  dominated  convergence  theorem  implies  that  W(t,co)  is  strongly  Cauchy  for 
every  a>  in  a  set  of  full  measure  £2(/)  that  depends  on  /  in  €2(ZV).  Since  €2(Zy)  is 
separable,  we  take  the  countable  dense  set  T>\  and  consider 


188 


M  Krishna  and  K  B  Sinha 


which  also  has  full  measure  being  a  countable  intersection  of  sets  of  full  measure.  For 
each  a>  e  ^3,  W(t,  to)  is  a  family  of  isometries  such  that  W(t  ,  &>)/  is  a  strongly  Cauchy 
sequence  for  each  /  e  X>i,  therefore  this  property  also  extends  by  density  of  T>\  to  all  of 
12(ZV)  point  wise  in  ^3.  Thus  it  is  enough  to  show  that  the  quantity  in  (5)  goes  to  zero  as 
t  and  r  go  to  +00. 

We  have  the  following  inequality  coming  out  of  Cauchy-Schwarz  and  Fubini,  for  an 
arbitrary  but  fixed  (j>  e  T>.  In  the  inequality  below  we  denote,  for  convenience  the  operator 
of  multiplication  by  the  sequence  an  as  A  and  in  the  first  step  we  write  the  left  hand  side 
as  the  integral  of  the  derivative  to  obtain  the  right  hand  side 

mW(t,a>)4>-W(r9a))4>\\}<E\\\  f  ds 

I     Jr 

<  I  ds  E{||VV-"%||} 

Jr 

/t 


ds 


(6) 


The  required  statement  on  the  limit  follows  if  we  now  show  that  the  quantity  in  the  integrand 
of  the  last  line  is  integrable  in  s.  To  do  this  we  define  the  number 


=  mfmf{\hfj(0j)\  :  #  e  supp0),   #  = 


0V). 


(7) 


We  note  that  since  the  support  of  <j>  is  compact  in  ~P  \  C,  hj  ',  j  =  1,  .  .  .  ,  v  (which  are 
continuous  by  assumption),  have  non-zero  infima  there,  so  v^  is  strictly  positive.  Then 
consider  the  inequalities 


\\<rAF(\nj\  >  v+  s/4  Vj  )e'isH 
+  \\aAF(\nj\  <  v<f>  s/4  for  some 


(8) 


for  some 


where  we  have  used  the  notation  that  F(S)  denotes  the  orthogonal  projection  (in 
given  by  the  indicator  function  of  the  set  S  and  used  the  function  g  as  in  the  Hypothesis 
2.3(2)  which  is  integrable  in  s,  so  the  first  term  is  integrable  in  s.  We  concentrate  on  the 
remaining  term. 


\\F(\nj\  <  v0  5/4,   for  some  j)e"/sHVI|. 


(9) 


To  estimate  the  term  we  go  to  the  spectral  representation  of  HQ  and  do  the  computation 
there  as  follows.  Since  \rij\  <  v^  s/4  for  some  j,  we  may  without  loss  of  generality 
set  7  =  1  and  proceed  with  the  calculation.  Let  us  denote  the  set  S\(s)  =  (n  :  \n\  \  < 
V(f)S/4,  HJ  e  Z,  7  ^  1}.  In  the  steps  below  we  pass  to  L2(TV)  via  the  Fourier  series, 
(where  the  normalized  measure  on  Tv  is  denoted  by  dcr  (tf  )). 


f 
JT> 


1/2 


Spectra  of  random  operators 


189 


f       f[ 

/P-1    - 


/" 

->T 


2)1/2 


(10) 


We  define  the  function  J(9,  s,n\)  =  n\9  +sh\(0).  When  &  is  in  the  support  of  0,  we  have 
that  1*1(61)!  >  ity,  by  eq.  (7).  This  in  turn  implies  that  when  #  =  (0i, . . . ,  0P)  €  supp0, 


=  \ni  +  5  &i(0i)|  > 


when  «i  <  v<f>  s/4. 


We  use  this  fact  and  do  integration  by  parts  twice  with  respect  to  the  variable  0\  to  obtain 


= 


f 

JT-I 


j=2 


We  note  that  the  quantity 
3 


1 


'(Oi,ni,s) 


dcr(0i) 


2 1  1/2 


(ID 


(70 


(/O6 


a  ~ 


(/') 


'3 


(12) 


is  in  I2(TV). 

The  assumptions  on  the  lower  bound  on  J'  (when  \n\  \  <  v^s/4)  and  the  boundedness 
of  its  higher  derivatives  by  Cs  (which  is  straightforward  to  verify  by  the  assumption  on  hj) 
together  now  yield  the  bound 

T  ~      m  +  " 


which  gives  the  required  integrability. 

We  proved  the  case  (1)  of  the  theorem  assuming  that  JJL  has  compact  support.  The  case 
when  IJL  has  infinite  support  requires  only  a  comment  on  the  function  Q~~lsHQ</>  being  in 
the  domain  on  V0*  almost  everywhere,  when  s  is  finite  and  for  fixed  0  e  T>.  Once  this  is 
ensured  the  remaining  calculations  are  the  same.  To  see  the  stated  domain  condition  we  first 
note  that  for  each  fixed  5,  the  sequence  (e~~lsH°<p)  (n)  decays  faster  than  any  polynomial,  (in 
|n|).  The  reason  being  that,  by  assumption,  0  is  smooth  and  of  compact  support  in  Tv  \  C, 
10  (n)  |  <  \n\~N  for  any  N  >  0,  as  \n\  ->  oo.  On  the  other  hand  for  \n  —  m\  >  ,$||/fo||,  we 
have 


1 


,-         v"'"»-  \n-m\N 
These  two  estimates  together  imply  that 


,    for  any  N  >  0. 


V. 


(13) 


1  90  M  Krishna  and  K  B  Sinha 

We  now  consider  the  events 

An  =  {co:  \q»(n)\  >  \n\2v+1} 
and  they  satisfy  the  condition 

Prob(An)  <  oo, 


by  a  simple  application  of  Cauchy-Schwarz  and  the  finiteness  of  the  second  moment 
jit.  Hence,  by  an  application  of  Borel-Cantelli  lemma,  only  finitely  many  events  An  occ 
with  full  measure.  Therefore  on  a  set  of  full  measure  all  but  finitely  many  q®(n)  satisi 
\q°*(n)\  <  |/i|2v+1.  Let  the  set  of  full  measure  be  denoted  by  QI.  Then  for  each  CD  e&i\ 
have  a  finite  set  S(co)  such  that  e~lsH®c/)  is  in  the  domain  of  the  operator  V®  =  V  ^  (I  —  PS(CO] 
where  PS  (a)  is  tne  orthogonal  projection  onto  the  subspace  i2(S(co)),  in  view  of  the  e 
(13).  Then  the  proof  that  the  a.c.  spectrum  of  the  operator 

H?  =  Ho  +  V?,  V<w  €  £2i  D  ^o 

goes  through  as  before.  Since  for  each  co  €  fii  fl  £2o,  H^  differs  from  H™  by  a  fin: 
rank  operator,  its  absolutely  continuous  spectrum  is  unaffected  (by  trace  class  theory 
scattering)  and  the  theorem  is  proved. 

The  statement  on  the  singular  part  of  the  spectrum  of  H^,  is  a  direct  corollary  of  t 
Theorem  5.2.  We  note  firstly  that  since  {<5n,  n  e  Zv}  is  an  orthonormal  basis  for  £2(ZV} 
is  automatically  a  cyclic  family  for  H®  for  every  co. 

Secondly,  by  assumption,  the  subspaces  H^n  and  H<y,m  are  not  mutually  orthogon 
so  the  conditions  of  Theorem  5.2  are  satisfied.  Therefore,  since  the  a.c.  spectrum  of  / 
contains  the  interval  (E_,  £+)  almost  every  CD  the  result  follows. 

(2)  We  now  turn  to  the  proof  of  part  2  of  the  theorem.  The  essential  case  to  consic 
again  as  in  (1)  is  when  /x  has  compact  support,  the  general  case  goes  through  as  befo 
The  proof  is  again  similar  to  the  one  in  (1),  but  we  need  to  choose  a  dense  set  T)\  in  1 
place  of  V  properly. 

The  operator  A+  is  self  adjoint  on  £2(Z+)  and  its  restriction  A+i  to  -£2(Z+  \  {0}) 
unitarily  equivalent  to  multiplication  by  2  cos(<9)  acting  on  the  image  of  t2  (Z+  \  {0})  unc 
the  Fourier  series  map.  We  now  consider  the  operator 


-f  HO 
in  the  place  of  #0+  and  show  the  existence  of  the  Wave  operators 


almost  every  co. 
We  take  the  set  D  as  in  eq.  (4),  T>i  as  in  Lemma  3.1  and  define 


i      T-,         e     ,        e     2, 


C 


We  then  define  the  minimal  velocities  for  </>  e  D+  with  w^  defined  as  in  Lemma  3.1 


Spectra  of  random  operators  191 


inf 

k 


=  infinfinf{\h'j(0j)\  : 
v0  =  min{iui,0,  u>2,0}- 
Calculating  the  limits,  as  in  eq.  (5) 


=  /" 

Jr 


ds  IKe^^CV^  -  F0A+  +  -A+PO  +  PoA+P0)e~/f//0+10||,       (16) 


where  PQ  is  the  operator  po  ®  /,  with  /?o  being  the  orthogonal  projection  onto  the  one 
dimensional  subspace  spanned  by  the  vector  <$o  in  -£2(Z+).  We  note  that  by  the  definition 
of  A+,  the  term  PQ  A+  PQ  is  zero.  The  estimates  proceed  as  in  the  proof  of  (1),  after  taking 
averages  over  the  randomness  and  taking  0  €  P+.  As  in  that  proof  it  is  sufficient  to  show 
the  integrability  in  s  of  the  functions 

\\aAe-isHW(t>\\,    \\\Si  }(  S0|  0^-^+1011,    \\\&Q)(&i\®Ie-isHM4>\\, 

respectively.  By  the  definition  of  X>+,  any  0  there  is  a  finite  sum  of  terms  of  the  form 
<pj(0\)\lsj(02,  .  .  .  ,  0v+i),  so  it  is  enough  to  show  the  integrability  when  0  is  just  one  such 
product,  say  0  =  0i  ^i-  Therefore  we  show  the  integrability  in  s  of  the  functions 


H,    \\\8i  ){So|®/ 
for  ^  large  we  are  done.  We  have 

F(\ni\  >  v<i>s/4)8i  =  0,    *  =  0,  1  and  ||crAF(|n/|  >  v0ly/4,  V/)ll  G  L^l,  oo), 

by  the  Hypothesis  2.3(2X)  on  the  sequence  an.  Therefore  it  is  enough  to  show  the  integra- 
bility of  the  norms 


for  each  y  =  l,...,y+l.  When  7  =  2,  .  .  .  ,  v  +1  ,  the  proof  is  as  in  the  previous  theorem, 
while  for  j  —  1,  the  proof  is  given  in  the  Lemma  3.1  below. 

The  statement  on  the  absence  of  singular  part  of  the  spectrum  of  H^-}-  in  (E-  —  2.E+  + 
2),  is  as  before  a  direct  corollary  of  the  Theorem  5.2,  since  the  set  of  vectors  [8n,  n  — 
(0,  m),  m  e  2V]  is  a  cyclic  family  for  H+,  for  almost  all  co  and  Ti^n  and  Ti^m  are  not 
mutually  orthogonal  for  almost  all  co  when  m,  n  are  in  {(0,  n)  :  n  e  Zv},  and  the  fact  that 
the  a.c.  spectrum  of  H™  contains  the  interval  (—2  +  £_,  2  H-  E+)  almost  every  ox 

The  lemma  below  is  as  in  Jaksic-Last  [13](Lemma  3.11)  and  the  enlarging  of  the  space 
in  the  proof  is  necessary  since  there  are  no  non-trivial  functions  in  ^2(Z+)  whose  Fourier 
series  has  compact  support  in  (0,  2n)  (all  of  them  being  boundary  values  of  functions 
analytic  in  the  disk). 

Lemma  3.  1.  Consider  the  operator  A+i  on£2(Z+).  Then  there  is  a  set  T>2  dense  in  i2  (Z+) 
and  a  number  w^  such  that  for  s  >  1, 


C|5|~,   V0 
with  the  constant  C  independent  ofs. 


192  M  Krishna  and  K  B  Sinha 

Proof.  We  first  consider  the  unitary  map  W  from  HQ  to  a  subspace  S  of  {/  e 
/(0)  =  0},  given  by 


(W/X/i)  =  (17) 

(~n)'  n  <  °" 


Then  the  range  of  W  is  a  closed  subspace  of  I2  (I.)  and  consists  of  functions 

S  =  {/  €  £2(Z)  :  /(n)  =  -/(-n)}. 
Under  the  Fourier  series  map  this  subspace  goes  to 


so  that  the  functions  here  have  mean  zero.  Then  under  the  map  from  £2(Z"1"  \  {0})  to  S  ob- 
tained by  composing  W  and  the  Fourier  series  map,  the  operator  AI+  goes  to  multiplication 
by  2  cos(#).  We  now  choose  a  set 


and  define  the  number 

W(f,  =  inf{|2sin(6>)|  :  0  e  supp(0)}, 

for  each  0  e  T>\  .  We  denote  by  T>2  all  those  functions  whose  images  under  the  composition 
of  W  and  the  Fourier  series  lies  in  V\  .  The  density  of  £>2  in  £2  (Z+  \  {0})  is  then  clear.  We 
shall  simply  denote  by  fa  elements  in  T>2  whose  images  in  T>\  is  0.  Given  a  0  e  T>\  and  a 
w  we  see  that 


f 

JJ 


2 

<C\S\ 


by  a  simple  integration  by  parts,  done  twice,  using  the  condition  that  \\n\  +  2s  sin(0)l  > 
w^s/4  in  the  support  of  6. 


Proof  of  Theorem  2.8.  The  proof  of  this  theorem  is  based  on  a  technique  of  Aizenman 
[1].  We  break  up  the  proof  into  a  few  lemmas.  First  we  show  that  the  free  operators  HO 
and  #0+  have  resolvent  kernels  .with  some  summability  properties,  for  energies  in  their 
resolvent  set. 

Lemma  3.2.  Consider  a  function  h  satisfying  the  Hypothesis  2. 1  and  consider  the  associated 
operators  HO  or  HQ+.  Then  for  all  s  >  v/(3v  +  3), 

sup  y~^  |  { 8n,  (#o  —  E)~l8m  )  \s  <  C(E), 

and  C(E)  ->  0,  \E\  — >  oo.  Similarly  we  also  have  for  all  s  >  v/(3v  +  3), 
sup    V ^  |  {  5, 


Proof.  We  will  prove  the  statement  for  HO,  the  proof  for  #0+  is  similar.  We  write  the 
expression  for  the  resolvent  kernel  in  the  Fourier  transformed  representation  (we  write 


Spectra  of  random  operators 


193 


the  Fourier  series  of  an  £2(Zy)  function  as  ii(&)  =  ^2neZv  em'^(/z)),  use  the  Hypothesis 
2.1(1),  and  integrate  by  parts  3v  +  3  times  with  respect  to  the  variable  Oj  (recall  that 
&  =  (#1  ,  .  .  .  ,  #„)),  to  get  the  inequalities 


/ 
•» 


JTU 


((m-n);)^3 
-E)-1,  (18) 


where  we  have  chosen  the  index  j  such  that  |(m  —  w)y|  >  \m  —  n\/v  and  assumed  that 
m  7^  n  (when  m  —n  the  quantity  is  just  bounded).  Let  us  set 


CQ(£)  =  max    \  sup 


a3v+3 


It  is  easy  to  see  that  since  the  function  h  is  of  compact  range  and  all  its  3v  +  3  partial 
derivatives  are  bounded,  by  hypothesis  CQ(£)  goes  to  zero  as  \E\  goes  to  oo.  We  then  get 
the  bound  for  any  s  >  v/(3v  +  3), 

3v+3 


m-n 


Given  this  estimate  we  have 


sup 


where  C(s)  is  finite  since  \m\  J(3v+3D ,  m  ^  0  is  a  summable  function  in  2V  when  s(3v  -f- 
3)  >  v. 

Proof  of  Corollary  2.9.  We  prove  the  theorem  only  for  the  case  Hta  the  proof  of  the  other 
case  is  similar. 

By  the  Hypothesis  2.3(2)  on  the  fmiteness  of  the  second  moment  of  /x  we  see  that 
/  d/x(x)  |jc|  <  oo,  so  that  we  can  set  r  =  1  in  the  Lemma  5.1.  Since  the  assumption  in  the 
theorem  ensures  the  boundedness  of  the  density  of  \JL  we  can  also  set  q  =  oo  in  the  Lemma 
5.1  with  then  Q1/14"^  =  ||d/i/dx||oo-  Then  in  the  Lemma  5.1  the  constant  C  is  given  by 

*  \       1       2/cQ 

-,  oo  I  =  1  +  • 


'  l-2/c'     7  l-/c 

The  condition  on  the  constant  K  becomes 

K  <  1/3. 
Below  we  choose  a  s  satisfying  (3^3)  <  -s  <  1/3,  and  consider  the  expression 


194  M  Krishna  and  K  B  Sinha 


where  we  take  z  =  E  -f-  i€  with  6  >  0.  Then  by  the  resolvent  equation  we  have 

G(co9z,n,m)  =  G(Q,z,n,m)-  £  G(a>,  z,  n,  /)V"(/)G(0,  z,  /,  m).    (2 

/€Z«M 

We  denote  by 


where  P/  is  the  orthogonal  projection  onto  the  subspace  generated  by  <$/.  Then  using  t 
rank  one  formula 


whose  proof  is  again  by  resolvent  equation,  we  see  that  eq.  (20)  can  be  rewritten  as 
G(a>,  z,  n,  m)  —  G(0,  z,  n,  m) 


+ 


(2 


Raising  both  the  sides  to  power  5-  (noting  that  s  <  I  so  the  inequalities  are  valid),  we  ge 


(2 


Now  observing  that  G;  is  independent  of  the  random  variable  V^C/),  we  see  that 


|G(0,z,/,m)|s. 


This  then  becomes,  integrating  with  respect  to  the  variable  q*>  (I) ,  remembering  that  V"  (/) 
=  |G((U,/i,m)|ff 

•) 


Gt(a>,z,l,t) 


which  when  estimated  using  the  Lemma  5.1  yields 

?(a>,z,nfZ) 


»,z,U) 


1 


Spectra  of  random  operators  195 

where  Ks  is  the  constant  appearing  in  Lemma  5.1  with  K  set  equal  to  s.  We  take  K  = 
^  \an\s)KS9  and  rewrite  the  above  equation  to  obtain 


,  z,  w,  m)|5)  =  |G(0,  z,  /i,  m)|*  +  ^  #E(|(G(a>,  z,  n,  /)T|G(0,  z,  /,  m)|J.      (26) 

> 

We  now  sum  both  the  sides  over  m,  set 


and  obtain  the  inequality 

|G(0,  z,  /i,  m)|J  4-  sup    T  AT/|G(0,  z, 


Therefore  when  there  is  an  interval  (a,  b)  in  which 

Ksup    Y    |G(0,z,/,m)|*  <  1,    £€(*,*),  (27) 


> 

we  obtain  that 


E(|G(<w,  E  +  fO,  n,  m)|5)  <  oo, 

by  an  application  of  Fatou's  lemma  implying  that  for  almost  all  E  e  (a,  b)  and  almost  all 
CD,  we  have  the  finiteness  of 

+  /0,/2,m)|2  <  oo, 

satisfying  the  Simon-Wolff  [19]  criterion.  This  shows  that  (the  proof  follows  as  in  Theo- 
rems II.5,  II.6  [18])  the  measures 

fit 

are  pure  point  in  (a,  b)  almost  every  CD.  This  happens  for  all  n,  hence  the  total  spectral 
measure  of  H0*  itself  is  pure  point  in  (a,  b)  for  almost  all  &>. 

There  are  two  different  ways  to  fix  the  critical  energy  £(AO  now-  Firstly  if  K  is  large, 
then  in  view  of  the  Lemma  3.2  (by  which  Co(£)  ->  0,  \E\  ->  oo)  and  the  fact  that  K  is 
finite  (by  Lemma  5.1) 


K  sup  |G(0,  z,  /,  m)\s  <  KCo(E)sC(s)  <  1,    \E\  ->  oo.  (28) 


Therefore  there  is  a  large  enough  E(IJL)  such  that  for  all  intervals  (a,  b)  in  (— oo,  —  1 
?  (E(jjL),  oo),  the  condition  in  eq.  (25)  is  satisfied. 

On  the  other  hand  if  the  moment  B  =  f  \x\  d/x(jc)  is  very  small,  then  we  can  choose 
i  by  the  condition, 


KC0(E)CS  <  1, 
even  when  C$(E)  >  1,  since  it  is  finite  for  E  in  the  resolvent  set  of  HQ  by  Lemma  3.2. 


196  M  Krishna  and  K  B  Sinha 

4.  Examples 

In  this  section  we  present  some  examples  of  the  operators  H$  considered  in  the  theorems. 
We  only  give  the  functions  h  stated  in  the  Hypothesis  2.1. 

®  Examples  of  operators  HQ 

1.  h($)  =  £7=1  2cos(0|),  corresponds  to  the  usual  discrete  Schrodinger  operator  and 
it  is  obvious  that  the  Hypothesis  2.1  are  satisfied.  The  Jaksic-Last  condition  5.2  on 
mutual  non-orthogonality  of  the  subspaces  generated  by  HQ  and  8n  for  different  n  in 
jy  are  also  satisfied,  by  an  elementary  calculation  taking  powers  of  HQ  depending 
upon  a  pair  of  vectors  3n  and  <5m,  since  the  operator  HQ  is  given  by  T  -f-  T~l,  with  T 
being  the  bilateral  shift  on  12(Z). 


2.  fc(tf)  =  E,LiAi(ftO,Ai(»i)  =  EcosOWj),    N(i)  <  co.  Clearly  each  A/  is  a 
smooth  function  in  Rv  and  each  hi  and  all  its  derivatives  are  2n  periodic.  Hence  the 
Hypothesis  2.3  is  satisfied.  Further  each  of  /i/  is  a  trigonometric  polynomial,  and  its 
derivative  is  also  a  trigonometric  polynomial  and  hence  has  only  finitely  many  zeros 
on  the  circle. 

The  condition  in  Jaksic-Last  condition  Theorem  5.2  on  mutual  non-orthogonality 
is  again  elementary  to  verify  in  this  case. 

3.  Consider  the  functions 

hi(0i)  =  0?v+4(2jr  -  0;)3v+4,    0  <  Ot  <  2n,    i  =  1,  .  .  .  ,  v 

and  take  h  =  XllLi  hi  W  extended  to  the  whole  of  Rv  periodically.  Clearly  these  are 
in  C3u+3(Tv),  by  construction. 

Examples  of  pairs  (an  ,  /z) 

We  give  next  some  examples  of  sequences  an  satisfying  the  Hypothesis  2.2  such  that 


=  a-suppO). 

We  consider  v  >2  and  the  sequence  an  =  (1  +  |«i  |)a,  a  <  -1.  Then  we  have  that 
kiy  n  {(0,  n)  :  n  e  Z^1}  =  {(0,  n)  :  n  €  kZv"1} 

and  a^Ca,  b)  =  (a,  b)  for  any  interval  (a,  b)  and  any  n  e  1v~l.  Therefore  for  any 
positive  integer  k,  we  have 


whenever  (Ji((a,  b))  >  0. 

•  Examples  of  measures  y,  with  small  moment 

We  next  give  an  example  of  an  absolutely  continuous  measure  of  compact  support  sucri 
that  the  Aizenman  condition  (in  Lemma  5.  1  is  satisfied.  We  use  the  notation  used  in  that 
lemma  for  the  example. 

We  consider  numbers  0  <  6,  8  <  1,  R  and  let  p  be  given  by 


Spectra  of  random  operators  1 97 

*.       8  <  jc  <  R  (29^ 

D ?  )         «     ^--.    .A,      _^     J.\  j  \^*^ ) 

0,    otherwise. 
Then  /z  is  an  absolutely  continuous  probability  measure  and 


*  -  5       jR-5 
We  take  T  =  1 ,  then  the  moment  B  is  bounded  by 

B  <  (1  -  6)5 +  (/?  H- 5)6/2. 
Now  if  we  fix  /?  large  and  choose  €  =  l/R3  and  5  =  I//?2,  we  obtain  an  estimate 


B*  <          and  j^g 


1"2* 


Taking  /c  =  s  in  the  lemma  and  noting  that  s  <  1/3  implies  2  —  6s  <  0  so  that  both  the 
terms  above  go  to  zero  as  R  goes  to  oo.  We  see  that  by  taking  IJL  with  large  support  but 
small  moment,  we  can  make  the  constant  K  in  the  Lemma  5.  1  as  small  as  we  want.  This 
in  particular  means  that  in  the  Theorem  2.8.  given  a  energy  E$  outside  the  spectrum  of 
HQ  we  can  find  a  measure  ^  which  is  absolutely  continuous  of  small  moment  such  that 
^  is  smaller  than  Co  (£o)5C5  in  the  proof  of  Theorem2.8.  andhence  £(/x)  <  \EQ\.  We 
can  use  such  measures  to  give  examples  of  operators  with  compact  spectrum  with  both 
a.c.  spectrum  and  pure  point  spectrum  present  but  in  disjoint  regions. 

•  Example  when  Jaksic-Last  condition  is  violated 

We  finally  give  examples  where  Jaksic-Last  condition  is  violated  and  yet  the  conclusion 
of  their  theorem  is  valid. 

Consider  v  =  1,  for  simplicity,  and  let  h(0)  =  2  cos(20).  Then  the  associated  HQ  has 
purely  a.c.  spectrum  in  [-2,  2]  and  we  see  that  the  operator  HQ  —  T2  +  7"~2  if  T  is  the 
bilateral  shift  acting  on  12(Z).  Then  if  we  consider  the  operators  H°>  =  HQ  +  Vw,  and 
the  cyclic  subspaces  HU,  i  ,  Ha),2  generated  by  the  H®  and  the  vectors  <$i  ,  £2  respectively, 
such  an  operator  satisfies 

Wo,fi  C  12({1}  +  2Z),    Ha>,2  C  12({1  +  1)4-  2Z),    almost  every  CD. 
We  then  have 

Wo,,!  C  I2({2n  +  l,/i€  /}),    W<y,2  C  -£2(2Z),    almost  every  co. 

The  subspaces  I2({n  :  n  odd})  and  £2(2Z)  are  generated  by  the  families  {<$&,  k  odd}  and 
{<$£,  k  even}  respectively.  (We  could  have  taken  any  odd  integer  k  in  the  place  of  1  to  do 
the  above) 

These  two  are  invariant  subspaces  of  H®  which  are  mutually  orthogonal,  a.e.  co. 
Therefore  the  Jaksic-Last  theorem  is  not  directly  valid.  However,  by  considering  the 
restrictions  of  H®  to  these  two  subspaces,  one  can  go  through  their  proof  in  these 
subspaces  to  again  obtain  the  purity  of  a.c.  spectrum  for  such  operators  when  they  exist. 

We  consider  two  examples  to  illustrate  the  point,  for  which  we  let  #w(n)  denote  a 
collection  of  i.i.d.  random  variables  with  an  absolutely  continuous  distribution  /x  of 
compact  support  in  R,  its  support  containing  0. 


198  M  Krishna  and  K  B  Sinha 

1.  If  V^(n)  =  anqw(n),  with  0  <  an  <  (1  H-  |n|)~~a,  a  >  0,  we  see  that  there  is  pui 
a.c.  spectrum  in  [-2,  2],  a.e.  &>  by  applying  trace  class  theory  of  scattering. 

2.  On  the  other  hand  if,  with  0  <  an  <  (1  +  N)~a,  a  >  1, 

(n),  n  odd 
),  n  even, 

then  there  is  dense  pure  point  spectrum  embedded  in  the  a.c.  spectrum  in  [-2,  2" 

We  can  give  similar,  but  non  trivial,  examples  in  higher  dimensions  but  we  leave  it  to  tl 
reader. 

5.  Appendix 

In  this  appendix  we  collect  two  theorems  we  use  in  this  paper.  One  is  a  lemma  of  Aizenm; 
[1]  and  another  a  theorem  of  Jaksic-Last  [14]. 

The  first  lemma  and  its  proof  are  those  of  Aizenman  [1] (Lemma  A.I)  which  reprodu 
below  (with  some  modifications  in  the  form  we  need),  with  a  slight  change  in  notation  (\ 
in  particular  call  the  number  s  in  Aizenman's  lemma  as  /c), 

Lemma  5.1  (Aizenman).  Let  IJL  be  an  absolutely  continuous  probability  measure  who 
density  f  satisfies  J^&x\f(x)\l+q  =  Q  <  ooforsomeq  >  0.  LetO  <  r  <  1  andsuppo 
B  =  fRdii(x)  \x\r  <  co.  Then  for  any 


r      2     n-1 

*<!+-+- 

L        r      q\ 


we  have 

JC  for  all   a  6  C, 


\x-a\K  JR  \x-a\ 

with  KK  given  by 


KK  =  fl    (2+^  +4)    51-    +  B    C(fi, 


,I=2« 


<  00. 


Remark.  We  see  from  the  explicit  form  of  the  constant  KK  that  the  moment  B  can 
made  sufficiently  small  by  the  choice  of  IJL  even  when  its  support  is  large.  This  will  ensi 
that  in  some  models  of  random  operators,  the  region  where  the  Simon-Wolff  criterion 
valid  extends  to  the  region  in  the  spectrum.  This  is  the  reason  for  our  writing  KK  in  ti 
form. 

Proof.  The  strategy  employed  in  proving  the  lemma  is  to  consider  the  ratio 


and  obtain  upper  bounds  for  the  numerator  and  lower  bounds  for  the  denominator. 

Note  first  that  B  finite  and  K  <  r  implies  that  |jc  -  a\K  is  integrable  even  if  a  is  pur 
real  and  we  have 

rb  i 

f(x)dx< 


/ 

Ja 


Spectra  of  random  operators  199 

by  Holder  inequality.  Hence 

f  i  r00  i 

I  dfji(x) <  1  +  /      dr  IJL({X  : >  r}) 

J  \x-a\K  Ji  \x-a\K 


—  K 

(31) 


where  the  integral  is  estimated  using  the  estimate  in  eq.  (30). 

Consider  the  region  \ot\  >  (25)  r :  We  then  estimate  for  fixed  a  the  contributions  from  the 
regions  \x\  <  \ot\/2  and  \x\  >  \ot\/2  to  obtain 


\x\" 

—  ±  r-rr  \  I  a^x)  \x  -  +  /  a/j,(x) ] 

*K        \a\K  \J  J  \x-a\K ) 

(32) 


/ 


with  K  chosen  so  that  K/(\  —  IK/T)  <  q/(\  +  q).  (Here  we  have  explicitly  calculated 
the  p  occurring  in  the  lemma  of  Aizenman  in  terms  of  K.  and  r).  For  a  fixed  T  and  q  this 
condition  is  satisfied  whenever  K  satisfies  the  inequality  stated  in  the  lemma. 

The  lower  bounds  on  /d/x(x)  l/\x  —a  \K  is  obtained  first  by  noting  that  5  <  oc  implies 

VL({X  :  \x\r  >  (25)})  <  i. 
Since  \a\  >  (25)^,  we  have  the  trivial  estimate 


;/, 

J\x\ 

*L 


-f 


,  IJ 

\x-a\ 


\x-  ct\K 

(33) 


2(M  +  (2B)ry 
Putting  the  inequalities  in  (32)  and  (33)  together  we  obtain,  (remembering  that  |a 

(25)b, 


(34) 


We  now  consider  the  region  \a\  <  (25) r;  Estimating  as  in  eq.   (32)  but  now  splitting 

the  region  as  \x\  <  (25)^  and  \x\  >  (25)  r ,  we  obtain  the  analogue  of  the  estimate  in 
/  eq.  (32),  in  this  region  of  a  as 


^—  <  -i-r  f  f 

-a\K       (2fl)r  \J 


x-a 


(35) 


200  M  Krishna  and  K  B  Sinha 

Similarly  the  estimate  for  the  denominator  term  is  done  as  in  eq.  (33), 


I 

J 


-  Ot 


>  I        , 

JU|<(2J5)r 


2((2B)r  +  (2B)r) 


(3 


4(2  JB) 
Using  the  above  two  inequalities  we  obtain  the  estimate, 


when  |«  |  <  (2B)  *  .  Using  the  inequalities  (34)  and  (36)  obtained  for  these  two  regions 
values  of  a  we  finally  get 


C(Gt 


for  any  a  e  R. 

We  next  state  a  theorem  (Corollary  1.1.3)  of  Jaksic-Last  [14]  without  proof  ,  its  proo 
as  in  Corollary  1.1.3  of  Jaksic-Last  [14].  We  state  it  in  the  form  we  use  in  this  paper. 

Theorem  5.2  [Jaksic-Lasi\.  Suppose  H  is  a  separable  Hilbert  space  and  A  a  boun 
self  adjoint  operator.  Suppose  {</>„}  are  normalized  vectors  and  let  Pn  denote  the  orth 
onal  projection  on  to  the  one  dimensional  subspace  generated  by  each  <t>n.  Let  q  (n. 
independent  random  variables  with  absolutely  continuous  distributions  //,„.  Consider 

A®  =  A  +  V"  q^(n)Pn,   almost  every  a). 

n 

Suppose  that  the  following  conditions  are  valid 

1.  The  family  {<£„}  is  a  cyclic  family  for  A™  a.e.  a). 

2.  Let  Ha>,n  denote  the  cyclic  subspace  generated  by  A™  and  0n.  Then  the  cyclic  subsp 
H^n  and  H^m,  are  not  orthogonal 

Then  whenever  there  is  an  interval  (a,  b)  in  the  absolutely  continuous  spectrum  of  A 
A  +  Yin  <?co(n)j?>n>  almost  all  a),  we  have 

o>(Aw)  H  (a,  b)  =  0,    almost  every  a). 

Acknowledgement 

This  work  is  supported  by  the  grant  DST/INT/US(NSF-RP014)/98  of  the  Departme 
Science  and  Technology. 


Spectra  of  random  operators  20 1 

References 

[1]  Aizenman  M,  Localization  at  weak  disorder:  Some  elementary  bounds,  Rev.  Math.  Phys.  6 

(1994)1163-1182 
[2]  Aizenman  M  and  Graf  S,  Localization  bounds  for  electron  gas,  preprint  mp_arc  97-540 

(1997) 
[3]  Aizenman  M  and  Molchanov  S,  Localization  at  large  disorder  and  at  extreme  energies:  an 

elementary  derivation,  Commun.  Math.  Phys.  157  (1993)  245-278 
[4]  Boutet  de  Monvel  A  and  Sahbani  J,    On  the  spectral  properties  of  discrete  Schrodinger 

operators,  C.  R.  Acad.  Sci.  Paris,  Series  1 326  (1998)  1145-1150 
[5]  Boutet  de  Monvel  A  and  Sahbani  J,    On  the  spectral  properties  of  discrete  Schrodinger 

operators:  multidimensional  case,  to  appear  in  Rev.  Math.  Phys. 
[6]  Carmona  R  and  Lacroix  J,    Spectral  theory  of  random  Schrodinger  operators    (Boston: 

Birkhauser  Verlag)  (1990) 
[7]  Cycon  H,  Froese  R,  Kirsch  W  and  Simon  B,  Topics  in  the  Theory  of  Schrodinger  operators 

(New  York:  Springer- Verlag,  Berlin,  Heidelberg)  (1987) 

[8]  Figotin  A  and  Pastur  L,  Spectral  properties  of  disordered  systems  in  the  one  body  approxi- 
mation (Berlin,  Heidelberg,  New  York:  Springer- Verlag)  (1991) 
[9]  Kirsch  W,  Krishna  M  and  Obermeit  J,  Anderson  model  with  decaying  randomness-mobility 

edge.  Math.  Zeit.  (2000)  DOI  10.1007/s002090000136 
[10]  Krishna  M,  Anderson  model  with  decaying  randomness  -  Extended  states,  Proc.  Indian. 

Acad.  Sci.  (Math.  Sci.)  100  (1990)  220-240 
[11]  Krishna  M,  Absolutely  continuous  spectrum  for  sparse  potentials,  Proc.  Indian.  Acad.  Sci. 

(Math.  Sci.),  103(3)  (1993)  333-339 
[12]  Krishna  M  and  Obermeit  J,  Localization  and  mobility  edge  for  sparsely  random  potentials, 

preprint  xxx.lanl.gov/math-ph/98050 1 5 

[13]  Jaksic  V  and  Last  Y,  Corrugated  surfaces  and  a.c.  spectrum  (to  appear  in  Rev.  Math.  Phys) 
[14]  Jaksic  V  and  Last  Y,   Spectral  properties  of  Anderson  type  operators,   Invent.  Math.  141 

(2000)561-577 
[15]  Jaksic  V  and  Molchanov  S,  On  the  surface  spectrum  in  dimension  two,  Helvetica  Phys.  Acta 

71  (1999)  169-183 
[16]  Jaksic  V  and  Molchanov  S,  Localization  of  surface  spectra,  Commun.  Math.  Phys.  208  (1999) 

153-172 
[17]  Reed  M  and  Simon  B,  Methods  of  modern  mathematical  physics:  Functional  analysis  (New 

York:  Academic  Press)  (1975) 
[18]  Simon  B,  Spectral  analysis  of  rank  one  perturbations  and  applications  in  CRM  Lecture  Notes 

(eds)  J  Feldman,  R  Froese,  and  L  Rosen,  Am.  Math.  Soc.  8  (1995)  109-149 
[19]  Simon  B  and  Wolff  T,    Singular  continuous  spectrum  under  rank  one  perturbations  and 

localization  for  random  Hamiltonians,  Comm.  Pure  Appl.  Math.  39  (1986)  75-90 
[20]  Stein  E,  Harmonic  analysis  -  real  variable  methods,  orthogonality  and  oscillatory  integrals 

(New  Jersey:  Princeton  University  Press,  Princeton)  (1993) 
[21]  Weidman  J,  Linear  operators  in  Hilbert  spaces,  GTM-68  (Berlin:  Springer- Verlag)  (1987) 


Multipliers  for  the  absolute  Euler  siimmability  of  Fourier  series 

PREM  CHANDRA 

School  of  Studies  in  Mathematics,  Vikram  University,  Ujjain  456  010,  India 
MS  received  30  December  1999;  revised  30  October  2000 

Abstract.  In  this  paper,  the  author  has  investigated  necessary  and  sufficient  condi- 
tions for  the  absolute  Euler  summability  of  the  Fourier  series  with  miltipliers.  These 
conditions  are  weaker  than  those  obtained  earlier  by  some  workers.  It  is  further  shown 
that  the  multipliers  are  best  possible  in  certain  sense. 

Keywords.  Multipliers;  absolute  summability;  summability  of  factored  Fourier 
series;  absolute  Euler  summability. 

1.  Definitions  and  notations 

Eoo 
_   wn  be  a  given  infinite  series  and  let  q  be  a  real  or  complex  number  such  that 

q  i=.  —  1.  Then  we  write 

-mwm;      w°n=wn.  (1.1) 


m=0 

Following  Chandra  [2],  Y^  wn  is  said  to  be  absolutely  summable  by  (E,  q)  means  (or 

Eoo 
_  an  €  |£,  q\  if 

CO 

£}       \W\\        <        00.  (1.2) 

n=0 

For  q  >  0,  a  reference  may  be  made  to  Hardy  ([9];  p.  237).  It  may  be  observed  that  the 
method  \E,q\  (q  >  0)  is  absolutely  regular. 

Let  LI-K  be  the  space  of  all  2n  -periodic  and  Lebesgue-integrable  functions  over  [—  TT,  re]. 
Then  the  Fourier  series  of  f  e  LI^  at  x  is  given  by 

*  00  00 

-ao  +  y^(fl/i  cos  nx  +  bn  sin  nx)  =  ^  An(x),  (1.3) 

n=l  n-0 

where  an  and  bn  are  the  Fourier  coefficients  of  /. 

Throughout  the  paper,  we  assume  that  the  constant  term  arj  =  0.  For  real  x,  q  >  0  and 
<5  >  0,  we  write 

-t)}9  (1.4) 

(1-5) 


=     -  I  0(iOd«, 
t  Jo 


204  Prem  Chandra 


cos  f)1/2,  (1. 


(1. 

r=l 

=    log-s(«  +  l),  (1.1 

n     =    dn-dn+\,  (1.1 

(1.1 


KO     =    tlog**,  (1-1 

,  (1.1 


nt 

where  0  <  c  <  n  and  k  is  a  suitable  positive  constant  taken  for  the  convenience  in  t 
analysis  and  possibly  depending  upon  8. 

2.  Introduction 

In  1968,  Mohanty  and  Mohapatra  [12]  began  the  study  of  absolute  Euler  summability 
Fourier  series  by  proving  the  following: 

Theorem  A.  Let 

0(f)log-eBV(0,c),     0<c<l.  (2 

Then 


Among  other  results  the  above  result  was  also  proved  by  Kwee  [10]  independently. 
also  proved  that  the  condition  (2.1)  cannot  be  replaced  by  the  weaker  condition 

0(0  log77-  eBV(CU),     0<7j<l,  C 

in  Theorem  A.  This  result  of  Kwee  [10]  was  further  improved  by  the  present  author  ; 
Dikshit  [7]. 

In  1978,  the  present  author  [4]  proved  the  following: 

Theorem  B.  Let 


Then 


n^l  10g(-n  ' 


Multipliers  for  Fourier  series  205 

Recently,  Ray  and  Sahoo  [15]  have  not  only  bridged  the  gap  in  between  Theorems  A  and 
B  but  they  have  also  improved  Theorem  B  by  proving  the  following: 

Theorem  C.  Let  0  <  8  <  I  and  let 

k 

0(01ogl~5-  €  BV(0,c),      0<c<l.  (2.6) 

t 

Then 

oo  A    (    \ 


-  €  \E,q\         (q  >  0).  (2.7) 

—  log" (/i  -t-  i) 

It  may  be  remarked  that  in  Theorem  C,  8  has  been  restricted  to  be  in  [0, 1  ]  since  for  8  >  1 , 
(2.6)  implies  the  absolute  convergence  of 


A  reference  may  be  made  to  Chandra  [1];  Theorem  2  on  page  6,  and  hence  (2.8)  is  neces- 
sarily summable  |£,  g|  (q  >  0). 

In  a  different  setting,  very  recently,  Dikshit  [8]  has  obtained  a  few  more  results  concerning 
the  absolute  Euler  summability  factors  for  Fourier  series. 

One  of  the  main  objects  of  the  present  paper  is  to  improve  Theorem  C  on  replacing  (2.6) 
by  the  following  weaker  condition: 


(i)  P(t)logl~s~         e      BV(0,c) 

(ii)  t-lP(t)log-**     e      L(0,c) 


(2.9) 


where  0  <  8  <  1  and  0  <  c  <  1.  The  above  claim  that  (2.9)  is  weaker  than  (2.6)  has  been 
settled  in  Lemma  1  of  the  present  paper. 

Secondly,  we  investigate  necessary  and  sufficient  conditions,  imposed  upon  the  gener- 
ating functions  of  the  Fourier  series  of  /  at  jc,  for  the  truth  of  (2.7).  Before  we  give  the 
statement  of  the  theorem  to  be  proved,  we  give  the  following  equivalent  form  of  (2.9), 
which  follows  from  Lemma  2  of  the  present  paper: 


(i)  I  log-     |dg(r)|   <  oo,      0  <  c  <  1 

t 


(2.1 
Precisely,  we  prove  the  following: 


(ii)  g(0+)      =      0 


^  Theorem.  Let  8  >  0  and  let  (2.10)  (i)  hold.   Then  in  order  that  (2.7)  should  hold,  it  is 

necessary  and  sufficient  that  (2.10)  (ii)  must  hold.  Further,  the  condition  (2.10)  (i)  is  best 
possible  in  the  sense  that  it  cannot  be  replaced  by 

7T 

j  log"  *     |dg(r)|   <  oo     (0  <  r,  <  1).  (2.11) 

0 


206  Prem  Chandra 

3.  Estimates 

To  prove  the  theorem,  we  shall  require  the  following  estimates  for  3  >  0  but  proved  fo 
real  5:  uniformly  in  0  <  t  <  c, 


Hn(t)      = 
Hn(t)      = 


1 


'  nt 


,-2. -2 


(3.1 
(3.2 
(3.2 


Proof  of  (3.1).  We  have 


sin  nt 

- 

nt 


f  sin  nu  „  /"  sin  n 

4-      /  -  d*(u)      -       / 

J       nu  J       nu 


Now,  since  fc(w)  is  monotonic  increasing  therefore 


r  r 

/sin  ww  f 
db(u)  <       I  db(u)      = 
^w                           J 


b(t) 


and 


c  n    L 

C  sin  nw  f 

/  db(u)  <       /  db(u 

J      nu  J 


)      + 


sn 


nu 


-db(u) 


d 
— 


Also  w     —  Z?(w)  decreases  therefore,  we  have,  by  the  second  mean  value  theorem 
dw 


r  sinnw  f  _j  d         1  ^    sin  nw  _,       /  _j 

/   -  db(u)     =      w   ^K")  /      -  dw     (n   L 

J       nu  I       du        lu^-iJn-i      n 


_j 
L  <  0  <  c 


Collecting  the  results,  we  get  (3.1). 


Proof  of  (3.2).  Since  u~l~  -b(u)  decreases,  therefore,   by  the  second  mean  val 


therorem 


du 


c  c' 

/sin  nu  _i  d  /* 
db(u)  =     (nt)   l—b(t)  /  sin  JIM dw         (t  <  cf 
nu                                  dt        J 


Using  this  estimate  in  the  definition  Hn(t),  we  get  (3.2). 


Multipliers  for  Fourier  series  207 

Proof 'of '(3.3).  We  have 

n 

sn (0  =  imaginary  part  of     ]T      V* (w)  exp(z Jkr), 

£=0 

where 


k=0 

n+1 

«-"  E 

m=l 

n+1 

e~" 

m=0 


(1+^)"  (1 


+ 

where/?  cos  9  =  q  +  cost,  RsinO  =  sin  rand 
sin<9 


=  tan 


)• 

Hence  imaginary  part  of 

/  ,  Vk  (n^  e^pO'^0  =  ( )        sin[(n  +  1)^  —  t]  +  (  — - —  )        sin/, 

where 


Hence 

I*.  (01     <     /(O     + 
Fhis  completes  the  proof. 

f.  Lemmas 

iVe  require  the  following  lemmas  for  the  proof  of  the  theorem: 


208 


Prem  Chandra 


Lemma  1.  For®  <  8  <  1, 

(2.6)  =»  (2.9)  (4.1 

but  its  converse  is  not  true  in  general. 

Proof.  It  has  been  observed  (Chandra  [5];  p.  19)  that  (2.6)  with  8  =  1  holds  if  and  only  i 
(i)     P(0  e  B  V(0,  c),    (ii)     t~l  Pt  6  L(0,  c),  (4.2 

which  is  stronger  than  (2.9)  with  8  =  1. 

We  now  consider  the  case  0  <  5  <  1.  In  this  case,  we  observe  that 


(2.6) 


0(0 


fiV(O.c) 
e      1(0,  c)     (see  (4.2)  (ii)) 

r_ 
*-& 


log"6-      e 


L(0,c). 
Hence  (2.9)  (ii)  holds.  Now  for  the  truth  of  (2.9)  (i),  we  write 


t 
h(t)  =  0(0  log1"5  -     and    hi(t)  =  -  /  M«0  dw. 


Then  /ii(r)  6  B  V(0,  c),  where 


Hence 


from  which  one  gets 
P(0     Ic 


01  (M)     log~d-dw. 


Observe  that 


7  / 
t  J 


BV(0,c)      =>      0i  (0     log"5-  €  BV(0,c) 


Multipliers  for  Fourier  series 
Hence  using  these  results  in  (3.3),  we  get 


209 


1"* 


P(0      log 


To  prove  that  converse  is  not  true  in  general,  let  /  be  even  function  and  x  =  0.  Then 
=  f(t)  in  [0,  jr].  We  define 


log- 


0 


in  (0,  c) 
elsewhere. 


Then  (2.6)  does  not  hold. 

On  the  other  hand,  since  0(0  e  B  V(0,  c),  therefore 


and  hence 


/ 


=  -(1  -«)- log  - 

2  t  J  u 


i/^T 


(a-3)/2 


r 


<  oo, 


d« 


r 

l  log1'5  (-)  / 
\t  /  J 


(4.4) 


which  proves  (2.9)  (ii).  Also  from  (4.4) 


P(0  log1"*      =     (1-  S)t~l  log1'5 


(1  - 

t  /        2, 


Now  it  may  be  observed  that  each  of  the  term  on  the  right  above  is  of  bounded  variation  on 
(0,  c)  and  hence 

.  %   P(t)logl~s  (-\   €  £V(0,  c), 
which  proves  (2.9)  (i). 


210  Prem  Chandra 

This  completes  the  proof  of  the  lemma. 

k 
Lemma  2  [11].  If  ij  >  0,  then  necessary  and  sufficient  conditions  that  (i)  h(t)  log  - 

£  V(0,  ?])  and  (ii)  t~lh(t)  e  L(0,  r?)  a 


*7 

//&\ 
log  I  -  j  \ 


log  I  -     \dh(t)\   <  oo. 
o 


Lemma  3  [15].  Let,  for  0  <  c  <  TT, 

7T 

2    / 

an  =  —  I  cj)(t)cosnt  dr. 
TT  J 

c 

Eoo 
n=1««4    €    |E,  0|  (4  >0). 


iw  w  really  proved  for  0  <  <5  <  1  but  the  same  arguments  hold  for  8  >  0. 
Lemma  4.  L*tf  0  <  0  <  JT  a«J  5  >  0.  T/zew  uniformly  in  0  <  f  <  j8 

ir)  =  O 

f  5  =  1  is  dealt  with  in  Lemma  2  of  Chandra  [4].   The  general  case  may  b 
obtained  similarly. 

Lemma  5.  For  0  <  c  <  JT  and  for  all  real  j$ 
c 

2    f  sinnw       *  k  B 

—  I  log^  —  du  ^  log^  n . 

TT  J       u  u 

0 


The  case  fi  =  1  wif&  c  =  TT  wa^  <iea/r  wfrA  ^3;  Mohanty  and  Ray  [13]  and  for  all  real  t 
with  c  =  n,  references  may  be  made  to  Ray  [14]  or  Chandra  [6].  Since  the  same  argumem 
hold  if  we  replace  n  bye  in  Ray  [14]  or  Chandra  [6],  therefore  one  can  get  the  above  resu 
from  either  Ray  [14]  or  Chandra  [6]. 

Lemma  6.  Uniformly  in  0  <  f  <  JT, 


m=l 

n  +-o{dnyi(o}  +  o 


Proo/.  Let  N  denote  the  integral  part  of  —  -  -  for  n>2q.  Then  we  first  observe  th 

1  +  q 


(n)  increases  monotonically  with  m  <  N  and  decreases  with  m  >  N.  And,  by  Abel 
transformation 


Multipliers  for  Fourier  series 


211 


n~\ 
v— \ 


m=l 

N 


m=l 


n-l 


V/(n)sinJk/ 


n=N+l 


=         ^Sm(t)Mm+dN+iSn(t)  - 
m=l 


n)  sin  fa 


m=N+l  k=m+\ 


-   E 

owever,  by  Abel's  lemma 


.+E2+E3' 


.(01     <     V*(n)       max 

l<m'<m"<m 


sin/:? 


(for  m  <  N) 


ence 


(4.5) 


m=l 


m=l 


=     0(t~l)Adn, 
ice  m2Adm  is  increasing  and 


(4.6) 


m=l 

id  by  (3.3) 


lally,  once  again  by  applying  Abel's  lemma  in  the  inner  sum  of  V  ,  we  get 


, 


n-l 


m=N+\ 


212  Prem  Chandra 


=     0(rl)Mn,  (4.8) 

Combining  (3.5)  through  (4.8),  we  get  the  required  result. 

Lemma  7.  There  exists  an  f  e  L^  for  which  (2.10)  (i)  and  (2.11)  hold  but  the  series 
(2.8)  at  x  =  0  diverges  properly  for  every  real  8  and  hence  not  summable  by  any  regular 
summability  method. 

Proof.  Let  /  be  even  and  let  x  -  0.  Then  0(r)  =  f(t).  Define  /  by  periodicity.  We  first 
consider  the  case  5  =  0  for  which  we  define 

I/k\ 
lOg  log  UJ,       Q<t<7T  (49) 

0,  t  =  0 

where  k  >  Tte2.  Then 


*®'i 

which  is  of  bounded  variation  and  g(0+)  =  0.  Hence 


^ 

/"rMog"-2*-^, 


o  o 

which  converges  whenever  0  <  rj  <  I.  Thisprovesthat(2.10)(i)and(2.11)hold.  However 

7t 


2  r 

=     —  I  log  log  ~  cos  nt  dt 

ft  J  t 

o 

2    ?  sin  nr       ^  fc 
=    -  /  -  log  !  -  dr 
7i  J       nt  t 


0 
1 


«  log  72  ' 
oo 
by  using  Lemma  5.  Thus  ^^  A"  (JC)  diverges  properly  and  hence  it  cannot  be  summable  by 


any  absolutely  regular  summability  method  and,  a  fortiori,  (2.8)  with  8  =  0  is  not  \E,  q  \ 
(q  >  0)  summable. 
In  the  case  when  8  is  non-zero  real  number,  we  define 


.  (4.10) 

0,         t=Q 


Multipliers  for  Fourier  series 
Then  since  (j)(t)  =  f(t),  we  have 


P(t) 


r 

=     8~  f 
t  J 


-     du 
u 


213 


and  hence 


=     P(t)log~ 


+ 


8(8  -1) 


/•*"(i 

J  \u 


which  shows  that  g(0+)  =  0  and 


,   2 

r  log' 


+ 


and  for  all  real . 

t 


-  }  du  <  Mt  log6 


where  M  is  a  positive  constant  not  necessarily  the  same  at  each  occurrence  and  possibly 
depending  upon  8.  Therefore 


which  converges  for  0  <  rj  <  1.  This  proves  that  (2.10)  (i)  and  (2.11)  hold  for  all  real 
S  ^  0.  But  for  the  function  defined  by  (3.10) 


n 

O       /*  /  7  \ 

=     -  /  logM  -  )  cos  nt  dt 

K  J  \t  J 


o 

j*-.. 


214 


Prem  Chandra 


by  Lemma  5  and  hence, 
An(x) 


\ogs(n  +  1)       n  log(n  +  l)' 

This  shows  that  for  every  real  5^0,  (2.8)  is  not  \E,  q\  (q  >  0)  summable  since 
1 


E 


•  =  oo. 


'  n  log(n  +  1) 
This  completes  the  proof  of  the  lemma. 

5.  Proof  of  the  theorem 

In  view  of  the  inclusion:  |£,  q\  c  |£,  qf\  (q!  >  q  >  -1)  (see  Chandra  [2];  Corollary  2] 
we  assume  0  <  q  <  I  for  the  proof  of  the  theorem,  without  any  loss  of  generality. 
Let  (2.10)  (i)  hold.  Then  proceeding  as  in  Chandra  ([3],  p.  388-9),  we  have  for  n  >  1 


n 

--[ 

K  J 


sin  nc 


/sin  nt 


dr 


(5.1 


and  integrating  by  parts,  we  get 


n,     3   /sin  nt 
tP(t)-    —  —  1  dr 


at  \    nt 


•/ 


3   /sin  nt 


dt\    nt 


df 


and  f  or  0  <  t  <  c 

c 


sm 


nu 


du  = 


sin  nc 


nc 


sin 


nt 


7- 


sin  nu 


(5.: 


Using  (5.3)  in  (5.2),  we  get 

f    n,     3   /sinwA 

/  tP(t)—    dr 

J  Bt\ntJ 


c                            ~| 

c 

,  f  ,  sin  nc       f  sin  nu 

h(r\                        1                   r\h(u} 

+/*« 

sin  nc 

nc         J      nu 

) 
nc 

L                       o                       J 

0 

c                                   c 
crffU  ^    1             HU  Ahd,\          1     TJ   ( 

+\  /-i«^*\ 

c 
-    /  Hn(f) 

J 


nu 


(5. 


Multipliers  for  Fourier  series 
And  using  (5.4)  in  (5.1),  we  get 


IT  n 


sin  nu 


=     a/i  +  pn  -  yn  -  5n,  say. 
Since  AQ  =  iflo  =  0,  therefore 


n(x)dn  e  \E,q\     (q  >  0) 


if  and  only  if 


<  00. 


However,  it  follows  from  Lemma  3  that 


n  €\E,q\      (q  >  0) 


and  since 


and 


n  =  —<j>(c)sinnc 1 

TT  L/i  +  1       /i(n  +  l)J 


E 


sm/zc 


<  oo. 


Therefore,  in  view  of  absolute  regularity  of  |£,  q\  (q  >  0)  method, 

00 

^Pndn€\E,q\     (q>0) 

n=l 
if 

oo 

i  x-^  /  n  \    „   ...      sin  nc 


m=l 


,    , 

m  +  1 


oo       .. 
""^n  +  1 


n=l 

which  holds  by  Lemma  4.  Now 

00 


]P  V^(m)^msinwc 


<  00, 


n=l 


215 


(5.5) 


(5.6) 


(5.7) 


(5-8) 


(5.9) 


216  Prem  Chandra 

if  and  only  if 


*     (9 


m 


Clearly 


<2<- 


and  since  by  (2.10)  (i), 


/  fc.ii 


dg(r)|  <  oo 


therefore  for  the  proof  of  (5.9)  it  is  suffiecient  to  prove  that 

oo  .  n 


m=l 


uniformly  in  0  <  t  <  c. 

For  T  =  [k/t],  the  integral  part  ofk/t,  we  write 


n<T      n>T 


By  (3.1),  we  get 


=      0(1)4(0  I>  E  V-(")  +  0(D  E 
n=l        m=l  n=l 

r  r 


uniformly  in  0  <  t  <  c,  since 


^     ^771  =  1 


<  1.  And  by  (3.2) 


Y^    _ 


2Lr 


- 


+  0 


<  oo. 


f-md, 


sinm£ 


(5.10) 


(5.11) 


m(m  H-  1) 


Multipliers  for  Fourier  series 


217 


where 


Now,  by  using  repeatedly  the  relation: 


m  -f-  1 


+"('-'  *  7)  EfrTW^W 


log*  -  +  O  It"1  log*  -        (r),     say, 


«\   ,_m      <4, 


m(m  +  1) 


m(m  H-  1) 


where  r  and  s  are  integers  such  that  r  >  s  >  0,  we  get 


n  +  l^\m+\r         m 

tn=l 


m-l 


m 


m+2 


(n 


n2(n  +  3)  ^ 

m=i 

however,  the  function  (jc  +  2)  log"5  x  increases  with  x  >  exp(3S),  therefore 

m=l 

m+3 


n=T 

oo 


(5.13) 


218 


Prem  Chandra 


n=T 


uniformly  in  0  <  t  <  c.  And,  by  Lemma  6, 

oo         , 


n=T  m=l 

oo      A    ,  oo 

^7  +  0U> 


=     0(1)  log 
uniformly  in  0  <  t  <  c,  since 


i-s 


and 


(5.14) 


(5.15) 


Combining  (5.11)  through  (5.15)  we  get  (5.10).   Also  in  view  of  (5.5)  through  (5.9), 
^ILi  An^n  €  \E>4\(4  >  °)  if  and  only  if 

oo 

y»dn€|£^|     («>0),  (5.16) 


where 


and,  by  Lemma  5, 


'-! 


HU 


2    f  sinnu 

-  /  db(u) 

7t  J      nu 


»  , 
n  +  1)  -  5 log^^n  +  1)] 


Multipliers  for  Fourier  series  219 

and  hence 

c 

sinnu  1 

db(u)  ^  -. 

nu  n 

0 

9 

Thus  in  order  that  (5.16)  should  hold  it  is  necessary  and  sufficient  that 

Y" —  €  i^i  (?  >  °) 

A—/      n 

tt=l 

for  which  it  is  necessary  and  sufficient  that  (2. 10)(ii)  must  hold,  since  V]       1/n  diverges 
strictly. 

The  fact  that  the  condition  (2.10)(i)  cannot  be  replaced  by  (2.1 1)  follows  by  Lemma  7. 

This  proves  the  theorem  completely. 

Acknowledgement 

2  The  author  is  thankful  to  the  referee  for  reading  the  manuscript  carefully  and  giving  his 

valuable  suggestions  to  improve  the  presentation  of  the  paper. 

References 

[1]  Chandra  P,  Absolute  summability  factors  for  Fourier  series,  Rend.  Accad.  Nazionale  dei  XL 

24-25  (1974)  3-23 

[2]  Chandra  P,  On  some  summability  methods,  Boll  Un.  Mat.  Ital.  (4)  12(3)  (1975)  21 1-224 
[3]  Chandra  P,  Absolute  summability  by  (£,  g)-means,  Riv.  Mat.  Univ.  Parma  (4)  4  (1978)  385- 

393 
[4]  Chandra  P,  On  the  absolute  Euler  summability  factors  for  Fourier  series  and  its  conjugate 

series,  Indian  J.  Pure  Appl.  Math.  9  (1978)  1004-1018 

[5]  Chandra  P,  On  a  class  of  functions  of  bounded  variation,  Jnandbha  8  (1978)  17-24 
[6]  Chandra  P,  Absolute  Euler  summability  of  allied  series  of  the  Fourier  series,  Indian  J.  Pure 

Appl  Math.  11  (1980)  215-229 
)  [7]  Chandra  P  and  Dikshit  G  D,  On  the  \B\  and  \E,q\  summability  of  a  Fourier  series,  its 

conjugate  series  and  their  derived  series,  Indian  J.  Pure  Appl.  Math.  12  (1981)  1350-1360 
[8]  Dikshit  G  D,  Absolute  Euler  summability  of  Fourier  series  J.  Math.  Anal  Appl  220  (1998) 

268-282 

[9]  Hardy  G  H,  Divergent  Series  (Oxford)  (1963) 
[10]  Kwee  B,  The  absolute  Euler  summability  of  Fourier  series,  J.  Austra.  Math.  Soc.  13  (1972) 

129-140 
[11]  Mohanty  R,  On  the  absolute  Riesz  summability  of  Fourier  series  and  allied  series,  Proc. 

London  Math.  Soc.  52  (1951)  295-320 
[12]  Mohanty  R  and  Mohapatra  S,  On  the  |£,  q\  summability  of  Fourier  series  and  allied  series, 

/.  Indian  Math.  Soc.  32  (1968)  131-139 

[13]  Mohanty  R  and  Ray  B  K,  On  the  convergence  factors  of  a  Fourier  series  and  a  differentiated 
;  Fourier  series,  Proc.  Cambridge  Philos.  Soc.  65  (1969)  75-85 

I  [14]  Ray  B  K,  On  the  absolute  summability  factors  of  some  series  related  to  a  Fourier  series,  Proc. 

)  Cambridge  Philos.  Soc.  67  (1970)  29-^45 

[15]  Ray  B  K  and  Sahoo  A  K,  Application  of  the  absolute  Euler  method  to  some  series  related  to 
Fourier  series  and  its  conjugate  series,  Proc.  Indian  Acad.  Sci.  (Math.  Sci.)  106  (1996)  13-38 


Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  1 1 1,  No.  2,  May  2001,  pp.  221-227. 
Printed  in  India 


On  a  Tauberian  theorem  of  Hardy  and  Littlewood 


TPATI 

10,  Bank  Road,  Allahabad  21  1002,  India 

Institute  of  Mathematics  and  Applications,  Bhubaneswar 

MS  received  17  April  2000 

Abstract.     In  this  paper,  we  give  a  simple  alternative  proof  of  a  Tauberian  theorem 
of  Hardy  and  Littlewood  (Theorem  E  stated  below,  [3]). 

Keywords.    Abel's  theorem;    Tauberian  theorem;    Hardy-Littlewood  Tauberian 
theorem;  divergent  series. 

1.  Introduction 

Let  Y1T=Q  an  be  an  infinite  series  of  real  terms.  Let 

0  <  XQ  <  A.I  <,...,  Xn  ->  00 
and  let  £  ane~knX  be  convergent  for  all  jc  >  0.  If 


as  x  —  >  0,  then  we  say  that  ]P  an  is  summable  (A,  Xn)  to  s.  When  Xn  =  n,  the  method 
(A,  Xn)  reduces  to  the  classical  method  summability  (A),  named  after  Abel. 

It  is  a  famous  result  due  to  Abel  that  if  £  an  is  convergent  to  s,  then  £  an  is  summable 
(A)  to  s.  That  the  converse  is  not  necessarily  true  is  evident  from  the  example  of  the 
series 

1-1  +  1-1--- 

which  is  summable  (A)  to  ^  ,  but  not  convergent.  The  question  naturally  arises  as  to  whether 
one  can  determine  a  suitable  restriction  or  restrictions  on  the  general  term  an  so  that  ^  an 
will  be  convergent  to  s  whenever  it  is  summable  (A).  The  first  answer  to  this  question  was 
given  by  Tauber  in  1897  in  the  form  of  the  following  theorem. 

Theorem  A  [7],  If^an  is  summable  (A)  to  s  andnan  =  0(1),  then  ^an  is  convergent 
to  s. 

A  generalization  of  Theorem  A  to  the  set-up  of  summability  (A,  Xn)  was  proved  by 
Landau  [4]. 

Another  significant  generalization  of  Theorem  A  was  obtained  by  Littlewood  in  1910  in 
the  form  of 

Theorem  B.  If^an  is  summable  (A)  to  s,  and  nan  =  0(1),  then  ^an  is  convergent 
to  s. 


222  TPati 

In  fact  Littlewood  proved  the  following  more  general  theorem. 
Theorem  C  [5].  If^l^n  is  a  series  of  positive  terms  such  that,  as  n  —  >  oo, 

^n  —  Ml  +  M2  H  -----  h  ^  ->  00,     tJLn/^n  ~>  0, 

Y^  an&~~XnX  ->  s  as  x  -*•  0, 


^  an  is  convergent  to  s. 

Littlewood  had  stated  that  Theorem  C  is  true  even  without  the  restriction:  /zn/A,tt  —  >- 
This  result  is  stated  below  as  Theorem  C*.  It  was  proved  in  1928  by  Ananda-Rau  [1]. 
simple  alternative  proof  was  supplied  by  Bosanquet  (see  Hardy  [2]). 


(1. 


Theorem  C*.  If£  [Ln  is  a  series  of  positive  terms  such  thatXn  =  A&i-h/^H  -----  \~f^n 
as  n  ->  oo,  5Z  anQ"^nX  -*  s  as  x  —  >  0  and 


then  ^2  &n  is  convergent  to  s. 

Littlewood  also  conjectured  [5]  that  the  following  theorem  is  true. 
Theorem  D.  Ifk\  >  0,  Xn+i/A,n  >  0  >  I  (n  =  1,  2, ...),  and 

CO 

n=l 

then  J^  an  converges  to  s. 

The  truth  of  this  conjecture  was  proved  by  Hardy  and  Littlewood  [3] l .  Theorems  of  t 
kind  are  called  'high  indices'  theorems,  as  distinguished  from  'Tauberian'  theorems,  sii 
in  such  theorems  no  restriction  is  needed  to  be  imposed  upon  the  general  term  an  of 
series  in  question,  excepting,  of  course,  that  Y^an^~XnX  is  convergent  for  every  x  > 
Such  a  theorem  shows  that  the  method  (A,  A.n)  with  the  type  of  A.n  involved  does  not  s 
any  series  which  is  not  convergent,  and  therefore  shows  the  'ineffectiveness'  of  the  metl 

Hardy  and  Littlewood  first  established  Theorem  D  in  the  special  case  in  which 
an  =  0(1) 

and  then,  by  further  analysis,  derived  Theorem  D  itself.  This  is  an  instance  of  a  Taubei 
theorem  leading  to  a  high  indices  theorem.  Thus  Hardy  and  Littlewood  first  establisl 
the  following  Tauberian  theorem. 

Theorem  E.  Ifk\  >  0,  An+i/Xn  >  0  >  I  (n  =  1,  2, . . .), 

oo 

an€~~^nX  ->  s  as  x  -»  0, 


1  For  a  proof  of  Theorem  D  due  to  A  E  Ingham,  see  [2],  proof  of  Theorem  114,  where  too,  the  result  has 
obtained  via  a  Tauberian  theorem. 


Tauberian  theorem  of  Hardy  and  Littlewood  223 

and  an  =  (9(1),  then  ]T  an  is  convergent  to  s. 

It  should  be  observed  that  Theorem  E  is  included  in  the  theorem  of  Ananda-Rau,  in  which 
no  extra  restriction  is  imposed  on  Xn,  in  view  of  the  fact  that  whenever  Xn+i  /Xn  >  0  >  1, 

and  an  =  (9(1),  an  =  O  \n~^~l\  On  me  omer  hand,  under  the  hypotheses  of  Theorem 
C*,  (1.1)  implies:  sn  =  (9(1),  and  hence  an  =  (9(1)  (see  Lemma  2  in  the  sequel). 

The  object  of  the  present  paper  is  to  give  an  alternative  proof  of  Theorem  E  which  is 
quite  straightforward,  not  requiring  Lemmas  1  and  2  of  Hardy  and  Littlewood  [3]. 

2.  Lemmas 

We  shall  need  the  following  lemmas. 

Lemma  I  [5].  If,  as  y  —  >  0,  i/s(y)  —  >•  s,  and  for  every  positive  integer  r, 


then  for  every  positive  integer  r,  yrty(r\y)  =  o(l). 

Lemma  2  [4].2  //  0  <  k\  <  X2  <,...,  Xn  ->  oo,  as  n  ->  oo,  /(jc)  = 
O(l)asx  ->  0, 


then 

sn=ai+a2  +  ----  h  an  =  (9(1). 

Lemma  3  [3].  ./jfA.1 


>  then,  for  r  =  1,2, 


3.  Proof  of  Theorem  E 

We  may  assume,  without  loss  of  generality,  that  s  =  0.  Thus  /(jc)  =  o(l)  as  x  —  >  0.  Also, 
since  aw  =  0(1),  for  r  =  1,  2,  ..., 


n=l 


2  As  remarked  by  Ananda-Rau  in  [1],  the  argument  in  Landau  [4],  pp.  13-14,  has  only  to  be  slightly  modified  to 
yield  the  result  of  Lemma  2. 


224  TPati 

by  Lemma  3.  Hence,  by  Lemma  1, 

Since 


11=1 
we  have 


*du 

'At       I 

n=l       JA«  n=l 

=     Vr-rVr_i,    say. 


Hence,  by  Lemma  1, 

Vr     =    rVr_i+o(l) 


=     r(r  -  l)(r  -  2)  Vr_3  +  o(r(r  -  1))  +  o(r) 

=     r\f(x)  +  o(r(r  -  1)  .  .  .  2)  +  •  •  •  +  o(l), 
so  that3 


(3.1) 
This  can  be  explicitly  written  as 

(3.2) 


as  x  -*  0.  If  $,!  does  not  converge  to  zero,  there  exists  a  positive  constant  h  such  that 
\sn\>  h  for  an  infinite  number  of  values  of  n.  Let  m  be  any  one  of  these  values.  We  shall 
show  that,  when  r  exceeds  a  sufficiently  large  positive  integer  ro, 


where  8  is  a  positive  constant.  This  will  contradict  (3.2),  and  hence  we  will  conclude  that 
£  an  converges  to  zero,  which  is  required  to  be  proved. 


3  In  Hardy  and  Littlewood  [3],  (2.41)  should  be  replaced  by  our  (3.1)  -y  =  0(1);  line  4  from  the  top  on  p.  225 

00  00  y 

should  be  replaced  by:  r\^^snwn  =  Vr  so  that  /_^^«^«  =  ~ -  =  o(l).  For  similar  alterations  needed  in  the 

n=0  n=0 

papers  [1],  [5]  and  [8],  see  Pati  [6]. 


Tauberian  theorem  of  Hardy  and  Littlewood  225 

Now,  by  the  hypotheses  of  Theorem  E  and  Lemma  2,  sn  =  (9(1)  and  hence 


_i  J^n 


where  K  is  a  positive  constant.  We  choose 

2r 


Then,  for  fixed  r,  x  -+  0  iff  m  -*  oo.  Since  ([5],  p.  440) 

lim  *H 
1 


m  ,-     .      v 

u  =  r- — —  =  r(l  +  ??), 

~r  >^m 


Xn+l  / 

^e-^dr  <  / 
n  h 


(3.3) 


fre~-"dr  >  /i.  (3.4) 

We  now  use  the  transformation  u  —  xt,  so  that,  for  t  =  A,m+i, 


where 

J"""~-^-  (3-5) 


Thus  the  second  term  in  (3.3)  gives 

°°  -    wv 

(3.6) 


The  third  term  in  (3.3)  gives 

iim^^o  *r+1  Y!  /         ^e^dr  <  /  wre~"dw,  (3.7) 


where  77  is  as  defined  in  (3.5). 
Combining  (3.4),  (3.6)  and  (3.7)  we  have 

TT-  ......  ,      _M.  .  ,      _,..          ,  ^^g) 


r°°  1 

urc~udu    .      (3.8 

r(l+77)  J 


We  show  below  that 

Ii=  f  ure~udu  <  Kirr6~r  (3.9) 

Jo 


226  T  Pati 

and 

»  CO 

h 


/CO 
(1+7?) 

where  the  K  in  each  inequality  denotes  a  positive  constant,  independent  of  r. 
Proof  of '(3.9).  We  have 

°  Jo 

Hence 

/  ( 1 )  ure~udu  =  rr(l  -  7})re~r(1~~77). 

Jo  v^        / 

Now,  since  0  <  r\  <  1,  u  <  r(l  —  r])  implies: 


u  I  —  t]' 

so  that 


since 

e77  <  1  +  n  H-  rr  +  -  -  •  = 
Thus 

where 


^  A.^+1  -  km         9—1 

Proof  of  (SAG).  We  have 

/CO 
M'-VdK. 
(1+7?) 

Hence 


since 


w  >  r(l  -F  ?y)    implies:    1 > 

u 


we  have 

n 


Tauberian  theorem  of  Hardy  and  Littlewood  227 

so  that 

72  <  AVV", 
where 

ft_i±!_*-iL_  __*_«»<„„. 

Xm        i  -  J*L_       0  -  1 


Hence,  from  (3.8),  (3.9)  and  (3.10),  we  have 


Since  by  Stirling's  theorem, 
1 


taking  r  >  ro,  a  sufficiently  large  positive  integer,  we  have 


which  contradicts  (3.2).  Hence  our  assumption  that  {$„}  does  not  converge  to  0  is  false. 
This  completes  the  proof  of  Theorem  E. 

References 

[1]  Ananda-Rau  K,  On  the  converse  of  Abel's  theorem,  /.  London  Math.  Soc.  3  (1928)  200-205 

[2]  Hardy  G  H,  Divergent  Series  (Oxford)  (1949) 

[3]  Hardy  G  H  and  Littlewood  J  E,  A  further  note  on  the  converse  of  Abel's  theorem,  Proc.  London 

Math.  Soc.  25(2)  (1926)  219-236 
[4]  Landau  E,  Uber  die  Konvergenz  einiger  Klassen  von  unendlichen  Reihen  am  Rande  des  Kon- 

vergenzgebietes,  Monatshefte  fur  Math,  und  Phys.  18  (1907)  8-28 
[5]  Littlewood  J  E,  The  converse  of  Abel's  theorem  on  power  series,  Proc.  London  Math.  Soc.  9(2) 

(1910)434-448 
[6]  Pati  T,  Remarks  on  some  Tauberian  theorems  of  Littlewood,   Hardy  and  Littlewood, 

Vijayaraghavan  and  Ananda-Rau  (forthcoming  in  /.  Nat.  Acad.  Math.,  Gorakhpur) 
[7]  Tauber  A,  Ein  satz  der  Theorie  der  unendlichen  Reihen,  Monatshefte  fur  Math,  und  Phys.  8 

(1897)  273-277 
[8]  Vijayaraghavan  T,  A  Tauberian  theorem,  J.  London  Math.  Soc.  2  (1926)  113-120 


Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  1 1 1,  No.  2,  May  2001,  pp.  229-239. 
Printed  in  India 


Proximinal  subspaces  of  finite  codimeesioe 
in  direct  sum  spaces 


V  INDUMATHI 

Department  of  Mathematics,  Pondicherry  University,  Kalpet,  Pondicherry  605  014, 

India 

E-mail:  indumath@md4.vsnl.net.in 

MS  received  6  April  2000;  revised  21  August  2000 

Abstract.    We  give  a  necessary  and  sufficient  condition  for  proximinality  of  a  closed 
subspace  of  finite  codimension  in  co-direct  sum  of  Banach  spaces. 

Keywords.    Proximinality  and  strong  proximinality. 

0.  Notation  and  preliminaries 

Let  X  be  a  normed  linear  space  and  A  be  a  closed  subset  of  X.  We  say  A  is  proximinal  in 
X  if  for  each  x  e  X  there  exists  an  element  a  €  A  such  that  ||;c  —  a  ||  =  d(x,  A). 

We  say  A  is  strongly  proximinal  in  X  if  A  is  proximinal  in  X  and  given  €  >  0,  there 
exists  8  >  0  such  that 

a  e  A,  ||*  -a  \\   <  d(x,  A)  +  6  =>  d(a,  PA  00)  <  €, 

where  PA(X)  =  {a  e  A  :  \\x  —  a\  =  d(x,  A)}. 

Proximinal  subspaces  of  finite  codimension  have  been  studied  by  various  authors  (see 
[1-4,  7-10]).  In  this  paper  we  obtain  a  necessary  and  sufficient  condition  for  proximinality 
of  subspaces  of  finite  codimension  in  CQ  -direct  sum  of  Banach  spaces  in  terms  of  the  proxim- 
inality of  the  corresponding  subspaces  of  finite  codimension  of  the  coordinate  spaces.  We 
also  give  an  example  to  show  that  similar  result  does  not  hold  in  l\  -direct  sum  of  Banach 
spaces. 

Let  X  be  a  real  normed  linear  space  and  X*  its  dual.  The  closed  unit  ball  of  X  is  denoted 
by  BX  and  the  unit  sphere  by  Sx  -  Let  Y  be  a  closed,  linear  subspace  of  codimension  n  in 
X.  For  a  set  /i,  /2,  .  .  .  ,  /H  of  linear  functionals  in  the  annihilator  space  Y^,  we  give  the 
following  definitions  from  [4].  We  have  modified  the  notation  used  in  [4]. 


{*  €  Bx  :fi(x)  =   \\fi\\] 
Jx(fl,  /2,  •  .  -  ,  ft)  =  {Jx(fl,  -  -  •  ,  fi-l)  '  fiW  =  sup,,  eJx(fi^..,fi-i) 


for/  =2,3,  ...,/i. 
Similarly  we  set 


230  V  Indumathi 

for  i  =  2,  3  .  .  .  n.  Since  71  is  finite  dimensional,  the  above  n  sets  are  nonempty.  We  als 
set 


||/i||   =  N(fi) 

M(/i,  ...,./})  =  sup{/K*)  :  *  €  Jx(fi,  .  .  .  ,  /i-i)} 
and 

#  (/i,  ...,/i)  =  max{<J>(/i)  :  d>  €  %±r(/i,  .,  .  ,  ^-i)}. 
We  also  need  the  following  Theorem  from  [4]. 

Theorem  A.  Let  X  be  a  normed  linear  space  and  Y  be  a  subspace  of  codimension  n  in  J 
Then  Y  is  proximinal  in  X  if  and  only  if  for  every  basis  f\,...,fnofYLwe  have 


--,fi)  ±  ®forl<i<n. 
2. 


We  shall  first  show  that  condition  2  of  the  above  theorem  can  be  reformulated  wi 
conditions  only  involving  the  normed  linear  space  X.  This  is  easily  done  using  the  weal 
density  of  BX  in  BX**  and  in  a  manner  similar  to  that  of  Vlasov  [10].  For  this  purpose  \ 
make  the  following  definitions  for  €  >  0  and  any  finite  subset  f\  ,  .  .  .  ,  fn  of  Y  . 

=  \\fi\\, 

,€)  =  [xeBx  :  fi(x)  >   ||/i||-€}, 
N(fi,  ...,/i,€)=  sup{/i  W  :  jc  €  Jx(/i,  .  .  .  ,  //-i,  €)}, 


and 


1.  Proximinality  of  subspaces  of  finite  codimension 

We  begin  with  the  following  proposition. 

PROPOSITION  1.1 

Let  X  be  a  normed  linear  space  and  Y  be  a  closed  subspace  of  codimension  n. 
every  finite  subset  /i,  ...,/«  ofY^  we  have 


Proof.  By  induction.  The  case  i  =  I  is  trivial.  Assume 

..,/*)  =  N(fi,...,fk)  for  l<t<i-l. 


Select  any  O  6  %±)*(/i,  /2  -..//)•  Since  B^  is  weak*  dense  in  BX**,  there  exists  a 
(jca)  in  BX  that  weak*  converges  to  <1>.  In  particular, 


lim  /jk(jca)  ==  O(/fc)  for  1  <  k  <  n. 


Proximinality  in  direct  sum  spaces  23  1 

Thus,  given  €  >  0,  3  ao  such  that 

fk(xa)   >  #(/i,...,  /*)-*Va  >  ao  and  1   <k  <n. 
This  together  with  the  induction  hypothesis  implies  that 

(xa)  €   /*(/!,  ••  .,.#-1.0  V  a  >  a0. 
and  so 


To  prove  the  other  inequality,  for  each  positive  integer  n,  select  an  element  (xn)  in 
Jx(/i,...,/i,).  Then 


Let  T/OZ  =  -Xfliy1"  for  each  n.  Then  T^W  is  in  B(yj.^*.   Since  F-1  is  finite  dimensional, 
w.l.o.g  we  assume  (\l/n)  converges  to  ^  in  B^^y  -  Note  for  1  <  A:  <  /, 


Again  by  induction  hypothesis  ^  e  /(yj_)*(/i,  /2,  .  .  .  ,  //-i)  and 


Hence 

I 


and  this  completes  the  induction  and  the  proof. 
The  above  Proposition,  along  with  theorem  A,  implies  as  follows: 

COROLLARY  1.2 

Let  Y  be  a  closed  subspace  of  finite  codimension  n  in  a  normed  linear  space  X.  Then  Y  is 
proximinal  in  X  if  and  only  if  far  every  basis  fi ,  . . . ,  fn  ofY1-  the  sets 

j 

EBX:  fox)  =  N(fi,...9fi)}?!aforl<j<n. 


Remark  1.3.  If  f\,  ...,/„  is  a  finite  subset  of  X*  and  /BI , . . . ,  fnk  is  a  maximal  linearly 
independent  subset  of  f\  ,...,/„  satisfying  n\  <  HI  <  ...  <  rik  then  fl/=i  (x  e  &x  '• 
MX)  =  Wi,.. .,/«)}  ?4  0  forl  <  7  <nifand  only  ifnjLi{Jt€Bx:  /„,(*)  = 
#(/»„•..,/,•,)}  5*0  for  1  <m<k. 


232  V  Indumathi 

We  now  recall  some  known  proximinality  results  that  are  needed  in  the  sequel.  For  an] 
normed  linear  space  X,  let  N  A(  X)  denote  the  set  of  norm  attaining  elements  of  X*.  Garkav 
[1]  has  characterized  proximinal  subspaces  of  finite  codimension  in  general  normed  linea 
spaces  and  the  following  is  an  easy  corollary  of  his  result. 

Lemma  B  [5].  Let  X  be  a  normed  linear  space  and  Y  be  a  closed  sub  space  offinit 
codimension  in  X.  Then  Y  is  proximinal  in  X  if  and  only  if  every  closed  subspace  Z  J2  ] 
is  proximinal  in  X. 

Now,  if  /  6  X*  and  H  is  the  kernel  of  /,  it  is  well-known  that  the  hyperplane  H  i 
proximinal  in  X  if  and  only  if  /  €  NA(X).  Thus  from  Lemma  B  we  have  the  following 

Remark  1.4.  If  Y  is  a  proximinal  subspace  of  finite  codimension  in  X,  then  Y1-  c  NA(X] 
However  F1  c  NA  is  only  a  necessary  but  not  a  sufficient  condition  for  proximinalit 
of  a  subspace  Y  of  finite  codimension.  (See  the  example  of  Phelps  in  [7],  p.  309.)  Bi 
the  behaviour  of  the  space  coCO  in  the  above  respect  is  rather  special.The  following  fa< 
is  well  known,  see  for  instance  [6]. 

Lemma  C.  Let  Y  be  a  closed  subspace  of  finite  codimension  in  co(F)  and  f\  ,...,/«  be 
basis  ofY1-.  Then  Y  is  proximinal  if  and  only  if  fi  e  NA(co(r))for  1  <  /  <  n. 

Finally  we  quote  a  characterization  of  strongly  proximinal  subspaces  of  finite  codimei 
sion  from  [3],  which  is  needed  in  the  proof  of  our  main  result. 

Theorem  B.  Let  X  be  a  normed  linear  space  and  Y  be  a  proximinal  subspace  ofcodime 
sion  n  in  X.  Then  Y  is  strongly  proximinal  in  X  if  and  only  if  the  fallowing  hold  for  eve 
basis  fi,...,fn  of  Y-L. 

Given  €  >  0  there  exists  8  >  0  such  that  for  each  i,l  <  i  <  n  and  for  each  x 

€. 


2.  Direct  sum  spaces 

We  now  consider  proximinality  in  co-direct  sum  spaces.  Let  A  be  an  index  set  and  J 
be  a  Banach  space  for  each  X  e  A.  Let  X  =  ®Co  Xx-  Then  X*  =  0/j  XJ.  Furtlr 
F  =  (A)xeA  is  in  N  A(X)  if  and  only  if  /x  =  0  but  for  finite  number  of  indices  a 
/x  €  NA(Xx)  whenever  /x  ^  0.  Also,  in  this  case 

Jx(F)  =  {(*x)eX:  ||*x|  =  1  and/x(*x)  =  ||A||VX€A}^  0.       i 
For  X  defined  as  above,  we  have  the  following  Proposition. 

PROPOSITION  2.1 


Let  Ft  6  X*  and  FI  =  (/^)  for  1  <  i  <  n.  Assume  further  that  for  each  i,  1    <  i 
H.  jfoi  =  0  but  for  finite  indices  X.  Then 


Remark.  Observe  that  the  above  sum  has  only  finite  number  of  nonzero  entries.  Also, 
condition  of  the  above  proposition  is  satisfied  if  F/  e  NA(X)  for  1  <  i  <  n. 


Proximinality  in  direct  sum  spaces  233 

Proof  of  the  Proposition.  Let 

A  =  U?=1U€A:   /U^O}.  (2) 

Then  card  A  =  /  <  oo.  Set 

a(€)  =  max  max  [N(fa,  .  .  .  ,  fa,  6)  -  N(fa,  .  .  .  ,  fa)]. 

A.6A 


For  €  >  0,  let 

$!   =  e,  e/  =  /<7(e/_i)  4-  e/_i  for  2  <i<n.  (3) 

Then  ei  <  62  <  -  .  .  <  €n.  Clearly,  a(e)  and  therefore  £/,  1  <i<n  tend  to  0  as  6  -*  0. 
Further, 


and 


x  6  /jr(Fi,  «)     •*>     (x  =  (JCA)  6 


(4) 


=     (x  =  (xx)  €  fix  :        /u(*x)   >  (/u)    -  €•    (5) 

A. 


Using  (4)  and  (5)  and  the  fact  that  /uOx)  <  N(fu.)  we  get 


€  5x  :  Ex 
-      V 


» 

and 


Inductively  assume  that  for  some  i  ,  2  <  z  <  n  we  have 

for 


0  ^x(/u,  -  -  -  ,  fi-u,  y)  c  yx(Fi,  .  .  .  ,  F/-I,  €)^Y[  Jxi(fa,  •  •  •  ,  //-u,  €f-i). 

AeA  X€A 

Now,  observing  that  the  summation  below,  over  A,  involves  only  finite  number  of  nonzero 
terms,  we  have 


<     inf€>0  sup  {^x  fafa)  :  /xx(/u,  -  -  -  ,  ^-u,  ^_i)  VX  €  A} 
=    infe>0  ^x  sup{/^(^)  :  Jx^(fa,  •  •  •  ,  /^-u,  ^-i)  VX  €  A} 
=    inf6>0  E 


,  .  -  •  ,  jfix)  for  €,-_!  ->  0  as  6  ->  0. 


234  VIndumathi 

Similarly  using  the  other  inclusion  we  conclude 


Hence 


=     {jc  =  (*x)  6  Jx(Fi,  ...,  FI-I,  e)  :  Ft(x)  >  N(Fi,  .  .  -  ,  F/)  - 


We  have 

a:  =  (*x)  e  A-  =*•  /a(^i)    < 

<    Wu,.-.,/a)  +  crfe_i)VX.  (6) 

Further 


X 

Let  j  =  fo)  e  A-  If  for  some  A.0>  /l-x0(^x0)  <  ^(/ix0,  •  -  -  ,  ^x0)  -  e,-  then  using  (6), 


=     Ex 
which  contradicts  (7)  as  orfe_i)  +  €,-_i  >  e.  Thus  *  =  (*x)  6  £»,•  implies 

/«i(*x)  >  JV(/u,  •  •  •  ,  /a)  -  «/  VX.  (8) 

Now  €,-_i  <  e,-  and  so 

JW/ix,  •  •  •  ,  //-ix,  <i-i)  c  JXl(/ix,  •  •  •  ,  /i-ix,  €,-)  V  A..  (9) 

Since  D,-  c  {x  =  (jcx)  e  f]x  /^(/ix,  .  .  -  ,  /i-ix,  e/-i),  using  (8)  and  (9)  we  conclude 
e  f]  /Xi(/u,  .  .  .  ,  /f_u,  6j)  v  A  }. 

X€A 


Proximinality  in  direct  sum  spaces  235 

But  Jx  (F\ , . . . ,  FI  ,  e)  c  Di  and  therefore 

JX(F{ , . . . ,  ft,  6)  c  {*  -  (*A)  €         /^(/u,  . .  - ,  //A,  £/)}•  (10) 


A.€A 

On  the  other  hand 
59  Jx(Fi,...,F/,€)     =     [x  = 


This  completes  the  induction  and  we  have 
/x(Fi,...,F/,O  D 


-f  v 

7)- 


for  1  <  i  <  n.  This  completes  the  proof  of  the  proposition.  We  now  prove  our  main  result. 

Theorem  2.2.  Let  Xx  be  a  normed  linear  space  for  each  X  in  an  index  set  A  and  X  be  the 
CQ-directsum  of  the  spaces  X^forX  €  A.  Let  Y  be  a  closed  subspace  of  finite  codimension 
n  in  X.  Then  Y  is  (strongly)  proximinal  in  X  if  and  only  if  the  following  two  conditions 
hold  for  every  basis  {F/  :  1  <  z  <  n}  ofYL,  where  F/  =  (fix)keA>far  1  <  i  <  n. 

1.  For  each  /,  1  <  i  <  n,  //A.  is  nonzero  only  for  finite  number  of  indices  X. 

2.  YX  =  Pi{Ker  fix  :  1  <  i  <  n}  is  (strongly)  proximinal  in  X^for  each  A  e  A. 

Proof.  Necessity.  First  we  observe  that  by  remark  1.4,  F/  €  NA(X)  for  any  basis  {F/  : 
1  <  i  <  n}  of  F-1.  This,  in  particular,  implies  condition  1  above.  Hence  FX  is  a  proper 
subspace  of  X\  only  for  finite  number  of  indices  X. 

To  prove  2,  for  each  X  such  that  7^  is  a  proper  subspace  of  XA.,  choose  any  basis 
gu,  -  .  .  ,  gnx  of  (FA)X.  If  d  =  (gix)  for  I  <  i  <  n  then  GI,  .  .  .  ,  Gn  is  a  basis  of 
F1-.  Since  F  is  proximinal  in  X,  we  can,  by  Corollary  1,  get  an  element  x  =  (xx)  €  X 
satisfying  G/(JC)  =  W(Gi,  .  .  .  ,  Gj)  for  1  <i  <n.  In  particular, 


A  A. 

We  have  ||jc||     =    |  supx  ||jcx|    <    1  and  so  the  above  inequality  implies  gufe)   = 
|gu  I  for  all  X.  Assume  inductively, 

for  1  <  fc  <  i  -  1  and  V  X. 


236  V  Indumathi 

Now  again  by  Remark  1.4,  G/  €  NA(X)  for  1  <  i  <  n.  Hence  by  Proposition  2. 
have, 


Gi  (x)  =        gixto) 

A.  A. 

Also  by  induction  hypothesis,  x^  €  A/Xgu,  .  .  .  ,  g/_u,  e)  for  every  £  >  0  and  so  we 

gttfo)  <  #(gu,  .  .  .  ,  gtt)  VA. 
This  with  (11)  implies 


and  completes  the  process  of  induction.  Hence  for  all  X,  JCA.  e  X^  satisfies 

,  -  -  -  ,  ga)  V  l<i<n. 


By  Corollary  1.2,  Y\  is  proximinal  in  X^  for  each  A.. 
If  7  is  strongly  proximinal  in  X,  then  given  6  >  0  there  exists  8  >  0  such  that  fc 


d(G,Jx(G\,...tGi))    <  €  V  1   <f  <n. 
It  is  easy  to  verify  using  (11)  and  (12)  that 


and 

Jx(Gi,...,Gi,8)  D 

for  1  <  z  <  n.  Now  using  (13)  and  (14)  we  conclude  that  for  any  A.  and  hi  in  Jx^  (gi. 
g/x,  f)  we  have 


for  1  <  z  <  w.  Hence  FX  is  strongly  proximinal  in  X^.  for  each  X. 

Sufficiency.  If  GI  ,  .  .  .  ,  Gn  is  any  basis  of  71  and  G/  =  (ga)ieA»  then  by  conditioi 
each  /,  1  <  z  <  n,  g^  =  0  except  for  finite  number  of  indices  X.  So,  Proposition  * 
be  applied  to  the  basis  {G;  :  1  <  i  <  n}  of  F1-. 

Since  7^  is  proximinal  for  each  X,  by  Remark  1.3  and  Corollary  1.2,  there 
X),  e  B(xk)  satisfying  for  each  A,, 


=  A(gu,  •  -  -  ,  gix)  V  1  <  i  <  n. 
Now  let  x  =  (JCA.)A€A-  Clearly  jc  e  ^x  and  Proposition  2.1  implies 

i'  -  -  -  G«)  for  1  - 


A.  A 

The  conclusion  now  follows  from  C9rollary  1.2. 


Proximinality  in  direct  sum  spaces  237 

Assume  now  F^  strongly  proximinal  in  Xx  for  each  A..  Let  6  >  0  be  given.  Since 
5  set  A  given  by  (2)  is  finite  we  can  get  8  >  0  such  that  for  each  A.  €  A  and  hi  in 
..,£/x,<5))  wehave 


d(hit  Jxx(gix,  .  -  -  ,  ft*))  <  6  for  1  <  i  <  /i.  (15) 

)w  choose  ?7  >  0  small  enough  so  that  YH  is  given  by 

m  =  *?>  m  =  Mm-i)  +  ty-i  for2  </  <w 
in  (3),  is  less  than  6.  We  have  from  (10) 

...^,*).    (16) 


Clearly  (14),  (15)  and  (16)  imply  that  ifx  =  Oa)  e  /x(Gi,  .  .  .  ,  G/,  ?/)  then 


d  this  completes  the  proof. 

We  now  give  an  alternate  shorter  proof  for  the  proximinality  of  Y  when  conditions  (1) 

d  (2)  of  Theorem  2.2  are  satisfied.  This  proof  avoids  the  use  of  Proposition  2.  1  and  uses 

immaB. 

Let  [Xj  :  I  <  j  <  1}  =  {Xx  :  A  e  A},  where  the  set  A  is  given  by  (2).  We  set 

G  =  X\  0oo  X2  ®oo  .  -  -  ®oo  Xi 


r  1  ^7  <  I-  We  have  Zj  to  be  proximinal  subspace  of  finite  codimension  in  Xj  for 
<j<l,  by  (2)  of  Theorem  2.2.  Further  if 

_  *7  .,  /r\        "7^  /T\  /TS        '7, 

—  -^1  tfoo  ^2  vPoo  •  •  •  vfoo  ^/» 

en  Z  is  a  proximinal  subspace  of  finite  codimension  in  G.  Set 


len  FI  is  a  subspace  of  G,  Z  c  Fj  c  G.  Now  we  use  Lemma  B  to  conclude  FI  is 
oximinal  in  G.  It  is  easily  verified  that  this,  in  turn,  implies  proximinality  of  F  in  X. 

zmark  2.1.  It  is  easy  to  see  that  the  above  proof  goes  through  when  X  is  taken  as  a  finite 
-direct  sum  of  normed  linear  spaces  and  condition  (2)  of  Theorem  2.2  is  satisfied.  The 
:ample  below  shows  that  this  is  no  longer  the  case  when  X  is  an  infinite  l\  -direct  sum. 

emark  2.2.  We  observe  here  that  the  necessity  of  Theorem  2.2  does  not  hold  even  for 
lite  /i  -direct  sums.  For  instance,  let  X  be  a  non-reflexive  Banach  space  and  pick  /  and  g 
(  the  unit  sphere  of  X*  such  that  there  exists  x  e  X  with  ||  x  \\  =  1  =  f(x)  and  g  does  not 
tain  its  norm  on  X.  Now  1  =  max{  |  /  1|  ,  ||  g  \\  }  =  \\  (/,  g)  \\  =  I  =  /(*).  Hence  (/,  g) 
tains  its  norm  at  (jc,  0)  and  Z  =  {(x,  y)  :  f(x)  +  g(y)  =  0}  is  a  proximinal  subspace  but 
er(g)  is  not  proximinal  in  X. 


238  VIndumathi 

3.  Example 

Theorem  2.2  is  not  true  if  we  replace  the  co-direct  sum  by,  for  instance,  the  l\  direct  sum 
as  the  following  example  shows. 

Example.  Let  X  =  e/,  Xn  where  Xn  =  c0  for  H  =  1,  2,  .  .  ..  Then  X*   =  0/^X*.  Select 
for  each  positive  integer  n,  //„  e/!  with||//n|    <   1  and  fin  e  NA(c<j).  Further  set 

/12  =      /21=0, 

ll/ii  II     =     I/all  =  1. 
/in         =     /2«  forn  >  3, 

|/m  |      <     Iforn  >  3,  i  =  1,2 
and 

.HSj/'-l-1- 

Define  F/  eX*,i  =  1,2,  as 


Let  Y  =  n{Ker  F/  :  i  =  1,  2}  and  7n  =  n{Ker  //„  :  i  =  1,  2}  for  n  =  1,  2,  .  .  ..  Since 
//„  6  NA(CQ)  for  /  =  1,2  and  for  all  n,  ^  is  proximinal  in  Xn  =  Co  for  all  rc.We  will 
now  show  that  Y  is  not  proximinal  in  X. 

Choose  */  e  C0,  /  =  1,  2  such  that  ||jc/  1  =  1  for  i  =  1,  2  and  /n  (xi)  =  /22(*2>  =  1- 
Consider  A:  =  (^  ,  jc2,  0,  0,  .  .  .)  in  X.  Then  ||  jc  ||  =  2.  Further  Ft  €  A^A(X)  as 


So,  ^U,  7)  =     JCJF-L  I  >  I.We  now  show  that  d(x,  Y)  =  1. 

To  see  this,  select  for  n  >  3,  xn  €  C0  satisfying  j^O^)    =    -1  for  /  =   1,  2  and 
limn-^oo  ^xn  ||  =  1.  Define  a  sequence  (j;0&>3  €  X  by 


0      otherwise. 
Then  F/(>'^)  =  /HUi)  +  fx(xk)  =  0  for  /  =  1,  2  and  so  yk  e  Y  for  all  k.  Further 

lk-^1   =     xk\\  ->  las/:  -^  oo. 
Hence  d(x,  Y)  =  1. 

We  recall  that  a  nearest  element  to  x  from  Y  exists  if  and  only  if  there  exists  3>  in  X 
satisfying 

Ft(y)  =  Ff(Jc)  =  1  for  i  =  1,2  |y[  =  d(x,  Y)  =  1. 

However  |y|  =  1  =  FI(>>)  implies  y  =  (yi,  0,  0.  .  .)  where  fn(yi)  =  1,  ||yi||  =  1.  But, 
m  this  case,  F2(y)  =  0  ^  F2(jc)  and  the  above  equality  can  not  hold.  Therefore,  Y  is  not 
proximinal  in  X. 

Acknowledgement 

The  author  would  like  to  thank  the  referee  for  suggesting  the  alternate  proof  given  in  the 
sufficiency  part  of  Theorem  2.2  and  also  the  Remarks  2.  1  and  2.2. 


Proximinality  in  direct  sum  spaces  239 

References 

[1]  Garkavi  A  L,  On  the  best  approximation  by  the  elements  of  infinite  dimensional  subspaces 

of  a  certain  class,  Mat.  Sb.  62  (1963)  104-120 
[2]  Garkavi  A  L,  Helley's  problem  and  best  approximation  in  spaces  of  continuous  functions, 

Izv.  Akad.  Nauk  SSSR  Ser.  Mat.  31  (1967)  641-656 
[3]  Godefroy  Gilles  and  Indumathi  V,  Strong  proximinality  and  polyhedral  spaces,  Revista 

Matemdtica  Complutense  (to  appear) 
[4]  Indumathi  V,  Proximinal  subspaces  of  finite  codimension  in  general  normed  linear  spaces, 

Proc.  London  Math.  Soc.  45(3)  (1982)  435-455 

[5]  Indumathi  V,  On  transitivity  of  proximinality,  J.  Approx.  Theory  49(2)  (1987)  130-143 
[6]  Pollul  W,  Reflexvitat  und  Existenz-Teilraume  in  der  Hnearen  Approximations  theorie,  Dis- 
sertation Bonn  (1971);  Schriften  der  Ges  fur  Math  und  Datenverarbeitung,  Bonn  53  (1972) 

1-21 
[7]  Singer  Ivan,  Best  approximation  in  normed  linear  spaces  by  elements  of  linear  subspaces, 

Die  Grundlehren  der  mathematischen  Wissenschaften,  Band  171  (Springer  Verlag)  (1970) 
[8]  Singer  Ivan,  On  best  approximation  in  normed  linear  spaces  by  elements  of  subspaces  of 

finite  codimension,  Rev.  Roumaine  Math.  Pure.  Appl.  17  (1972)  1245-1256 
[9]  Vlasov  L  P,  Elements  of  best  approximation  relative  to  subspaces  of  finite  codimension,  Mat. 

Zametki  32(3)  (1982)  325-341 
[10]  Vlasov  L  P,  Subspaces  of  finite  codimension:  Existence  of  elements  of  best  approximation, 

Mat.  Zametki  37(1)  (1985)  78-85 


Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  1 1 1,  No.  2,  May  2001,  pp.  241-247. 
Printed  in  India 


Common  fixed  points  for  weakly  compatible  maps 

RENU  CHUGH  and  SANJAY  KUMAR 

Department  of  Mathematics,  Maharshi  Dayanand  University,  Rohtak  124  001,  India 

MS  received  31  January  2000;  revised  1 1  December  2000 

Abstract.  The  purpose  of  this  paper  is  to  prove  a  common  fixed  point  theorem, 
from  the  class  of  compatible  continuous  maps  to  a  larger  class  of  maps  having  weakly 
compatible  maps  without  appeal  to  continuity,  which  generalizes  the  results  of  Jungck 
[3],  Fisher  [1],  Kang  and  Kim  [8],  Jachymski  [2],  and  Rhoades  [9]. 

Keywords.    Weakly  compatible  maps;  fixed  points. 

1.  Introduction 

In  1976,  Jungck  [4]  proved  a  common  fixed  point  theorem  for  commuting  maps  generalizing 
the  Banach's  fixed  point  theorem,  which  states  that,  'let  (X,  d)  be  a  complete  metric  space. 
If  T  satisfies  d(Tx,  Ty)  <  kd(x,  y)  for  each  jc,  y  e  X  where  0  <  k  <  1,  then  T  has  a 
unique  fixed  point  in  X\  This  theorem  has  many  applications,  but  suffers  from  one  draw- 
back -  the  definition  requires  that  T  be  continuous  throughout  X.  There  then  follows  a 
flood  of  papers  involving  contractive  definition  that  do  not  require  the  continuity  of  T.  This 
result  was  further  generalized  and  extended  in  various  ways  by  many  authors.  On  the  other 
hand  Sessa  [11]  defined  weak  commutativity  and  proved  common  fixed  point  theorem  for 
weakly  commuting  maps.  Further  Jungck  [5]  introduced  more  generalized  commutativity, 
the  so-called  compatibility,  which  is  more  general  than  that  of  weak  commutativity.  Since 
then  various  fixed  point  theorems,  for  compatible  mappings  satisfying  contractive  type 
conditions  and  assuming  continuity  of  at  least  one  of  the  mappings,  have  been  obtained  by 
many  authors. 

It  has  been  known  from  the  paper  of  Kannan  [7]  that  there  exists  maps  that  have  a 
discontinuity  in  the  domain  but  which  have  fixed  points,  moreover,  the  maps  involved  in 
every  case  were  continuous  at  the  fixed  point.  In  1998,  Jungck  and  Rhoades  [6]  introduced 
the  notion  of  weakly  compatible  and  showed  that  compatible  maps  are  weakly  compatible 
but  converse  need  not  be  true.  In  this  paper,  we  prove  a  fixed  point  theorem  for  weakly 
compatible  maps  without  appeal  to  continuity,  which  generalizes  the  result  of  Fisher  [1], 
Jachymski  [2],  Kang  and  Kim  [8]  and  Rhoades  et  al  [9]. 

2.  Preliminaries 
DEFINITION  2.1  [6] 

A  pair  of  maps  A  and  S  is  called  weakly  compatible  pair  if  they  commute  at  coincidence 
points. 

Example  2.1.  Let  X  =  [0,  3]  be  equipped  with  the  usual  metric  space  d(x,  y)  —  \x  —  y\. 


242  Renu  Chugh  and  Sanjay  Kumar 

Define  /,  g  :  [0,  3]  ->  [0,  3]  by 


x     if  jc  6  [0,1) 
3     if*  €[1,3] 


and    g(x)  = 


3  -  jc     if  x  e  [0,  1) 

3  if  x  e  [1,3] 


Then  for  any  x  G  [1,3],  fgx  =  gfx,  showing  that  /,  g  are  weakly  compatible  maps  or 
[0,  3]. 

Example  2.2.  Let  X  =  R  and  define  f,g:R->Rbyfx  =  x/3,  x  e  R  and  gx  = 
jc2,  x  €  R.  Here  0  and  1/3  are  two  coincidence  points  for  the  maps  /  and  g.  Note  tha 
/  and  g  commute  at  0,  i.e.  /g(0)  =  g/(0)  =  0,  but  /g(l/3)  =  /(1/9)  =  1/27  anc 
g/(l/3)  =  g(l/9)  =  1/81  and  so  /  and  g  are  not  weakly  compatible  maps  on  R. 

Remark  2.1.  Weakly  compatible  maps  need  not  be  compatible.  Let  X  =  [2,  20]  and  t 
be  the  usual  metric  on  X.  Define  mappings  B,  T  :  X  ->  X  by  Bx  =  x  if  x  =  2  o 
>  5,  Bx  =  6  if  2  <  jc  <  5,  TJC  =  x  if  x  =  2,  Tx  =  12  if  2  <  jc  <  5,  Tx  =  %  -  : 
if  jc  >  5.  The  mappings  B  and  T  are  non-compatible  since  sequence  {xn}  defined  b; 
xn  =  5  +  (1/n),  n  >  1.  Then  Tz,z  -»  2,  Bzn  =  2,  T5jcn  =  2  and  5T;cn  =  6.  But  the 
are  weakly  compatible  since  they  commute  at  coincidence  point  at  x  =  2. 

3.  Fixed  point  theorem 

Let  R+  denote  the  set  of  non-negative  real  numbers  and  F  a  family  of  all  mappings  </> 
(R+)5  ->  /?+  such  that  <p  is  upper  semi-continuous,  non-decreasing  in  each  coordinal 
variable  and,  for  any  t  >  0, 

<p(t,  t,  0,  at,  0)  <  0r,  0(r,  f,  0,  0,  at)  <  /3r, 
where  ft  =  1  for  of  =  2  and  ^  <  1  for  a  <  2, 

y(0  =  0(/,r,air,a2^a30  <  r, 
where  y  :  T?4"  ->  jR+  is  a  mapping  and  «i  4-  0^2  4-  ^3  =  4. 

Lemma  3.1  [12].  For  every  r  >  0,  y  (f)  <  t  if  and  only  z/lim^oo  yn(t)  =  0,  w/zere  > 
denotes  the  n  times  composition  ofy. 

Let  A,  B,  S  and  T  be  mappings  from  a  metric  space  (X,  d)  into  itself  satisfying  ti 
following  conditions  : 


A(X)  c  7XX)  and  B(X)  C  S(X),  (3. 

,  By)  <  0(J(Sjc,  Ty),  d(Ax,  S:c),  J(J5y,  Ty),  d(Ajc,  Ty),  ^(jBy,  5;c))        (3. 


for  all  x,  y  e  X,  where  0  €  F.  Then  for  arbitrary  point  JCQ  in  X,  by  (3.1),  we  choose 
point  x\  such  that  Tx\  =  AXQ  and  for  this  point  jci,  there  exists  a  point  X2  in  X  such  tt 
5*2  =  Bx\  and  so  on.  Continuing  in  this  manner,  we  can  define  a  sequence  {yn}  in  X  su 
that 


72«  =  Ax2n  =  ^2/7+1  and  y^+i  =  Bx2n+i  =  Sx2n+2,  n  =  0,  1,  2,  3,  ____          (3 
Lemma  3.2  lim^oo  d(yn,  yn+\)  =  0,  where  [yn]  is  the  sequence  in  X  defined  by  (3.3), 


Fixed  points  for  weakly  compatible  maps  243 

Droof  Let  dn  =  d(yn,  3>n+i)»  n  =  0,  1,  2,  —  Now,  we  shall  prove  the  sequence  [dn]  is 
ion-increasing  in  R+,  that  is,  dn  <  <4-i  for  n  =  1,  2,  3,  .  .  ..  From  (3.2),  we  have 


<  <l>(d(y2n-\  , 


,  ^2n-l,  ^2n,  0,  J2rc  +  ^2n-l).  (3.4) 

Suppose  that  dn~.\  <  dn  for  some  n.  Then,  for  some  a  <  2,  d^-i  +  <4  =  ^<^n-  Since  0  is 
ion-increasing  in  each  variable  and  ft  <  I  for  some  a.  <  2.  From  (3.4),  we  have 

,  d2n,  0, 


Similarly,  we  have  6?2n+i  <  ^2n+i-  Hence,  for  every  n,t/n  <  j6dn  <  <s?n,  which  is  a 
contradiction.  Therefore,  [dn]  is  a  non-increasing  sequence  in  J?"1".  Now,  again  by  (3.2), 
ve  have 


,  >Jo), 


[n  general,  we  have  dn  <  yn(d§),  which  implies  that,  if  d§  >  0,  by  Lemma  3.1, 
lim  dn  <  lim  yn(dQ)  =  0. 

«->oo  «.-H>-OO 

Fherefore,  we  have   lim  dn  =  0.    For  JQ  =  0,  since  {Jrt}  is  non-increasing,  we  have 

AZ-»OO 

lim  dn  =  0.  This  completes  the  proof. 

1-+00 

Lemma  3.3.  The  sequence  {yn}  defined  by  (3.3)  is  a  Cauchy  in  X. 

Proof.  By  virtue  of  Lemma  3.2,  it  is  a  Cauchy  sequence  in  X.  Suppose  that  {y2n}  is  not  a 
Cauchy  sequence.  Then  there  is  an  €  >  0  such  that  for  each  even  integer  2k,  there  exist 
even  integers  2m(k)  and  2n(k)  with  2m  (k)  >  2n(k)  >  2k  such  that 


For  each  even  integer  2&,  let  2;n(/:)  be  the  least  even  integer  exceeding  2n(k)  satisfying 
(3.  5),  that  is, 


)  <  €    and  ^(3^^),  y2m(£))  >  €.  (3.6) 

Then  for  each  even  integer  2k,  we  have 

€  <d(y2n(k),y2m(k)) 


244  Renu  Chugh  and  Sanjay  Kumar 

By  Lemma  3.2  and  (3.6),  it  follows  that 

d(y2n(k),  J2m(k]}  -»  €  as  *  -»  oo.  (3.7) 

By  the  triangle  inequality,  we  have 

\d(y2n(k],  y2m(k}~\)  —  d(y2n(K),  J2m(k})\  < 

and 


From  Lemma  3.2  and  eq.  (3.7),  as  k  ->  oo, 

d(y2n(k},  y2m(k}~\)  ->  €  and  d(y2n(k)+\>  y2m(k}~\)  ->  c.  (3.8) 

Therefore,  by  (3.2)  and  (3.3),  we  have 


Since  0  is  upper  semi  continuous,  as  fc  ->•  oo  as  in  (3.8),  by  Lemma  3.2,  eqs  (3.7),  (3.8) 
and  (3.9)  we  have 


which  is  a  contradiction.  Therefore,  [y2n]  is  a  Cauchy  sequence  in  X  and  so  is  {jn}.  This 
completes  the  proof. 

TheoremS.L  Let(A,  S)and(B,  T)  be  weakly  compatible  pairs  of  self  'maps  of  a  complete 
metric  space  (X,  d)  satisfying  (3.1)  and  (3.2).  Then  A,  B,  S  and  T  have  a  unique  common 
fixed  point  in  X. 

Proof.  By  Lemma  3.3,  {yn}  is  a  Cauchy  sequence  in  X.  Since  X  is  complete  there  exists  a 
point  z  in  X  such  that  lim  yn  =  z.  lim  Ax2n  =  Km  rx2w+i  =  z  and  lim  Bx2n+i  = 

«-»oo  n-»oo  /i-*oo  /i-voo 

lim  5^2/1+2  =  z  i.e., 

n-^oo 

lim  Ax2n  =  lim  T^n+i  =  lim  #*2n+i  =  lim  Sx2n+2  =  2- 

n->oo  n->oo  n-»oo  «->oo 

Since  B(X)  c  S(X),  there  exists  a  point  u  e  X  such  that  z  =  5w.  Then,  using  (3.2), 
d(Au,z)  <d(Au, 


d(Au,  Tx2n-i)d(Bx2n-i,  Su)). 


Fixed  points  for  weakly  compatible  maps  245 

king  the  limit  as  n  -»  oo  yields 

d(Au,  z)  <  0(0,  d(Au,  Su),  0,  d(Au,  z),  d(z,  Su)) 

=  0(0,  d(Au,  z),  0,  d(Aw,  z),  0)  <  £d(Aw,  z), 


lere  ft  <  1.  Therefore  z  =  Aw  =  Sw. 

Since  A(X)  C  T(X),  there  exists  a  point  i;  €  X  such  that  z  =  TV.  Then,  again  using 

2), 

z,  Bv)  =  d(Aw,  Bu)  <  <p(d(Su,  TV),  d(Au,  Su),  d(Bv,  TV),  d(Au,  Tv),d(Bv,  Su)) 
=  0(0,  0,  d(Bv,  z),  0,  d(Bv,  z))  <  0(f  ,  t,  t,  t,  t)  <  t, 

lere  r  =  d(z,  Bu).  Therefore  z  =  fiu  =  TV.  Thus  Aw  =  Su  =  #u  =  Tu  =  z.  Since 
ir  of  maps  A  and  5  are  weakly  compatible,  then  ASu  =  SAu  i.e,  Az  =  Sz.  Now  we 
ow  that  z  is  a  fixed  point  of  A.  If  Az  ^  z,  then  by  (3.2), 

d(Az,  z)  =  d(Az,  Bv)  <  </>(d(Sz,  TV),  d(Az,  Sz),  d(Bv,  TV), 

d(Az,Tv),d(Bv,Sz)) 
=  0(J(Az,  z),  0,  0,  d(Az,  z),  d(Az,  z)) 
<  <p(t,  t,  t,  t,  t)  <  t,  where  t  =  d(Az,  z). 

lerefore,  Az  =  z.  Hence  Az  =  Sz  =  z. 

Similarly,  pair  of  maps  B  and  T  are  weakly  compatible,  we  have  Bz  =  Tz  =  z,  since 

d(z,  Bz)  =  d(Az9  Bz)  <  4>(d(Sz,  Tz),  d(Az,  Sz), 

d(Bz,  Tz),  d(Az,  Tz),  d(Bz,  Sz)) 
=  4>(d(z,  Tz),  0,  0,  d(z,  Tz),  d(z,  Tz)) 
<  4(t,  t,  t,  t,  t)  <  t,  where  r  =  d(z,  Tz)  =  d(z,  Bz). 

ms  z  =  Az  =  Bz  =  Sz  =  Tz,  and  z  is  a  common  fixed  point  of  A,  B,  S  and  T. 
Finally,  in  order  to  prove  the  uniqueness  of  z,  suppose  that  z  and  w,  z  ^  w  ,  are  common 
ced  points  of  A,  B,  S  and  T.  Then  by  (3.2),  we  obtain 


(z,  w)  =  d(Az,  Bw)  <  0(J(Sz,  Tw),  d(Az,  Sz),  d(Bw,  Tw),  d(Az,  Tw),  d(Bw,  Sz)) 
=  4>(d(z,  w),  0,  0,  d(z,  w),  d(z,  w)) 
<  c/)(t,  t,  t,  t,  t)  <  t,  where  t  =  d(z,  w). 

tierefore,  z  =  w.  The  following  corollaries  follow  immediately  from  Theorem  3.1. 

OROLLARY  3.1 

et  (A,  S)  and  (B,  T)  be  weakly  compatible  pairs  of  self  maps  of  a  complete  metric  space 
t,  d)  satisfying  (3.1),  (3.3)  and  (3.10) 

d(Ax,  By)  <  hM(x,  y),Q<h<l,x,y  €  X,  where 

f  (jc,  y)  =  max  [d(Sx,  Ty),  d(Ax,  Sx),  d(By,  Ty),  [d(Ax,  Ty)  +  d(By,  Sx)]/2}. 

(3.10) 


246  Renu  Chugh  and  Sanjay  Kumar 

Then  A,  B,  S  and  T  have  a  unique  common  faced  point  in  X. 

Proof.  We  consider  the  function  0  :  [0,  oo)5  ~>  [0,  oo)  defined  by 

2,*3,*4,*5)  =  A  max  {*i,*2,*3 
Since  0  €  F,  we  can  apply  Theorem  3.1  and  deduce  the  Corollary. 

COROLLARY  3.2 

Let  (A,  S)  and  (B,  7")  be  weakly  compatible  pairs  of  self  maps  of  a  complete  metric  spat 
(X,  d)  satisfying  (3.1),  (3.3)  and  (3.1 1). 

d(Ax9  By)  <  h  max  {d(Ax,  Sx),  d(By,  7»,  l/2d(Ax,  Ty), 

l/2d(By,  Sx),d(Sx,  Ty)}forallx,  y  in  X,  where  0  <  h  <  1.         (3.1 

Then  A,  5,  S  and  T  have  a  unique  common  fixed  point  in  X. 

Proof.  We  consider  the  function  </>  :  [0,  oo)5  -»  [0,  oo)  defined  by  <p(x\ ,  *2,  *3,  *4,  *s) 
/z  max  {x\,X2,  *3>  */2  *4»  l/2  x$}.  Since  0  6  T7,  we  can  apply  Theorem  3.1  to  obtain  tt 
Corollary. 

Remark  3.2.  Theorem  3.1  generalizes  the  result  of  Jungck  [3]  by  using  weakly  compatit 
maps  without  continuity  at  5  and  T.  Theorem  3.1  and  Corollary  3.2  also  generalize  t 
result  of  Fisher  [1]  by  employing  weakly  compatible  maps  instead  of  commutativity 
maps.  Further  the  results  of  Jachymski  [2],  Kang  and  Kim  [8],  Rhoades  et  al  [9]  are  al 
generalized  by  using  weakly  compatible  maps. 

Acknowledgements 

The  authors  are  thankful  to  the  referee  for  giving  useful  comments  and  suggestions  for  1 
improvement  of  this  paper. 

References 

[1]  Fisher  B,  Common  fixed  points  of  four  mappings,  Bull  Inst.  Math.  Acad.  Sd.  11  (19 

103-113 
[2]  Jachymski  J,  Common  fixed  point  theorems  for  some  families  of  maps,  J.  Pure  AppL  Mi 

25  (1994)  925 
[3]  Jungck  G,  Compatible  mappings  and  common  fixed  points  (2),  Int.  J.  Math.  Math.  Sci. 

(1988)  285-288 

[4]  Jungck  G,  Commuting  maps  and  fixed  points,  Am.  Math.  Mon.  83  (1976)  261 
[5]  Jungck  G,  Compatible  mappings  and  common  fixed  points,  Int.  J.  Math.  Math.  Sci.  9(19 

771-779 
[6]  Jungck  G  and  Rhoades  B  E,  Fixed  point  for  set  valued  functions  without  continuity,  India 

Pure  AppL  Math.  29(3)  (1998)  227-238 

[7]  Kannan  R,  Some  results  on  fixed  points,  Bull.  Cal  Math.  Soc.  60  (1968)  71-76 
[8]  Kang  S  M  and  Kim  Y  P,  Common  fixed  points  theorems.  Math.  Japonica  37(6)  (1992)  10 

1039 
[9]  Rhoades  B  E,  Park  S  and  Moon  K  B,  On  generalizations  of  the  Meir-Keeler  type  contrac 

maps,  J.  Math.  Anal.  AppL  146  (1990)  482 
[10]  Rhoades  B  E,  Contractive  definitions  and  continuity,  Contemporary  Math.  72  (1988)  233- 


Fixed  points  for  weakly  compatible  maps  247 

[11]  Sessa  S,  On  a  weak  commutativity  condition  of  mappings  in  fixed  point  considerations,  Pub. 

Inst.  Math.  32(46)  (1982)  149-153 
[12]  Singh  S  P  and  Meade  B  A,  On  common  fixed  point  theorems,  Bull.  Austral.  Math.  Soc.  16 

(1977)49-53 


Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  1 1 1,  No.  3,  August  2001,  pp.  249-262. 
©  Printed  in  India 


On  totally  reducible  binary  forms:  I 

C  HOOLEY 

School  of  Mathematics,  Cardiff  University,  Senghennydd  Road,  PO  Box  926,  Cardiff 
CF24  4YH,  UK 

MS  received  18  October  2000;  revised  7  February  2001 

Abstract.  Let  v(n)  be  the  number  of  positive  numbers  up  to  a  large  limit  n  that  are 
expressible  in  essentially  more  than  one  way  by  a  binary  form  /  that  is  a  product  of 
t  >  2  distinct  linear  factors  with  integral  coefficients.  We  prove  that 


•  =  o\ 

where 

l/.£2,  if  £  =  3, 


£-1),  if£  >  J, 

thus  demonstrating  in  particular  that  it  is  exceptional  for  a  number  represented  by  / 
to  have  essentially  more  than  one  representation. 

Keyword.    Binary  forms. 

1.  Introduction 

In  this  publication  and  its  sequel  we  shall  fulfil  the  undertaking  given  in  our  earlier  paper 
[3]  to  resolve  the  following  problems  for  binary  forms  /  of  degree  t  >  2  that  are  totally 
reducible  as  a  product  of  t  (disjoint)  linear  factors  with  integral  coefficients: 
(i)  to  find  an  asymptotic  formula  for  the  number  T(/i)  =  T^(n)  of  positive  integers  that 
are  expressible  by  /  and  do  not  exceed  n,  each  such  integer  being  counted  just  once 
regardless  of  multiplicity  of  representations  (no  generality  is  lost  by  debarring  negative 
numbers  because  they  can  be  treated  by  changing  the  sign  of  one  of  the  linear  factors 
in/); 
(ii)  to  find  an  upper  bound  for  the  number  v(n)  =  i^(n)  of  such  integers  that  are  represented 

in  essentially  more  than  one  way. 

We  thus  shall  extend  to  a  special  class  of  binary  forms  of  arbitrary  degree  the  results 
obtained  for  cubics  and  certain  other  binary  forms  in  former  papers  and,  in  particular,  [3], 
to  the  last  of  which  the  reader  is  referred  for  a  history  of  the  problem  and  the  relevant 
citations. 

In  interpreting  the  second  quest,  on  which  the  first  will  be  seen  to  depend,  we  must 
anticipate  a  later  discussion  by  saying  that  representations  of  a  number  by  the  form  are 
regarded  as  being  inherently  distinct  if  they  be  not  associated  with  each  other  in  an  obvious 
way  through  an  automorphic  of  the  form.  With  this  appreciation,  we  shall  shew  here  that 


v(n)  =  O    n-Vt+t    ,  (1) 

040 


250  CHooley 

where 


if  t  =  3, 

t~  1),  if*  >  3, 


from  which  it  will  be  easily  demonstrated  that  it  is  extremely  rare  for  a  number  representable 
by  /  to  be  represented  in  essentially  more  than  one  way. 

The  derivation  from  this  result  of  an  asymptotic  formula  for  T(n)  principally  depends 
on  the  properties  of  the  automorphics  of  the  form.  We  therefore  reserve  the  treatment  of 
item  (i)  for  a  second  paper,  especially  as  an  exhaustive  treatment  of  the  structure  of  the 
automorphics  occupies  some  space,  is  in  itself  an  interesting  study,  and  involves  ideas  that 
are  somewhat  alien  to  those  used  in  the  present  work.  Suffice  it  then  for  the  time  being  to 
say  that  we  shall  ultimately  obtain  an  asymptotic  formula  of  the  type 

T(n)  -  A(/)n2/*,      (A(/)  >  0) 


with  a  remainder  term  similar  to  the  right-hand  side  of  (1). 

We  should  mention  that  the  advantage  of  our  present  methods  -  in  contrast  with  those 
often  used  in  problems  of  this  type  -  is  that  they  are  also  applicable  to  an  inhomogeneous 
situation  in  which  the  subject  of  study  is  a  completely  reducible  polynomial  of  degree  I 
consisting  of  factors  of  the  type  hx  +  ky  4-  q  .  It  is  hoped  to  give  an  account  of  this  extensi  or 
to  our  work  in  due  course. 

2.  Notation  and  conventions 

As  is  often  the  case  in  the  algebra  of  substitutions  as  applied  to  forms  or  quantics,  eacl 
symbol  for  a  variable  therein  will  denote  an  indeterminate  on  some  occasions  and  a  special 
ization  of  this  on  other  occasions.  With  this  agreement,  when  not  denoting  indeterminates 
r,  s,  p,  or  are  integers  and  m,  /x  with  or  without  distinguishing  marks  are  non-zero  inte 
gers;  p,  m  are  positive  prime  numbers.  The  letters  Ap  A2,  .  .  .  denote  suitable  positiv 
constants  depending  at  most  on  the  form  /  under  consideration;  €  is  an  arbitrarily  smai 
positive  number  that  is  not  necessarily  the  same  at  each  occurrence;  the  constants  irnplie 
by  the  O  -notation  are  of  type  A.  save  when  they  may  also  depend  on  €. 

Since  negative  integers  may  frequently  occur,  we  should  mention  that  they  may  b 
moduli  in  congruences.  The  terms  size,  magnitude,  modulus  are  used  as  synonyms  fc 
absolute  value  when  applied  to  real  numbers.  The  notation  (/z,  &)  indicates  the  positiv 
highest  common  factor  (when  defined)  of  integers  h,  k  save  when  it  designates  a  point  wit 
coordinates  ft,  k\  d(m)  is  the  number  of  positive  divisors  of  m,  while  dr(m)  is  the  numb* 
of  ways  expressing  |m|  as  a  product  of  r  positive  factors. 

3.  Prolegomena 

Being  totally  reducible  over  the  rationals  with  no  repeated  factors,  the  binary  form  /  - 
f(x,y)  of  degree  t  >  3  under  consideration  is  expressed  as 

n   (h.x+k.y)=    U    I/(jr,y),  say,  (: 


where  the  coefficients  of  the  linear  forms  L.(JC,  y)  are  integers  and  where,  even  apart  fro 
order,  there  is  a  slight  but  acceptable  ambiguity  in  their  definitions  when  /  is  imprimiti\ 


Totally  reducible  binary  forms  25 1 

Of  the  invariants  of  the  form,  the  only  one  that  will  be  needed  is  the  discriminant 

/  \2 

D  =  D(f)  =       n      (hikj  ~  hjki)    >  0,  (3) 

which  has  the  familiar  property  that,  if  /  (x ,  >•)  be  transformed  into  F(X,  Y)  by  a  substitu- 
tion of  modulus  M,  then 

D(F)  =  M^  Z>(/).  (4) 

Also,  since  our  investigation  only  concerns  the  representations  of  numbers  by  /  without 
regard  to  the  size  of  the  variables  in  /,  we  may  equally  well  work  with  any  form  /' 
equivalent  to  /  through  a  rational  integral  substitution 


(5) 
with  modulus 

erf-yj8=l.  (6) 

This  means,  in  particular,  that  we  may  certainly  assume  that 

/i1,/i2,...,/i£^0  (7) 

because1  ,  in  the  opposite  instance,  having  chosen  relatively  prime  numbers  a,  y  and  then 
/?,  8  to  satisfy  /(a,  y)  ^  0  and  (6),  we  find  through  (5)  a  form  with  non-zero  leading 
coefficient  /(a,  y)  that  is  equivalent  to  /. 

Closely  associated  with  our  study  of  the  representations  by  /  (x  ,  y)  of  positive  numbers 
up  to  a  large  limit  n,  the  curve  C  =  C(n)  defined  by  the  equation 

/(*,>')  =  "  (8) 

will  be  encountered  together  with  its  asymptotes 

I1(jc,;y)  =  0,...,Ij(*,;y)  =  0,  (9) 

which  both  here  and  in  our  second  paper  will  play  a  not  unimportant  role  in  the  elucidation 
of  lattice  point  problems  involving  regions  bounded  by  C(n)  .  Forming  2£  semi-infinite  rays 
emanating  from  the  origin,  these  asymptotes  divide  the  plane  into  2t  semi-infinite  domains, 
in  each  of  which  f(x,  y)  has  a  constant  sign  opposite  to  that  pertaining  to  its  neighbours. 
Moreover,  from  an  examination  of  the  configuration  formed  by  (8)  and  (9),  it  would  be 
foreseen  that  the  major  influence  on  our  situation  would  be  exerted  by  those  x  and  y  having 
absolute  values  not  substantially  larger  than  n  {/i,  which  expectation  prompts  us  at  once  to 
write 

N  =  n[^  (10) 

for  notational  convenience.  Next,  elaborating  on  this  line  of  thought  analytically  (a  geo- 
metrical approach  is  more  intuitive  but  harder  to  describe),  we  note  by  linear  relationships 
that,  if 


max 


:=  Q>  0  (11) 


1  With  a  little  more  effort  we  can  shew  that  we  may  suppose,  in  addition,  that  fcp  . . . ,  kt  ^  0. 


252  C  Hooley 

for  integer  values  of  x  and  y,  then  we  always  have 

\x\,\y\<A{Q.  (12) 

Also,  more  significantly,  if  here 

Q  >  A2N  (13) 

for  a  sufficiently  large  positive  constant  A2  and  /  (x  ,  y)  obey  the  usually  assumed  inequality 
0  <  /(jc,  y)  <  n,  we  see  first  that  at  least  one  form  Lv(x,  y)  for  v  ^  it  has  magnitude  not 
less  than  1  and  not  greater  than 

(n/fi)  W-l>  <  A~  1/("-1}  W  <  A-</(£-1}  Q  (14) 

and  then  deduce  from  linear  relationships  and  (11)  that  this  form  Lv(x,  y)  is  unique,  all 
other  forms  L.(x,  y)  having  magnitudes  greater  then  A3  Q.  Hence  A^~  l  Q^  }  <  n  so  that 

2<A4n1/(^D,  (15) 

while  also  the  bound  (14)  is  improved  to 


Some  amplification  of  an  introductory  remark  about  the  automorphics  of  the  form  is 
needed  at  once  even  though  a  full  examination  of  their  structure  will  be  delayed  until  our 
second  paper.  Let  now 


x  =  otX  +  pY,  y  =  y  X  4-  SY  (17) 

be  a  rational  automorphic  of  /,  namely,  a  substitution  with  rational  coefficients  a,  p,y,8 
with  the  property  that  /(*,  >')  =  /(X,  Y)  and,  as  we  confirm  from  (4),  the  consequencial 
property  that  its  modulus  otS  -  yfi  is  equal  to  ±1.  Points  (x,  y),  (X,  Y)  with  integral 
coordinates  that  are  connected  by  means  of  an  automorphic  of  type  (17)  will  be  said  to 
be  associated,  the  property  of  association  being  denoted  by  (x,  y)  ~  (X,  7).  Then,  since 
associated  points  give  rise  to  linearly  connected  representations  of  the  same  number,  we 
shall  agree  that  representations  of  a  number  as/  (x,  y)  and/0*:7,  yf)  are  deemed  essentially 
different  if  (x,  y)  ~  (#',  y).  Thus  an  unmistakable  meaning  has  been  attached  to  v(n)  ,  to 
whose  estimation  we  now  attend. 


4.  The  sum  T(n)  and  the  equation  m  F(m,  s)  =  MG(^»  °r) 
The  treatment  depends  on  an  analysis  of  the  sum 

Hit)  =          £         1,  (18) 

t)</(r,jr)s=/(/j,<T)<n 
(/-.*)?£(/».*) 

through  which  v(n)  is  bounded  by  the  obvious  inequality 

v(n)  <  T(n).  (19) 


Totally  reducible  binary  forms 


253 


First,  to  dissect  the  sum  into  parts  that  can  be  appropriately  assessed,  let  T,  (n)  be  that 
portion  of  T(n)  that  is  yielded  by  values  of/-,  s,  p,  a  in  the  conditions  of  summation  for 
which  a  linear  factor  of  maximal  size  in  the  constituents  of  the  equation 

f(r,s)=     [I     L.(r,s)=     fl 


>.  a)  = 


,  a) 


occurs  on  the  left.  Then,  allowing  7^  (/?  ,  M)  to  denote  the  contribution  to  7^  (n)  correspond- 
ing to  values  of  r,  j,  p,  a  for  which  the  size  of  this  maximal  linear  factor  lies  between  M 
inclusive  and  2M  exclusive,  we  have 

T(n)  <2T,(n) 

and  complete  the  first  phase  of  the  calculations  by  deducing  that 
T(n)  <2][]rl(/i,  Af,), 


in  which 


is  less  than 


{  =  21'         (i  >  0) 


(20) 


(21) 


(22) 


by  (15). 

In  further  preparation  for  the  estimation  of  T(n)  we  examine  the  solutions  of  the  inde- 
terminate equation 


f(r,s)  =  f(p,a) 
that  are  constrained  by  the  conditions 

Lu(r,  s)  =  m,         Lv(p,  or)  =  /t 


(23) 


(24) 


for  given  subscripts  M,  i>  and  non-zero  integers  m,  /i.  For  this  purpose,  recalling  (7),  we 
employ  the  substitutions2 


m  =  hur  4-  kus,         s  =  s, 


(25) 
(26) 


to  transform  (23)  into 


which  equation  for  brevity  we  express  as  either 


or 


(28) 
(29) 


2  Remember  the  remaiks  in  §2  about  entities  appearing  in  linear  transformations.  We  should  also  comment  that 
it  would  be  pendantic  and  unhelpful  here  to  introduce  symbols  5',  a'  to  substitute  for  st  a  in  (27)  and  in  the  left 
sides  of  the  right  hand  members  of  (25)  and  (26). 


254  C  Hooley 

where  F(m,s)  =  Fu(m,s)  and  G(/z,<r)  =  Gy(/z,a)(=  Fy(^,cr))  are,  respectively,  of 
exact  degree  £  —  1  in  5  and  cr. 

Needing  to  know  when  the  curve  defined  by  (27)  for  given  non-zero  integer  values  of  m 
and  IJL  is  irreducible  over  Q,  we  set 

5  =  ms',  a  =  i^cr1  ',  X  =  (m/^  ^  0 
to  form  the  equivalent  curve 

A.jF(l,j/)~G(l1cr/)  =  0,  (30) 

which  is  certainly  irreducible  (indeed  absolutely  irreducible)  when  its  projective  completion 


is  non-singular  and  hence  when  the  simultaneous  equations 


3s'  '  acr'  '  3z  dz 

have  no  non-zero  solution.  But,  if  z  =  0,  the  first  two  equations  only  hold  when  sf  =  a'  =  0,  f 

whereas  otherwise,  since  jp(z,  5*0  and  G(z,  ax)  are  each  products  of  real  distinct  factors, 

they  are  only  satisfied  when  s'  /z,  af/z  each  take  t  —  2  real  values,  each  combination  of 

which  determines  A.  through  the  last  equation  because  neither  both  3F/dsf,  dF/dz  nor 

both  3G/da\  3G/dz  can  (non-trivially)  simultaneously  vanish.  Thus  reducible  curves  of 

type  (30)  answer  to  at  most  (t  —  2)2  values  of  X,  and  we  therefore  infer  that  (29)  can  only 

be  reducible  if  •. 

£m  =  C/z  (31)  ; 

for  one  of  0(1)  sets  of  relatively  prime  (bounded)  non-zero  integers  B  =  BUjU,  C  =  Cw  v. 
Of  special  importance  is  the  case  where  the  left  side  of  (29)  has  a  rational  linear  factor 
in  s,  cr,  in  which  event  for  some  pair  B,  C  the  left  side  of  the  corresponding  equation  (30)  } 

with  A.  =  (C/B)1  contains  a  linear  factor  s'  =  Da'  +  E  with  rational  coefficients  D  ^  0  f 

and  E  that  do  not  depend  on  m  and  fji.  In  this  situation,  we  deduce  from  (31)  that  (28)  f 

holds  identically  whenever  r 

i 
Bs  =  Ems'  =  C/x/  =  CD  via  +  CE^  =  CDa  -f  CE/ji  \ 

and  hence  that  the  rational  substitution 

! 

Bs  =  CDa  +  CE\i,  Em  =  C/x- 


transforms  mF(m,  s)  into  ^GG-i,  a).  Therefore,  compounding  this  substitution  with  (26) 

and  the  inverse  of  (25)  in  the  obvious  order,  we  are  provided  with  a  rational  automorphic  « 

of  /  that  takes  r,  s  into  p  ,  a  ,  whence  any  solutions  of  (29)  arising  in  this  way  flow  from  / 

associated  points  (r,  s),  (p,  a)  and  are  of  a  type  not  counted  in  T(ri)  and  its  constituent  \ 

parts. 

In  combination  with  the  special  features  just  identified,  the  main  instrument  in  our  treat- 
ment of  equation  (29)  is  an  important  theorem  due  to  Bombieri  and  Pila  [1]  that  we  state 
here  as  follows. 


Totally  reducible  binary  forms  255 


Lemma  1.  Let  ^C^,  n)  be  an  irreducible  polynomial  of  degree  8  with  rational  coefficients. 
Then  the  number  of  solutions  ofty(%,  r?)  =  0  in  integers  of  size  not  exceeding  z  is 


O   z+<  (z  >  1), 

where  the  constants  implied  by  the  O  -notation  are  independent  of  the  coefficients  o 


Proved  by  Bombieri  and  Pila  when  the  condition  of  absolute  irreducibility  is  imposed, 
the  result  of  the  lemma.remains  true  if  *!>(£  ,  77)  be  irreducible  but  not  absolutely  irreducible 
because  then  any  integer  solution  is  a  zero  of  an  absolutely  irreducible  factor  of  ^(^,  rj)  of 
the  form 


?)  +  •••  +  *>ff^«,  17)     (^(§,  >?)€Q  B,  >?]), 

where  <y  j  ,  .  .  .  ,  coe  is  a  basis  of  the  field  of  degree  e  >  1  over  which  the  factor  is  defined. 
In  fact  the  zeros  of  this  are  the  common  zeros  of  the  system 


which  belong  to  a  variety  of  dimension  zero  and  limited  degree  since  clearly  ^  ,  .  .  .  ,  ^ 
have  no  common  factor.  This  confirms  our  extension  of  the  Bombieri-Pila  theorem  in  the 
context  of  the  present  work. 

For  the  case  t  =  3  we  shall  need  to  augment  our  armoury  with  an  elementary  estimate 
that  is  sharper  than  Lemma  1  when  *!>(£,  if)  is  a  special  type  of  quadratic.  This  is  as  follows. 

Lemma  2.  Let 


be  an  irreducible  quadratic  polynomial  with  rational  coefficients  having  bounded  denomi- 
nators and  size  not  exceeding  zAl.  Then  the  number  of  zeros  ofW(i-,  rj)  of  size  not  exceeding 


Supposing  first  that  *!>(?,  77)  is  absolutely  irreducible  and  noting  that  we  may  restrict 
attention  to  the  case  where  it  has  integer  coefficients,  multiply  it  by  4a}a2  to  transform  it 
into 


a2  (20  ,f  +  b{)2  +  a{  (2a2rj  +  b2)2  -  (a2b 


with  the  implication  that  a2b\  +  a}b2  -  4a^2c  ^  0.    Hence,  since  the  solutions  of 
])  =  0  are  contained  in  those  of  an  equation  of  the  type 


a3X   +  a4Y   =  a2b   +  a\b   -  4ala2ct 
we  deduce  that  the  solutions  to  be  counted  have  cardinality 
O  |  d  (a2b\  +  afil  -  cj  Iog2^:|  =  O  (z€) 

by  a  familiar  application  of  the  theory  of  quadratic  forms  as  used  for  example  in  our  paper 

[2]. 

The  case  where  ^  is  irreducible  but  not  absolutely  irreducible  is  catered  for  by  the 
argument  in  the  proof  of  Lemma  1  or,  alternatively,  is  easily  handled  a  priori  in  the  present 
framework  by  obvious  reasoning. 


256  C  Hooley 

5.  Estimation  of  T(n)  and  the  first  theorem 

In  treating  the  sum  T(n)  in  (17),  which  we  now  rejoin,  we  shall  first  primarily  address  the 
case  where  i  >  3  and  shall  delay  until  later  a  modified  argument  for  t  =  3  that  largely 
depends  on  Lemma  2  instead  of  Lemma  1,  although  it  should  be  stressed  that  nothing  in 
the  earlier  stages  of  the  reasoning  is  actually  invalid  for  the  latter  case. 
Having  indicated  the  sphere  of  operation,  we  first  suppose  that 

M  <  A2N  (32) 

in  the  notation  of  (13)  and  consider  the  contribution  to  T\(n,  M)  due  to  those  values  of 
r,  s,  p,  a  meeting  its  conditions  of  summation  for  which 

max  |L.(r,j)|  =  |LM(r,j)|andLM(r,j)=7n  (33) 

i      i   M 


for  some  specified  integer  m  of  a  size  between  M  inclusive  and  2M  exclusive.  In  these 
surroundings  the  requirement  that  0  <  /(r,  s)  =  /(/>,  a)  implies  that 


m   = 


<      O      (Lu(r,s),  L  .(p,  a)) 
i</<*  V  7  / 

so  that  at  least  one  factor  (LM(r,  5),  Lv(p,  a))  on  the  right  above  is  not  less  that 

(34) 


the  value  IJL  of  Ly(p,  or)  being  governed  by  the  condition  0  <  |/z|  <  |m|  <  2M  through 

the  definition  of  T[(ri).  Hence,  since  |s|,  \a\  <  2A{M  —  A^M  by  (12),  we  deduce  that  / 


Ti(n,M)<  rB|U(nfM;mf/i),  (35) 

1<W,U<^     0<|m|,|/«|<2Af 
(m./x)^1/* 

where  rw  v(n,  M;  m,  /x  is  the  number  of  solutions  of  (29)  in  integers  s,  a  of  size  not  ex- 
ceeding  A6M  that  do  not  appertain  via  (25)  and  (26)  to  the  association  (r,  s)  ^  (p,  or). 

Let  us  first  depose  of  the  contribution  T*  (n  ,  M)  to  the  right-hand  side  of  (35)  that  relates 
to  values  of  w,  v,  m,  /z  for  which  the  polynomial  mF(m,^)—/AG(/z,  or)  is  reducible.  In  this 
case,  by  (31),  m  and  /x  are  connected  by  an  equation  Bu  ym  =  Cu  v/z  for  one  of  a  finite  set 
of  pairs  of  coprime  non-zero  integers  Bu  v,  CM  y  .  Also,  the  zeros  ofmF(m,s)  —  /xG(/x,  cr) 
are  distributed  among  all  its  irreducible  factors  with  rational  coefficients,  each  such  factor 
of  degree  2  or  more  having  O  (  M^€  \  zeros  in  the  chosen  domain  of  s  and  a  by  Lemma  1. 
On  the  other  hand,  the  zeros  of  any  linear  factors  are  inadmissible  because  we  have  shewn 
earlier  that  they  would  not  meet  the  stipulation  that  (r,  s)  jk  (p,  a).  Consequently 

r,*(n,M)  =  o(M^)  (36) 

by  (35). 
If  we  write 

m  =  dm',  n.  =  dn',  where(m',  /i')  =  1  and3  d  >  M1/*-  (37) 


3  When  rf  >  2A/  all  our  subsequent  calculations  are  true  but  trivial,  the  underlying  sums  being  of  course  empty. 


Totally  reducible  binary' forms  257 

in  the  conditions  of  summation  for  the  remaining  portion  7^  (n,  M)  of  the  sum  in  the  right 
of  (35),  equation  (28)  takes  the  form 


0,  (38) 

which  with  (27)  and  (37)  implies  both  the  congruences 

<$>(m, s)  =  n  [h^n  +  (huki  —  hfi^ s\  ==  0,  mod  /z",  (39) 

and 

F(/x,  CT)  =   n  \h:iji+  (h^k:  —  h:k.\  a\  ==  0,  mod  m"  (40) 

y^   I     '  \          7  J       /        J 

for  certain  coprime  moduli 

m"  =  m'/0,      A*-"  =  At'/a 

derived  from  the  division  of  w'  and  //,  respectively,  by  certain  (small)  positive  divisors 
a  and  or  of  (m;,  h^~l)  and  (/u/,  /zf,'"1)-  Even  though  4>(m,  j1)  and  f(/z,  a)  are  products 
of  rational  linear  factors,  a  full  discussion  of  these  congruences  for  general  composite 
moduli  having  repeated  prime  factors  entails  the  same  sort  of  difficulties  that  attend  the 
general  theory  of  polynomial  congruencies  in  one  variable  as  expounded  by  Nagell  ([4],  ch. 
Ill);  these  difficulties  at  the  present  juncture  would  in  fact  involve  the  prime  divisors  of  the 
discriminants  of  <I>(w,  s)  and  F(/z,  a)  qua  polynomials  ins  and  a  and,  therefore,  ultimately 
and  especially  those  of  the  number  d.  However,  at  the  expense  of  a  balancing  slight 
lengthening  in  procedure,  we  are  able  here  to  circumvent  these  congruential  entanglements 
by  reducing  our  situation  to  one  where  the  moduli  are  square-free. 

Accordingly,  for  integers  m"  and  \JL"  in  (40)  and  (39)  whose  expressions  in  terms  of 
prime  factors  are  stated  as 

m"  =  ±  n  ph,     n"  =  ±  n  urft ,  say, 

p  w 

we  shall  first  use  the  positive  square-free  numbers 


while  later  we  shall  need  numbers  W4,  /Z4,  that  similarly  originate  from  m',  \jt!  and  bear  no 
relation  to  a  and  a;  finally,  for  each  given  number  of  type  m$  or  /Z4,  we  let  m$  =  ms(m^) 
or  ju<5  =  ^5(^4)  denote  positive  numbers  whose  prime  factors  are  divisors  of  77x4  and  /X4, 
respectively.  Then,  the  procedure  being  amply  illustrated  by  reference  to  the  congruence 
(40),  all  solutions  of  this  in  a  satisfy  the  corresponding  congruence  taken  to  the  modulus 
m3,  the  number  of  incongruent  solutions  of  which  we  denote  by  /c(m^).  Since  K(m^)  is 
multiplicative,  it  suffices  to  consider  K  (p)  when  p  /  D  because  in  the  contrary  instance  we 
are  content  with  the  trivial  estimate  *:(/?)  <  p.  Thus  we  may  assume  that  the  coefficient 
of  a  in  each  factor  of  F(/z,  a)  in  (40)  is  indivisible  by  p  and  deduce  that  each  such  factor 
is  divisible  by  p  when  a  belongs  to  just  one  residue  class,  mod  p.  Consequently,  we  see 
that  tc(p)  does  not  exceed  p  or  t  —  \  according  as  p\D  or  p  /  D  and  conclude  both  that 


=  0    (€  -  I)w(m3)    =  0  (m€3)  (41) 


258 


CHooley 


and  that  a  similar  result  holds  for  the  other  congruence  (39). 

In  the  current  circumstances  the  solutions  of  (38)  in  s,  a  have  been  shewn  to  be  distributed 
into  O(M€)  sets,  each  of  which  consists  of  pairs  of  numbers  of  the  type 

s=sQ+s{iJL39     a=aQ  +  a{m3  (42) 

for  certain  positive  numbers  $0,  aQ  not  exceeding  £iv  m3  respectively.  The  relevant  con- 
tribution to  TUtV(n,  M;  m,  /x)  corresponding  to  each  set  is  then  obtained  by  substituting 
(42)  in  (38)  to  obtain  an  irreducible  equation  is  s{,  cr{  of  degree  I  —  1,  of  which,  being 
constrained  by  the  inequalities 


<  < 

/x3  /i3 

the  number  of  qualifying  pairs  of  zeros  is 
O    M  max     —  ,  —  =  O 


< 


m 


m 


/  1       1  N 
(  —  ,  —  ) 

V/^3    m3/ 


max 


/  1       1   xVtf-iA 
(  —  ,  —  )  ) 

\M4     m4/  / 


by  Lemma  1.  Therefore,  taking  stock  after  this,  (41),  (35),  and  (37),  we  conclude  that 

max,  (43) 


/  i       1  \ 
(JL,JL) 

\m4     ^4/ 


from  which  the  estimate  for  7^(n,  Af)  will  flow  by  the  way  of  the  simple 

Lemma  3.  Let  q  denote  any  positive  integer  composed  entirely  of  prime  factors  (possibly 
repeated)  that  divide  a  given  positive  number  (possibly  I)  not  exceeding  z.  Then  the  number 
ofq  not  exceeding  z  is  O  (zf). 

This  is  a  special  case  of  the  Lemma  4  in  [3].  Evidently  the  inner  sum  in  (43)  does  not 
exceed 


2     E 

Q<m',ii'<2M/d 


0<m'<2M/d 


0<m4<2M/d 


-i/C^-D+A 
-i/(^-D     ) 


with  the  implication  that 


Totally  reducible  binary  forms  259 

wherefore  on  taking  this  with  (36)  we  have 

(44) 


for  M  <  A2N  as  in  (32). 

Similar  principles  are  successful  for  the  estimation  of  T{(n,  M)  in  the  complementary 
range  A^N  <  M  <  A4n1/^~"1)  but  are  less  straightforward  to  apply.  Now,  by  the  definition 
of  T\(n,  M)  and  (16),  we  first  modify  (33)  by  using  the  (unique)  subscript  u  for  which 
Lu(r,  s)  equals  a  non-zero  number  m  whose  size  does  not  exceed 

MI  =  A5n/Mt-1  <  M,  (45) 

even  though  the  previously  used  inequalities  for  s,  a  are  still  valid.  Next,  following  previous 
thinking,  we  find  there  is  a  subscript  v  for  which  the  number  JJL  =  Ly(/>,  a)  possesses  the 
properties 

m\l/t     and     \fi\<2M.  (46) 

This  clears  the  way  for  a  reconsideration  of  T^(nt  M)  because  the  assessment 

(47) 
\  / 

is  a  corollary  of  (3 1). 

The  new  surroundings  affect  the  sums  bounding  Z^(n,  M)  more  in  regard  to  the  condi- 
tions of  summation  than  the  summands  therein.  In  the  former  we  still  have  the  first  parts 
of  (37)  but  replace  the  last  part  by  d  >  \m\{fi  with  the  result  that 

0<in'<Mf/d,     0<mf<di~i1     0<^<2M/d. 

Hence,  emulating  the  derivation  of  (43),  we  have 

/ 

r/(n,M)     =     O 


=     O          (*~1)+€J'  say,  (48) 

V  d         dl 

and  then  go  on  to  treat  ^    for  the  two  cases  d  >  M\I<L  and  d  <  M\ft  .  In  the  earlier 
instance,  by  Lemma  3  and  then  (45),  we  get 

1 


E 
d- 


2Mj 
~~~d~ 


260 


C  Hooley 


=  O 


and  in  the  latter  instance  similarly  obtain 


after  replacing  M//d  by  d£~3  as  a  limit  for  yu/  in  the  summation.  Therefore  equation  (48) 
can  be  developed  into 


Af)  =  O 


which  in  combination  with  (47)  furnishes  us  with  the  estimate 
r,(n,  Af)  =  O 


(49) 


(50) 

that  is  the  complement  of  (44)  for  the  range  A2N  <  M  <  A4nly/(£~"l\ 

The  first  part  of  our  initial  theorem  follows  at  once  because  the  exponent  of  M  in  (50)  is 
negative  when  t  >  3.  Indeed,  by  embodying  (44)  and  (50)  in  (20)  and  then  recalling  (10), 
we  deduce  at  once  that 


or 


T(n)  =  O 

=  o 

T(n)  = 


Q 


and  so  estimate  v(n)  because  of  (19). 

When  t  =  3  it  is  only  the  last  part  of  the  analysis  leading  to  (50)  that  fails  to  be  effective. 
Yet,  if  we  take  the  opportunity  that  arises  here  to  use  Lemma  2  instead  of  Lemma  1  ,  we  can 
not  only  produce  a  workable  alternative  to  (50)  but  find  all  the  relevant  revised  estimates 
in  the  work  combine  to  yield  the  board 


7»  =  O 


(52) 


that  is  better  than  what  would  be  got  by  formally  putting  -£  =  3in(51).  Moreover,  although 
the  general  structure  of  the  previous  method  is  retained,  there  is  the  important  simplification 
that  all  references  to  the  congruences  (39)  and  (40)  and  to  Lemma  3  are  avoided. 


Totally  reducible  binary  forms  261 

To  indicate  briefly  what  is  to  be  done,  we  note  that  revisions  are  only  needed  when  the 
polynomial  m'F(m,  s)  —  //GO,  a)  in  5-,  a  is  irreducible  and  hence  when  it  has  O  (M6) 
zeros  of  size  not  exceeding  2  A  {M  by  Lemma  2.  Hence  we  can  improve  (43)  to 


=  0     Me 

t/>A/V«  0<m',ii'<2M/d 

and  thus  (44)  to 


Similarly  (48)  is  replaced  by 


=  o 

()<me<Mf/d,cle-1 


which  leads  to  the  counterpart 

!•*(»,  M)  =  o 


of  (49).  The  exponent  of  M  in  this  being  —  ~,  we  then  sum  over  M  as  before  to  obtain  (52) 
in  place  of  (5  1  )  for  t  =  3  and  thus  complete  the  proof  of 

Theorem  1.  Let  f(x,  y)  be  a  totally  reducible  binary  form  of  degree  t  with  integral  coef- 
ficients and  non-zero  discriminant.  Then,  if\)(n)  be  the  number  of  positive  integers  up  to 
a  large  number  n  that  have  essentially  more  than  one  representation  by  /,  we  have 


v(n)  ==  O 

where 

l/^2,  if  I  =  3, 


(I  -  2)/t2(t  -  1),  //  t  >  3. 

6.  The  second  theorem 

To  shew  it  is  exceptional  for  a  number  to  be  represented  by  /(jc,  y)  in  essentially  more  than 
one  way  we  must  foreshadow  a  simple  aspect  of  our  following  paper  by  defining  r  (m)  to 
be  the  number  of  ways  of  expressing  the  positive  number  m  by  /(*,  y),  where  of  course 

r(m)  =  O  {dt(m)}  =  O  (m6)  .  (53) 

Let  us  now  take  one  of  the  semi-infinite  triangular  regions  described  in  §3  in  which  /(jc,  y) 
is  positive  and  consider  points  (x  ,  y  )  within  having  integral  coordinates  for  which  |  x  \  ,  |  y  |  < 
A8nl/€  for  a  suitably  small  positive  constant  A8.  ThenO  <  /(*,  y)  <  n  for  all  these  points, 
the  cardinality  of  which  exceeds  A9n2^  by  a  standard  lattice  point  argument.  Consequently 

r(m)  >  A8n2/* 


262  C  Hooley 

and  thus,  by  (53), 

T(n)  >  A(0/r€     T  r(m) 


whence,  on  comparing  this  with  Theorem  1,  we  gain  the  following. 

Theorem  2.  Almost  all  the  positive  numbers  represented  by  the  form  /(;c,  y)  in  Theorem 
1  are  represented  thus  in  essentially  only  one  way. 

References 

[1]  Bombieri  E  and  Pila  J,  The  number  of  integral  points  on  arcs  and  ovals,  Duke  Math.  J.  59 

(1989)337-357 
[2]  Hooley  C,  On  the  representation  of  a  number  as  the  sum  of  a  square  and  a  product,  Math. 

Zeitschr.  69  (1958)  21  1-227 

[3]  Hooley  C,  On  binary  cubic  forms:  II,  J.  ReineAngew.  Math.  521  (2000)  185-240 
[4]  Nagell  T,  Introduction  to  Number  Theory  (Stockholm:  Almquist  and  Wiksell)  (1951) 


Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  Ill,  No.  3,  August  2001,  pp.  263-269. 
©  Printed  in  India 


Stability  of  Picard  bundle  over  moduli  space  of  stable  vector 
bundles  of  rank  two  over  a  curve 


INDRANIL  BISWAS  and  TOMAS  L  GOMEZ 

School  of  Mathematics,  Tata  Institute  of  Fundamental  Research,  Homi  Bhabha  Road, 

Mumbai  400  005,  India 

E-mail:  indranil@math.tifr.res.in;  tomas@math.tifr.res.in 

MS  received  14  September  2000 

Abstract  Answering  a  question  of  [BV]  it  is  proved  that  the  Picard  bundle  on  the 
moduli  space  of  stable  vector  bundles  of  rank  two,  on  a  Riemann  surface  of  genus  at 
least  three,  with  fixed  determinant  of  odd  degree  is  stable. 

Keywords.     Picard  bundle;  Hecke  lines. 

0.  Introduction 

Let  X  be  a  compact  connected  Riemann  surface  of  genus  g,  with  g  >  3.  Let  £  be  a 
holomorphic  line  bundle  over  X  of  odd  degree  d,  with  d  >  4g  —  3.  Let  M  denote  the 
moduli  space  of  stable  vector  bundles  E  over  X  of  rank  two  and  f\"E  =  £.  Take  a 
universal  vector  bundle  E  on  X  x  M.  Let  p  :  X  x  M  — >  M  be  the  projection.  The 
vector  bundle  P  :=  p^£  on  M  is  called  the  Picard  bundle  for  M.  In  [BV]  it  was  proved 
that  the  Picard  bundle  P  is  simple,  and  a  question  was  asked  whether  it  is  stable.  In  [BHM] 
a  differential  geometric  criterion  for  the  stability  of  P  was  given.  But  there  is  no  evidence 
for  this  criterion  to  be  valid. 
In  Theorem  3.1  we  prove  that  the  Picard  bundle  P  over  M  is  stable. 

1.  Preliminaries 

In  this  section  we  prove  some  lemmas  that  will  be  needed. 

A  vector  bundle  E  of  rank  two  and  degree  d  is  called  superstate  if  for  every  subline 
bundle  L  of  E  the  inequality 

deg(L)  <  -  -  I 

is  valid.   Clearly,  a  superstable  bundle  is  stable.   The  first  lemma  ensures  existence  of 
superstable  bundles. 

Lemma  1.1.      There  is  a  nonempty  open  subset  U  of  M  corresponding  to  superstable 
bundles. 

Proof.  Here  we  need  g  >  3.   Let  T  be  the  subset  of  M  of  vector  bundles  that  are  not 
superstable,  i.e.,  E  €  T  if  and  only  if  there  exists  a  subline  bundle  L  such  that  deg(L)  > 


264  Indranil  Biswas  and  Tomds  L  Gomez 

(d  -  l)/2.  Since  E  is  stable,  deg(L)  <  d/2,  and  since  d  is  odd,  deg(L)  =  (d  -  l)/2. 
There  is  a  short  exact  sequence 

0  _*  L  —  *  £  —  *  f  (g)  L"1  —  *  0. 

Note  that  the  quotient  is  torsion  free  (hence  a  line  bundle)  because  E  is  stable  and  L  has 
degree  (d-  l)/2. 

Therefore,  all  vector  bundles  in  7  can  be  constructed  by  choosing  a  line  bundle  L 
of  degree  (d  —  l)/2  together  with  an  extension  class  in  ExtJ(f  <8>  L"1,  L).  It  follows 
immediately  that  T  is  a  closed  subset  of  M  with  dimension 

dim(T)  <  g  +  h[(r{  <8>  L2)  -  1  =  £  -  x(rl  0  L2)  -  1 
=  2g  -  I  <  3g  -  3  =  dim(M), 

and  hence  the  complement  U  :=  M\T  is  open  and  nonempty.  D 

Lemma  1.2.  Choose  m  distinct  points  {x\,  ...txm]  C  X,  with  m  >  d/2.  Let  E  €  M 
be  a  vector  bundle  and  0  ^  s  €  7/°(E)  a  nontrivial  section.  Then  s  cannot  simultaneously 
vanish  at  all  the  chosen  points  {*!,...,*«,}. 

Proof.  If  s  vanishes  at  all  chosen  points  x\  ,  .  .  .  ,  ;c,n  ,  then  5-  :  O  —  >  E  factors  as 
s  :  O  —  >  £(-!>)  <-»  £, 

where  D  is  the  divisor  D  —  x\  H  -----  h  */rt  .  Since  deg  E(—D)  =  d  —  2m  <  0,  the  stability 
condition  of  E  forces  s  to  be  the  zero  section.  D 

2.  Hecke  lines 

Let  U  C  M  be  the  open  subset  of  superstable  vector  bundles  (Lemma  1.1).  Take  a  point 
x  e  X.  Let  E  e  U  and  /  C  Ex  a  line  in  the  fiber  of  E  at  x  (equivalently,  /  e  f(Ex)). 
Define  the  vector  bundle  W  by 

0  —  *  W(-jc)  —  •>  £  —  *  EJC//  —  >  0. 

The  vector  bundle  W(—x)  is  called  the  Hecke  transform  of  £  with  respect  to  x  and  /.  The 
exact  sequence  implies  A2^  =  ?  ®  £>(*).  The  vector  bundle  W  is  stable.  Indeed,  a 
line  subbundle  L  of  W  is  realized  as  a  subline  bundle  of  E(x)  using  the  homomorphism 
W  —  >•  £(*):  Now  the  superstability  condition  of  E  says 


In  other  words,  W  is  stable. 

We  can  reconstruct  back  E  from  W  by  doing  another  Hecke  transform,  and  E  is  given 
as  the  middle  row  of  the  following  commutative  diagram: 


Stability  ofPicard  bundle  265 

0  0 

T  t 

0     — *       Ex/l       -+        Wx        — >     C,     — »     0 

T  t  I 

0     — *         E         — *         W         -^V     Cx     — ^     0  (1) 

T  T 

W(-jc)     =     W(-x) 

T  T 

o  o 

Here  Cx  is  the  skyscraper  sheaf  at  x  with  stalk  C.  Instead  of  /o  :  W  — >  Cx  we  may 
consider  an  arbitrary  nontrivial  homomorphism 


/  e  Hom(W,C^)  =  Hom(Wx,Cx)  =  W* 
and  define  Ef  as  the  kernel 

Q—+Ef—»W-£+Cx-+.0.  (2) 

This  way  we  obtain  a  family  of  vector  bundles  parametrized  by  the  projective  line  P(W^), 
with  EfQ  =  E.  More  precisely,  there  is  a  short  exact  sequence  on  X  x  P(W/), 

o^£->  *jw  -4  0^,^(1)  -*  o, 

where  TTX  :  X  x  P(  W^)  —  >-  X  is  the  projection  to  X.  It  has  the  property  that  if  /  e  W^ 
and  we  restrict  the  exact  sequence  to  the  subvariety  X  x  [/]  =  X  of  X  x  P(W^),  then  a 
sequence  isomorphic  to  (2)  is  obtained. 

For  every  /  e  W^9  the  vector  bundle  Ef  is  stable.  Indeed,  if  L  is  a  subline  bundle  of 
Ef,  then  by  composition  with  the  homomorphism  Ef  —  >  W  in  (2)  it  is  a  subline  bundle 
of  W.  The  stability  condition  for  W  says  that  deg(L)  <  (d  +  l)/2.  Since  d  is  odd  this  is 
equivalent  to 


Note  that  if  £  is  stable  but  not  superstable,  then  W  is  semistable  but  not  necessarily  stable. 
The  semistability  condition  is  not  enough  to  ensure  the  stability  of  Ef  for  each  /. 

The  universal  property  of  the  moduli  space  M  gives  a  morphism  <p  :  P(W^)  —  >_  M 
for  the  family  E. 

DEFINITION  2.1 

The  data  consisting  of  the  pair  (P(W^),  <p)  is  called  the  Hecke  line  associated  to  the  triple 
(£,*,/)• 

Since  <p  is  determined  by  W  and  P(W^),  the  projective  line  P(W^)  will  also  be  called  a 
Hecke  line.  The  Hecke  line  P(  W/)  will  also  be  denoted  by  PE%  *,  /  or  simply  by  P  if  the  rest 
of  the  data  is  clear  from  the  context.  Note  that  there  is  a  distinguished  point  [/o]  E  P(W^) 
that  maps  to  E  e  M. 

For  any  /  e  P(W^),  let  //  denote  the  kernel  of  the  homomorphism  (£/)*  —  >  Wx  of 
fibers  in  (2).  Clearly,  the  images  of  the  two  Hecke  lines  Pgf  Xy  /  and  P£/>  Xi  if  in  M  coincide. 


266  Indmnil  Biswas  and  Tomds  L  Gomez 

Therefore,  for  each  E  e  M  ,  there  is  a  three  parameter  family  of  Hecke  lines  whose  image 
contains  E.  On  the  other  hand,  if  we  identify  two  Hecke  lines  if  their  images  in  M  coincide, 
then  through  each  point  of  M  there  is  a  two  parameter  family  of  rational  curves  defined  by 
Hecke  lines. 

Since  the  morphism  (p  is  given  by  the  universal  property  of  the  moduli  space,  the  pullback 
of  the  universal  bundle  8  on  X  x  M  to  X  x  P  by  the  map  idx  x  <p  is  isomorphic  (up  to  a 
twist  by  a  line  bundle  coming  from  P)  to  E.  In  other  words,  there  is  an  integer  k  such  that 

0  —  »  (idx  x<p)*£  —  >  Wm0p(k)  —  +  OXXP(k+l)  —  >  0  (3) 

is  an  exact  sequence  of  sheaves  on  X  x  P;  Op  (I)  is  the  tautological  line  bundle  on 
p  =  P(W^).  Applying  (7rp)*,  where  np  is  the  projection  of  X  x  P  to  P,  the  following 
sequence 

0  —  >  p*P  —  *  ff°(1V)  ®  £?/>(*)  —  »  £/>(£  +  1)  —  »  0  (4) 

on  P  is  obtained,  where  P  is  the  Picard  bundle.  Since  d  >  4g  —  3,  the  stability  condition 
ensures  that  H  l  (X,  E'}  vanishes  for  every  E'  e  M.  -* 

Let  W  denote  the  rank  of  P.  The  following  proposition  describes  the  pullback  (p*P.  | 

PROPOSITION  2.2  1 

The  pullback  (p*P  of  the  Picard  bundle  P  to  the  P  =  PE,  Xj  /  satisfies  ' 

<p*P  ^  0P(k)®N~l  ©  Op(Jk  -  1).  (5)  ' 

Hence  (p*P  has  a  canonical  subbundle 
0P(k)®N~l  ^  V  <->  <p*P. 

LetV  C  HQ(X,  E)  be  the  fiber  of  this  subbundle  over  the  distinguished  point  [fo]  e  P. 

Thens  e  Vifandonlyifs(x)  el.  If 

Proof.  Grothendieck's  theorem  [Gr]  says  that  a  vector  bundle  on  P1  is  holomorphically  i 

isomorphic  to  a  direct  sum  of  line  bundles.  Hence  » 

(p*P  ^  0P(a{)®"-®0P(aN). 

The  sequence  (4)  gives  A°(  W)  =  1  +  N,  5Z  ai  =  Nk  —  1  and  a/  <  A:  for  all  i  .  Combining  f 

these,  (5)  is  obtained  immediately. 

Now  we  are  going  to  identify  the  subbundle  V.  From  (3)  the  following  commutative 
diagram  is  obtained 

f 

0    —  *        (idxx?>)*£        —  >        W®0p(k)        —  *     Oxxp(k+l)    —+    0  * 

T  T 


T  T 

0  0 


Stability  ofPicard  bundle  267 

applying  (JT^)*  we  obtain  the  following  commutative  diagram  on  P: 
—  *  <P*P  —  >       HQ(W)®Op(k)     —  >    Op(k+l)    —  >     0 

T  T 


T  T 

0  0 


SSnce  <p*P  =  CM*)®"-1  0  OP(A:  -  1),  we  deduce  that 
V  =  H*(W(-x))®0P(K)  C  ^*P. 


Let  V  denote  the  fiber  of  V  at  the  point  [/0]  €  P.  So,  V  c  #°(£).  Now,  s  e  V  if  and 
if  j  e  //°(W(-jc))  C  #°(£).  Finally,  taking  global  sections  for  the  diagram  (2)  it  is 
e«y  to  see  that  this  is  equivalent  to  the  condition  that  s(x)  e  I.  This  completes  the  proof 
o:flfthe  proposition.  D 

Proposition  2.2  has  the  following  corollary. 

OOOROLLARY  2.3 

TWe  morphism  <p  is  a  nonconstant  one. 

Indeed,  if  (p  were  a  constant  map,  then  the  vector  bundle  (p*P  would  be  trivial. 

3,    Main  theorem 

In  ;  this  section  we  will  prove  the  main  theorem  of  this  paper. 


.1.  Let  P  be  the  Picard  bundle  on  the  moduli  space  M  of  stable  bundles  of  rank 
IM£B  and  fixed  determinant  of  odd  degree  d  with  d  >  4g  —  3.  Then  P  is  stable. 

fwxof.  Since  P  is  a  vector  bundle,  to  check  stability  it  is  enough  to  consider  reflexive 
subosheaves  of  P.  Let 


le^areflexive  subsheaf  ofrankr  <  N  =  rankCP).  Fix  m  distinct  points  x\,  ...  ,xm  in  X, 
ifiCHim  >  d/2. 
We  need  the  following  lemma  for  the  proof  of  the  theorem, 

\s.mma  3.2.  There  is  a  nonempty  open  set  of  M  such  that  ifE  is  a  vector  bundle  cor  re- 
to  a  point  of  that  open  set,  then  E  has  the  following  four  properties: 


(i)    E  is  superstable\ 
(ii)    ?  is  locally  free  at  E\ 
(iii)    TE  -*  PE  is  an  injection', 

(nr)    Letxi  be  one  of  the  fixed  points  andl  any  line  on  EXi  .  Let  P  =  PE,  Xi  ,  /  be  the  associated 
Hecke  line.  Then  T  is  locally  free  at  all  points  of  the  image  of<p:P  —  >  M. 


268  mdranilBiwas  and  TomdsL  Gomez 

**,  m  k  satisfied  is  open  and  nonempty  by  Lemma 

Proof.  Tne  subset  U  of  M  ^^^^l  satisfied  and  U"  C  U>  the  subset 
1 T.  Let  U'  C  U  be  the  subset  where  ^PW**^  .g  a  nonempty  open  subset  of  M. 
where  furthermore  property  (in)  is  sausn  .  ^  ^^^  ^  since  ^  ig  reflexive> 

s  c  M  denote  me  subvanety^wn^^ofaiiHeckei.nesp£^;)WhenEruns 


codimSj  >  3  -  1 
Finally  consider  the  union 


From  the  fixed  set 
section  s  does  not  ;  vamsh  at  «.  The 
Let  Z  C  Ex/  be  a  line  such  that  s  (x  ,) 
with  this  data. 


^ 


by  1-2 

p  =  ^  defined 


free  on  all  points  of  the  image  of 


.g 


0  (B»"-'  -  V  c 


Propo 


By  Grothendieck's  theorem 


j>O.Now, 


Stability  ofPicard  bundle  269 

References 

[BY]  Balaji  V  and  Vishwanath  P  R,  Deformations  ofPicard  sheaves  and  moduli  of  pairs,  Duke 

Math.  7.76(1994)773-792 

[BHM]  Brambila-Paz  L,  Hidalgo-Solis  L  and  Muci  no-Ray  mo  ndo  J,  On  restrictions  of  the  Picard 
bundle.  Complex  geometry  of  groups  (Olmue,  1998)  49-56;  Contemp.  Math.  240;  Am. 
Math.  Soc.  (Providence,  RI)  (1999) 
[Gr]  Grothendieck  A,  Sur  la  classification  des  fibres  holomorphes  sur  la  sphere  de  Riemann. 

Am.  J.  Math.  19  (1957)  121-138 

[NR]  Narasimhan  M  S  and  Ramanan  S,  Moduli  of  vector  bundles  on  a  compact  Riemann  surface. 
Ann.  Math.  89(1969)19-51 


Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  Ill,  No.  3,  August  2001,  pp.  271-291. 
©  Printed  in  india 


Principal  G-bendles  on  nodal  curves 


USHA  N  BHOSLE 

School  of  Mathematics,  Tata  Institute  of  Fundamental  Research,  Homi  Bhabha  Road, 

Colaba,  Mumbai  400  005,  India 

E-mail:usha@math.tifr.res.in 

MS  received  2  June  2000;  revised  30  October  2000 

Abstract.  Let  G  be  a  connected  semisimple  affine  algebraic  group  defined  over 
C.  We  study  the  relation  between  stable,  semistable  G-bundles  on  a  nodal  curve  Y 
and  representations  of  the  fundamental  group  of  Y.  This  study  is  done  by  extending 
the  notion  of  (generalized)  parabolic  vector  bundles  to  principal  G-bundles  on  the 
desingularization  C  of  Y  and  using  the  correspondence  between  them  and  principal 
G-bundles  on  Y.  We  give  an  isomorphism  of  the  stack  of  generalized  parabolic 
bundles  on  C  with  a  quotient  stack  associated  to  loop  groups.  We  show  that  if  G  is 
simple  and  simply  connected  then  the  Picard  group  of  the  stack  of  principal  G-bundles 
on  Y  is  isomorphic  to  ©mZ,  m  being  the  number  of  components  of  Y. 

Keywords.     Principal  bundles;  loop  groups;  parabolic  bundles. 


0.  Introduction 

Let  G  be  a  connected  semisimple  affine  algebraic  group  defined  over  C.  Let  Y  be  a  reduced 
curve  with  only  singularities  ordinary  nodes  yj  ,  j  =  1,  ...,/.  Let  Yit  i  =  1,  .  .  .  ,  /  be  the 
irreducible  components  of  Y  and  C/  the  desingularization  of  F/  .  Let  C  denote  the  disjoint 
union  of  all  C,-.  We  introduce  the  notions  of  stability  and  semistability  for  principal  G- 
bundles  on  Y  (§2).  If  Y  is  reducible  these  notions  depend  on  parameters  a  =  (a\,  .  .  .  ,  a/). 
The  study  of  G-bundles  on  Y  is  done  by  extending  the  notion  of  (generalized)  parabolic 
vector  bundles  [Ul  ]  to  generalized  parabolic  principal  G  -bundles  (called  GPGs  in  short)  on 
the  curve  C  and  using  the  correspondence  between  them  and  principal  G  -bundles  on  Y  (2.4, 
2.  1  1).  We  study  the  relation  between  stable,  semistable  G-bundles  and  representations  of 
the  fundamental  group  of  Y  .  Let  p  :  n\(Y)  ->  G  be  a  representation  of  the  fundamental 
group  jr^F)  of  F  in  G.  For  i  =  1,  ...,/,  let  //  :  ^i(F/)  ->  JTx(F)  be  the  natural  maps, 


Theorem  1.  (I)  IfY  is  irreducible  andp  \  it\  (C)  is  unitary  (resp.  irreducible  unitary)  then 
the  principal  G  -bundle  on  Y  associated  to  p  is  semistable  (resp.  stable).  The  converse  is 
not  true. 

(II)  IfY  is  reducible  then  there  exist  infinitely  many  I-tuples  of  positive  rational  numbers 
a\,  .  .  .  ,  a  i  with  Y^ai  =  1»  depending  only  on  the  graph  of  Y  and  g(Cj)  such  that  for 
a  =  (a\  ,  .  .  .  ,  a/)  the  following  statements  are  true. 

(1)  IfPd  ==  Pi  I  ^iCQ)  ore  unitary  representations  for  all  i,  then  the  principal  G-bundle 
T  on  Y  associated  to  p  is  a  -semistable. 


272  Usha  N  Bhosle 


(2)  Ifpci  are  irreducible  unitary  representations  for  all  i,  then  the  principal  G  -bundle  jP  < 

associated  to  p  is  a  -stable.  : 
Let  Aff/k  be  the  flat  affine  site  over  the  base  field  k  =  C,  i.e.    the  category  of 
fc-algebras  equipped  with  fppf  topology.  Let  R  denote  a  ^-algebra,  C^R  :=  C{  x  spec  R 

and  C*  =  C*x  spec  R.  For  each  i,  fix  a  point  /?/  6  C;-  such  that  /?/  maps  to  a  smooth  *  I 

point  of  Y.  Let  q\  be  a  local  parameter  at  the  point  Pi,i  =  1,  ...,/.  Let  LG,;  denote  I 


the  fc-group  defined  by  associating  to  R  the  group  G(tf(0/)).   Let  L+  f  (resp.   L§')  be  • 

the  &-group  defined  by  associating  to  R  the  group  G(#[[^]])  (resp.  G('r  (C?^,  Oc^))). 

Define  LG  =  n^W-  ^  =  UtLc.r  LG  =  H4'  -  Let 

f[G(C).  ! 


The  indgroup  L£  acts  on  G^.  Let  L^\Q^^  be  the  quotient  stack.  Let  Bun^P^  denote 
the  stack  of  GPGs  on  C  (this  is  isomorphic  to  the  stack  of  principal  G -bundles  on  F.) 

Theorem  2.  There  exists  a  canonical  isomorphism  of  stacks 


Moreover  the  projection  map  Q^^  ~~^  Bun^^  w  locally  trivial  for  etale  topology. 

Theorem  3.I/G  is  a  simple,  connected  and  simply  connected  affine  algebraic  group  then 
(1) 

Pic(Bun^)  «  e/  Z. 

(2)  IfY  is  irreducible  and  C  has  genus  >  2,  then 
ss  «  Z, 


where  ss  denotes  semistable  points. 

The  moduli  spaces  of  principal  G  -bundles  on  singular  curves  are  not  complete.  In  case  ^  - 

G  =  GL(n)  (resp.   G  =  O(ri)t  Sp(2n))  the  compactifications  of  these  moduli  spaces  IT 

were  constructed  as  moduli  spaces  of  torsionfree  sheaves  (resp.  orthogonal  or  symplectic 
sheaves)  on  Y  .  For  a  general  reductive  group  G  neither  the  moduli  spaces  nor  the  compacti- 
fications have  been  constructed  on  Y  yet.  One  way  to  construct  (normal)  compactifications 
of  these  moduli  spaces  is  to  use  GPGs  on  C  ,  for  this  one  needs  a  good  compactification  of  [ 

G.  IncaseG  isGL(n),  SL(n),  0(n)  or  Sp(2n)  we  use  a  compactification  F  of  G  obtained  ! 

by  using  the  natural  representation  and  construct  the  normal  compactifications  of  moduli  J 

spaces  ([Ul,  U2,  U4]).  In  case  G  is  of  adjoint  type  we  use  the  good  compactification  F  of 
G  defined  by  Deconcini  and  Procesi.  We  define  'a  compactification'  Bun^P"  of  Bun§^ 
using  F  and  show  that  it  is  isomorphic  to  the  quotient  stack  L^\QG  c  x  ]~]  .  F.  We  prove 
that  if  further  G  is  simple  and  simply  connected  then  Pic  Bung*"  «  0;Z  0  0;-Pic  F  ^ 

(Theorem  4).  '  f 

1.  Quasiparabolic  bundles 

1.1.  Notations.  Let  the  base  field  be  C  (or  an  algebraically  closed  field  of  characteristic 
0).  Let  /,  /  be  natural  numbers.  Let  F  be  a  connected  reduced  (projective)  curve  with 


Principal  G  -bundles  on  nodal  curves  273 

ordinary  nodes  as  singularities.  LetF/,  /  =  !,...,  7  be  the  irreducible  components  of  Y.  Let 
y'  =  y  -  {singular  set  of  Y },  Y!  ~  Y!  O  Y-t  for  all  i.  Let  C  be  the  partial  desingularization 
of  Y  obtained  by  blowing  up  nodes  }'y ,  j  =  1,  . . . ,  J.  Assume  that  C  =  []{C;  (a  disjoint 
union).  Let  C(  =  C/  —  sing(C/).  Fix  an  orientation  of  the  (dual)  graph  of  Y.  In  the  graph 
of  Y,  >'y  corresponds  to  an  edge.  The  initial  and  terminal  points  of  the  edge  correspond  to 
curves  F/Q-)  and  Yt^  respectively,  one  has  /(;')  =  t(j)  if  the  edge  is  a  loop.  Let*/  6  C/(y) 
andzy  6  Q  (y)  be  the  two  points  of  C  mapping  to  y>j  e  Y  and  Dj  =  Xj  +Zjt  j  =  1, ...,/. 
For  each  j,  Dj  is  an  effective  Cartier  divisor  on  C  supported  outside  the  singular  set  of  C, 
We  remark  that  the  parabolic  structure  we  shall  define  in  1.2,  1.4  depends  only  on  these 
divisors  and  not  on  the  choice  of  orientation.  Let  G  denote  an  affine  connected  semisimple 
algebraic  group  over  C  (or  an  algebraically  closed  field  of  characteristic  zero).  Let  g  denote 
the  Lie  algebra  of  G ,  n  =  dim  g.  A  principal  G-bundle  E  on  C  is  an  /-tuple  (Z^)>  E\  being 
a  principal  G-bundle  on  Cr . 

DEFINITION  1.2 

A  quasiparabolic  structure  GJ  on  E  over  the  divisor  Dj  consists  of  a  G -isomorphism 
aj  :  Ei(n,xj  -*  Et(j),Zj  where  E/iJC  denotes  the  fibre  of  E{  at*.  Let  or  be  the  ./-tuple  (ay)y, 
then  (J5,  a)  is  called  a  quasiparabolic  G-bundle,  called  a  QPG  in  short. 

Remark  1.3.  A  family  (£,  (cry))  of  QPGs  consists  of  a  family  of  principal  G -bundles  £  -> 
C  x  T  together  with  an  isomorphism  of  G -bundles  oy  :  £  \XjxT~*  £  \zjxT  for  each 
j  =  1, . . . ,  J.  Given  a  family  of  QPGs  (5,  (ay))  -»  C  x  r  and  a  representation  p  : 
G  ~>  GL(V)  one  can  associate  to  it  a  family  (£(V),  F/(V))  -»•  C  x  T  of  generalized 
parabolic  vector  bundles  [Ul]  as  follows.  £(V)  =  £  xp  V  is  a  family  of  vector  bundles. 
For  each  7,  or,-  induces  OTV|I/  :  £(V)  |  ^  x  T  ->  f  (V)  |  zy  x  T.  Let  F;-(V)  =  graph 
j  in  f  (V)  I  xj  x  T  e  f  (V)  |  zj  x  T.  Then  F/(V)  and  (2j(V)  =  (f  (V)  I  ^  x  T 
zy  x  T)/Fj(V)  are  vector  bundles  on  T  of  rank  =  dim  V. 

1.4.  Let  a  be  a  real  number,  0  <  a  <  1.  Taking  p  the  adjoint  representation  of  G  we 
get  the  associated  vector  bundle  £(g).  Then  £(g)  is  the  adjoint  bundle  of  E  and  we  often 
denote  it  by  Ad£.  The  isomorphism  GJ  gives  an  isomorphism  E(g)Xj  — >  E(g)z.  and  hence 
determines  an  n -dimensional  subspace  of  E(g)Xj  ®  E(g)Zj  =  g  ©  g  again  denoted  by  or,-. 
Let  tj  €  Endc(g  ®  g)  such  that  T;-  acts  on  cr;-  by  ot.Id  and  ty  restricted  to  a  complement 

of  cry  in  g  0  g  is  zero.  With  respect  to  a  suitable  basis,  TJ  =  f    nn     n  j ,  In  being  the  unit 

matrix  of  rank  n.  We  fix  a  conjugacy  class  of  ty .  (This  is  an  analogue  of  weights  in  case  of 
(generalized)  parabolic  vector  bundles,  the  weights  in  this  case  being  (0,  or)  for  the  vector 
bundle  E(g)  with  induced  (generalized)  parabolic  structure). 

We  want  to  define  the  notions  of  stability  and  semistability  for  QPGs.  Since  the  def- 
initions are  rather  complicated  in  the  general  case,  we  first  define  these  notions  on  an 
irreducible  smooth  curve  C  (1.5,  1.6)  and  later  extend  these  notions  to  the  general  case 
(1.7,1.8,1.9). 

Assume  that  C  is  a  nonsingular  irreducible  curve.  Let  P  be  a  maximum  parabolic 
subgroup  of  G  and  p  its  Lie  algebra.  Let  E/P  =  E(G/P)  be  the  associated  fibre  bundle 
with  fibres  isomorphic  to  G/P.  Let  s  :  C  -+  E/P  be  a  section  i.e.  a  reduction  of 
the  structure  group  to  the  maximum  parabolic  subgroup  P.  Let  Qj  be  the  stabilizer  in 


274  Usha  N  Bhosle 

GL(g  e  g)  of  the  subspace  E(p)Xj  ©  E(p)Zj  =p©pCg©g=  E(g)x.  ©  E(g)Zj.  Let 
Hi  denote  the  determinant  of  the  action  of  Qy  on  g/p  ©  g/p.  Let  Jtj  be  the  form  on  the 
Lie  algebra  L(gy)  of  gy  corresponding  to  My.  LetTy  be  a  conjugate  of  Ty  inL(G/)- 

DEFINITION  1.5 

A  QPG  (E,  (cr,-))  is  a-stable  (resp.  a-semistable)  if  for  every  maximum  parabolic  P  of  G 
and  every  reduction  s  :  C  -»  £/P,  one  has 

.  >)«7.  ranks*r(G/P). 


Here  r  (G/P)  is  the  tangent  bundle  along  the  fibres  of  £/P  -»  C  . 

i 
Lemma  1.6.  Wfr/z  f&e  <2&0v£  notations,  the  condition  (*1)  w  equivalent  to  the  following 

par  deg  E(p)  <  (resp.  <)aJ.  rank  £(p),  (*2)  ) 

where  par  deg  £  (p)  denotes  the  parabolic  degree  of  the  subbundle  E  (p)  of  the  (generalized)  I 

parabolic  vector  bundle  (E(g),  (a))  with  weights  (0,  a),  £<2C/z  weight  being  of  multiplicity  n.  f 

I 

Proo/.  One  has  s*T(G/P)  =  E(g/p),  EyFyOr/)  =  parabolic  weight  of  the  quotient  ) 

bundle  £(g/p)  of  (£(g),  (0y)).   Thus  (*1)  can  be  restated  as  par  deg£(g/p)  >  (resp.  I 

>)  a  J  rank£(g/p).  Since  G  is  semisimple,  deg£^(g)  =  0  ([Rl],  Remark2.2)  and  hence  par  ; 

deg£^(g)  =  otJ  rank  E(g).  The  result  now  follows  from  the  exact  sequence  0  -*  E(p)  -»  r 

E(g)  ->  E(g/p)  ->  0  using  the  additivity  of  parabolic  degrees  for  exact  sequences.  < 

1.7.  Semistdble  QPGs  on  reducible  curves.  Let  the  notation  be  as  in  1.1.  We  consider 
QPGs  (E,  (cry))  on  C  with  parabolic  structure  over  Dj  =  Xj  +  Zj,  j  =  1,  .  .  .  ,  J.  Let 
{tf/l,  {Ty},  a,  a  be  as  in  1.4.  For  i  =  1,  .  .  .  ,  7  let  P/  denote  either  a  maximum  parabolic 
subgroup  of  G  or  the  trivial  group  e  or  the  group  G  itself.  We  need  to  consider  the 
cases  P  =  {e}  or  G  also,  because  a  sub-object  N  =  (fy)  of  E  =  (£*/)  may  have  the 
property  that  for  some  i,  NI  =  £/  or  N/  is  trivial.  For  an  /-tuple  P  =  (Pi,  .  .  .  ,  P/),  let 
ri  =  dim  p/,  n/  =  dim  g/p/  for  all  z.  For  j  =  !,...,/  denote  by  2y  the  stabilizer  in 
GL(g0  g)  of  the  subspace  p/(y)  ©  pt^  C  g©  g.  Let  /xy  be  the  determinant  of  the  action  of 
Qj  on  g/p/  (y)  0  g/pf  (y-)  and  /Iy  the  form  on  the  Lie  algebra  L(Qj)  of  2  /  corresponding  to 
Aty.  Letfy  be  a  conjugate  of  ty  in  L(Qj).  LetCf'  =  C/-sing(C/),^  :  C[  -*  £(G/P/)  |  C| 
any  section,  5-  =  (jlt  .  .  .  ,  j/).  Let  S^  =  the  largest  subsheaf  of  Ad  E  \  Cf  such  that 
SSi  I  q  =  ^(BCpf))  I  C/.LetS,  =  (5Jl,...,5,/)fx(SJ)  =  ^.x(^),x(Ad£)= 
(Ad  £*  |  C/).  Let  QiV/  be  the  (smallest)  torsion  free  quotient  sheaf  of  Ad  E  \  Ct  with 

c  =  j*(£(g/P/))  I     .  Let  G,  =  (gj) 


DEFINITION  1.8. 


A  QPG  (£,  (o-y))  is  (a,  oO-semistable  (resp.  (a,  a)-stable)  if  for  every  reduction  s  of  the 
structure  group  to  P  such  that  P/  ^  G  for  all  i  and  P/  ^  {2}  for  all  i  one  has 


Principal  G-bundles  on  nodal  curves  275 

;«/*,•>  (>>X(AdE)//i-a.7.     (*!') 


Lemma  1.9.  (a)  The  condition  (*!')  <at&0ve  w  equivalent  to  the  following  condition 

(*20 


z/r/  <  (<)x(Ad  £)/n  -  «J, 


(b)  /f  C  is  irreducible  and  smooth,  then  (*27)  is  same  as  (#2). 

Proof,  (a)  The  quotient  Qs  of  E(g)  has  induced  parabolic  structure  overDy,  7  =  1,  .  .  .  ,  J 
given  by  (£*)*;  ©  (Qs)Zj  D  Fj(Qs)  D  0  with  weights  (0,a),  where  Fj(Qs)  is  the 
image  of  then-dimensional  subspaceo/  of  (E(g)Xj  0  E(g)Zj)  in  ((Qs)Xj  0  (Q.v)zy-)-  Let 


fj(Qs)  =  dim  Fj(Qs).  By  definition,  the  parabolic  weight  of  £^  =  or       .fj(Qs)-  Define 
Qj(Qs}  =  «/(j)  +  n^(/)  ~~  fj(Qs),  it  is  additive  for  exact  sequences.  Then  one  has 


parabolic  weight  of  Qf     =     a(n/(y)  4-  n/(y)  - 


-  n 


Note  that  since  Qs  &  E(g/p)  outside  sing(C)  and  all  Dj  avoid  sing(C),  one  has  parabolic 
weight  of  Qs  —  the  parabolic  weight  of  £(g/p)  =  /"^  ./£/(*./)•  Hence,  VJ  J!J(TJ)  — 

^2  .a(n/(y)  H-nr(y))  =  a^  .qj(Ss)  -  a/n.  Using  this  equality  and  £  0,-r/  =  w  —  ^  fli-n; 
the  first  part  of  the  Lemma  follows. 


(b)  If  C  is  a  smooth  irrreducibie  curve  then  one  has/  =  1,  5^  = 

=  n»X(-Sj)  =deg(J5(p))-fri(l— g),y"^  a0/(Sy)  =  Y^  c 

^— 'y  z— './ 

parabolic  weight  (&) .  Hence  the  left  hand  side  of  (*2')  becomes  equal  to  par  deg  £(p)/rank 

E(p)  -  2a /  +  (1  ~  g).  The  right  hand  side  of  (*27)  =  (1  -  g)  -  aJ.  Hence  the  result 
follows. 

2.  Principal  G-bundles  on  a  singular  curve  F 

2. 1 .  We  want  to  introduce  the  notions  of  stability  and  semistability  for  principal  G-bundles 
on  singular  curves.  On  a  smooth  curve  there  are  different  definitions  of  stability  and 
semistability  of  a  principal  G-bundle,  but  they  all  coincide  [Rl].  The  problem  is  that 
this  is  not  true  on  a  singular  curve.  The  choice  of  a  representation  of  G  used  to  define 
semistability  does  not  matter  on  a  smooth  curve  essentially  because  the  associated  bundles 
(tensor  products  etc.)  of  semistable  vector  bundles  (in  characteristic  0)  are  semistable. 
This  fails  if  the  curve  has  singularities.  For  example,  if  F\  is  the  semistable  vector  bundle 
of  rank  2,  degree  0  (on  an  irreducible  nodal  curve  7)  constructed  in  Proposition  2.7  of 


276  Usha  N  Bhosle 

[U3]  then  F\  ®  F\  and  S2F\  are  not  semistabie  [U5].  This  is  seen  by  checking  that  the 
corresponding  generalized  parabolic  vector  bundles  on  C  are  not  semistabie.  Similarly  one 
can  show  that  if  F2  is  the  stable  vector  bundle  of  rank  2m  constructed  in  Proposition  2.9, 
[U3]  then  FI  <g>  FI  is  not  semistabie  for  all  m  >  2  [U5]. 

We  give  here  a  notion  of  sernistability  for  principal  G-bundles  on  singular  curves  (see 
Definitions  2.2,  2.3,  2.9,  2.10)  which  is  intrinsic  and  seems  most  useful.  We  first  assume 
that  Y  is  irreducible  (the  case  of  a  reducible  curve  will  be  dealt  with  later).  Let  Yf  =  Y  — 
{singular  set  of  F},  i  :  Y'  -»  Y  inclusion  map.  Let  G  be  a  connected  reductive  algebraic 
group.  Let  P  be  a  maximum  parabolic  subgroup  of  G  and  p  the  Lie  algebra  of  P.  Let 
f  be  a  principal  G-bundle  on  Y  and  jF/P  =  F(G/P)  the  associated  fibre  bundle  with 
fibres  isomorphic  to  G/P.  Let  sf  —  Y'  -»  (F/P)  \  Y'  be  a  reduction  of  the  structure 
group  to  P  (i.e.  a  section  of  F/P  restricted  to  Y1).  Let  T(G/P)  denote  the  tangent 
bundle  along  the  fibres  of  f/P  ->  Y.  Let  Qs>  be  a  torsion  free  quotient  of  ^(g)  such  that 
Qs>  |  Y1  =  (s'Y(T(G/P))  |  F'  and  no  further  quotient  of  Qs>  has  this  property.  Let  Sj  be 
the  maximum  subsheaf  of  f(g)  containing  (s'^Ffo). 

DEFINITION  2.2 

F  is  stable  (resp.  semistabie)  if  for  every  reduction  sf  of  the  structure  group  to  a  maximum 
parabolic  P  (over  y'), one  has  degree  Q,y/  >  0  (resp.  >  0). 

Lemma  2.3.  The  above  definition  is  equivalent  to  the  following:  F  is  stable  (resp.  semi- 
stable)  if  for  every  sf  as  above,  degree  Ssf  <  0  (resp.  <  0). 

Proof.  The  exact  sequence  0  -»  p  ->  g  ~>  g/p  ->  0  gives  an  exact  sequence  0  -* 
j'*.F(p)  -*  AdJ7  |  r  -»  s'*T(G/P)  -+  0  and  hence  0  -»  S,/  -»  AdJ^  ->  Q5/  ->  0. 
Noting  that  Ad  f  has  degree  zero,  the  lemma  follows. 

We  now  assume  that  Y  has  only  ordinary  nodes  yi, . . . ,  yj  as  singularities  and  p  :  C  -+  Y 
is  the  normalization  map,  Dj  =  p~l(yj)  =  Xj+Zjt  j  =  1, ...,/.  Then  giving  a  principal 
G-bundle  J7  on  F  is  equivalent  to  giving  the  principal  G-bundle  p*  J1"  =  £  on  C  together 
with  a  G-isomorphism  or/  of  the  fibres  EXj  and  £Z;.  of  E  for  each  7 .  The  isomorphisms  ay- 
induce  isomorphisms  E(g)Xj  ->  E(g)ZJ .  We  denote  the  graph  of  these  isomorphisms  also 
by  a/. 

PROPOSITION  2.4 

(E,  (cry ))  M-  l-stable  (resp.  l-semistable)  if  and  only  if  the  corresponding  G-bundle  T  on 
7  w  stable  (resp.  semistabie). 

Proof.  Suppose  that  f  is  stable  (resp.  semistabie).  Let  s  :  C  -»•  E/P  be  a  reduction 
to  a  maximum  parabolic  subgroup  P.  Since  C  —  U/D/  «  y  —  U/y/,  under  p  and 
£  «  p*jF,  the  section  s  gives  a  reduction  s1  :  Yf  =  y  —  Uy-yy  ->  (F/P)  |y.  One  has  the 
exact  sequences  0  ->  ^(g)  ~>  p*s*E(g)  ->  ®jQjE(g)  ->  0,  0  ->  S,/ 
®jQjE(p)  ->  0  where  2y(^(g))  =  (**£&)*,- e**E(g)zy)/ay, 
J*£(p)Zy)/(a/n(j*£:(p)^ 

free  sheaf  obtained  from  ,y*£(g/p)  with  induced  parabolic  structure  (viz.  the  image  of  a/ 
in  E(g/p)xj  0  E(g/p)Zy ,  V/).  The  second  sequence  implies  that  par  deg  s*E(p)  -  J  rank 


Principal  G  -bundles  on  nodal  curves  271 

s*E(p)  =  deg  (Ss<).  Since  f  is  stable,  deg(5y/)  <  0.  The  result  follows  from  Lemma  1.6. 
The  converse  follows  similarly  working  backwards  in  the  above  argument.  One  has  only 
to  note  that  if  sf  :  Y'  -»  CF/P)  \'Y  is  a  reduction  to  a  maximum  parabolic  P,  then  sf  gives  a 
reduction  s  :  C  ->  E/P  (as  G/P  is  complete).  In  case  of  semistability  one  has  to  replace 
strict  inequalities  in  the  above  proof  by  inequalities. 

2.5  Bundles  associated  to  representations 

The  fundamental  group  n  \  (F)  of  Y  is  isomorphic  to  H  =  n  \  (C)  *  Z  *  ...  *  Z,  a  free  product 
of  TTi  (C)  and  J  copies  of  Z  (3.5,  [U3]).  To  a  representation  p  :  H  ->  G  we  associate  a  QPG 
(Ep,  (a/))  as  follows.  Ep  is  the  principal  G-bundle  on  C  associated  to  the  representation 
pc  =  p  \  7t\(C).  If  C  is  the  universal  covering  of  C,  then  Ep  =  C  xp  G.  Fixing 
suitably  points  x'j  ,  z'.  of  C  lying  over  *y  ,  zy  respectively,  the  fibres  (Ep)Xj  and  (Ep)Zj  can 
be  identified  to  G.  Letgy  =  p(ly),  ly  denoting  the  generator  of  the  y'thf  actor  Z  inH,  Then 
gy  gives  an  isomorphism  A';.  :  (£p)^.  =  (Ep)zy  and  hence  Ay  :  (Ep(g))Xj  =  (£p(g))2r 
Define  ay  =  graph  of  A/.  If  ,F  is  the  principal  G-bundle  on  Y  obtained  by  identifying 
fibres  of  Ep  at  jcy  and  zy  by  gyVy  ,  then  one  has  T  =  J^,,  the  G-bundle  associated  to  the 
representation  p  of  7Ti(F)  and  Ep  =  /?*^>. 

PROPOSITION  2.6 

7f  pc  z^  irreducible  unitary  (resp.  unitary)  then  Tp  is  stable  (resp.  semistable). 


Proof.  If  pc  is  unitary,  so  is  Ado^c  and  hence  J^Adop  =  -^(g)  is  semistable  ([U3], 
Proposition  2.5).  Therefore  jFp  is  semistable. 

If  pc  is  irreducible  unitary,  then  by  Theorem  7.1  of  [Rl]  (in  our  case  E(p,c)  =  Ep,  c  = 
Id)Ep  is  a  stable  G-bundle.  We  check  below  that  (Ep  ,  (a,  ))  is  1-stable,  then  Tp  is  stable  by 
Proposition  2.4.  Let  5-  be  a  reduction  of  the  structure  group  of  Ep  to  a  maximum  parabolic 
subgroup  P.  The  stability  of  Ep  implies  that  deg  (s*Ep(p))  <  0.  Note  that  cry  maps 
isomorphically  onto(£/))JC;,  ;  =  !,...,/.  Hence  orj(Ep(p))  =  crjn(Ep(p)XJ®Ep(p)zj) 
maps  injectively  into  Ep(p)Xj.  Therefore  dim  aj(Ep(p))  <  rank  (J^p(p))  for  all  y.  It 
follows  that  par  deg  (s*Ep(f))  =  deg  (s*Ep(p))  +  £;  dim  ay  (£p(p))  <  /  rank  (JEp(p)). 
Thus  (J?p,  (ay))  is  1-stable. 

Remark  2.7.  There  may  exist  stable  principal  G-bundles  on  Y  which  are  not  associated  to 
any  representations  of  jri(F).  For  examples  in  case  G  =  GL(ri)  see  [U3],  similar  examples 
can  be  constructed  in  case  G  =  O(n),  Sp(2ri)  also. 

Principal  G  -bundles  on  a  reducible  curve  Y 

Notations  2.8.  Let  the  notation  be  as  in  1.1.  Assume  further  that  Y  has  nodes  yy,  j  = 
1,  .  .  .  ,  J  as  only  singularities.  Let  F  be  the  graph  obtained  from  the  (dual)  graph  of  Y  by 
omitting  loops.  Let  yi,  .  .  .  ,  y#  be  the  nodes  of  Y  such  that  each  yy  lies  on  two  different 
components  of  Y.  Then  K  =  the  number  of  edges  of  F,  /  =  the  number  of  vertices  of  F. 
For  i  =  1,  ...,/,  let  P/  denote  either  a  maximum  parabolic  subgroup  of  G  or  the  trivial 
group  [e]  or  the  group  G  itself.  Let  ,F  denote  a  principal  G-bundle  on  7.  For  each  i, 
let*;  :  Y[  ->  F(G/Pi)  |r;  be  a  section.  Let  P  =  (P;)/,/  =  O^  be  /-tuples.  We 
call  sf  a  reduction  of  the  structure  group  to  P  over  Yf  .  Let  T(G/PZ)  denote  the  tangent 


278  Us  ha  N  Bhosle 

bundle  along  the  fibres  of  f(GfPi)  \yr  If  PI  =  M  then  s^(T(G/Pi))  ^  Ad  JF  |K/.  If 
Pi  =  G,  then  ,F(G/PO  |y,^  ^  and  the  Euler  characteristic  x(s*(T(G/Pt)))  =  0.  Let 
gj/  be  the  smallest  torsionfree  quotient  of  Ad  J7  such  that  Qs>  \Y>  «  s'?(T(GfP})  |y/ 
for  all  i.  Let  p/  denote  the  Lie  algebra  of  P,-  and  J^(p/),  ^"(g),  «F(g/pi)  the  fibre  bundles 
(with  fibres  p/,  g,  g/p/  respectively)  associated  to  the  P/  -bundle  J7  -»  f(GjP{)  via  the 
adjoint  representation.  Thus^'^g)  -  Ad  ^  |K/,  J>,F(g/Pi)  -  s?T(G/Pi).  Let  S,/  be 
the  maximum  subsheaf  of  Ad  f  such  that  Sst  \Y^  s^f(p).  Leta  =  (ai,  .  .  .  ,  a/),  where 
{a/  }  are  positive  rational  numbers  with  ]T  #/  =  1.  Recall  that  for  a  vector  bundle  V  on  7, 
a-rank  V  =  T*i  rank(V  |y.). 


DEFINITION  2.9 

The  principal  G-bundle  J^"  on  7  is  <2-semistable  (resp.  <7-stable)  if  for  every  reduction  sf 
of  the  structure  group  to  P  with  P,  7^  {^}  for  all  /  and  P/  ^  G  for  all  f  one  has  (in  the 
notations  of  2.8) 


-  rank  Qs'  >  (resP-  >)X  (Ad  F)/a  -  rank  Ad  *:r- 
Lemma  2.10.  .T7  w  a-semistable  (resp.  a-stable)  if  for  every  reduction  s1  as  above, 

s'  <  (resP-  <)X  (Ad  ^)/a  ~  rank 


/.  As  in  Lemma  2.3,  we  have  the  exact  sequences  0  ->  ^^^"(p)  ->  Ad  J7  |  7/  -> 
s*T(G/P)  ->  0  for  all  z  and  so  0  ->  Sv/  ->  Ad  J7  ->  QJ'  ^  0.  The  lemma  follows  using 
the  fact  that  both  the  Euler  characteristic  and  a-rank  are  additive  for  an  exact  sequence. 

PROPOSITION  2.  11 

For  i  =  1,  ...,/,  let  Q  be  a  partial  desingularization  of  F/  and  C  —  JJ  Q.  Suppose 
that  C  is  obtainedby  blowing  up  nodes  y\,...,  yj',  J'  <  J  ofY.  Let  (E,  (<jj))  denote  a 
QPG  with  quasi-parabolic  structure  or/  over  Dj,  I  <  j  <  J'.  Then  a  QPG  (E,  (cr;0)  is 
(a,  I)  -stable  (resp.  (a,  l)-semistable)  if  and  only  if  the  corresponding  principal  G-bundle 
on  Y  (obtainedby  identifying  fibres  of  E  by  ay)  is  a-stable  (resp.  a-semistable)  . 

Proof.  The  proof  is  exactly  on  same  lines  as  that  of  Proposition  2.4.  Starting  with  f 
<2-stable  (resp.  semistable)  and  a  reduction  sf  to  P,  one  gets  an  exact  sequence  0  ->  Ss>  -> 
P*SS  ->  ®jQj(Ss)  ->  0,  with  qj(S5)  =  dim  Qj(Ss).  Then  Lemma  1.9  gives 
(a,  Instability  (resp.  semistability)  of  (£,  (cr;-)).  The  converse  is  proved  by  reversing  the 
argument. 

2.12.  G  -bundles  associated  to  representations 

Let  p  :  7ti(Y)  ~>  G  be  a  representation  of  the  fundamental  group  n\(Y)  of  Y  in  G.  For 
i  =  1,  ...,/,  let  fi  :  n\(Yi)  ~»  ni(Y)  be  the  natural  maps,  p/  =  p  o  ft.  Let  f  be  the 
G-bundle  on  Y  associated  to  p.  Let  p*F  =  E  =  (£/)/.  Then  £/  is  the  G-bundle  on 
YI  associated  to  p/.  The  principal  G-bundle  T  corresponds  to  a  QPG  (E,  (cr/))  on  []f  F/ 
where  {cry},  7  =  1,  .  .  .  ,  K  are  G  -isomorphisms  of  fibres  of  E.  Finally  let  C/  denote  the 


Principal  G-bundles  on  nodal  curves 

279 


true 


For  ,,,e  proof  of  ,  he  lheoreilli  we  need 
PROPOSITION  2.  13 


-, 

that  for  every  I  -tule  r  =   r,  ,-       ,.  '   '  ~  [  '  '  '  '  '  I  Wlth  E  */  =  1 


v,=,    '"    •  <SS> 


//"m  addition  r;  =  0  /br  ^m^  /  0  ^  v 

for  some  ,,  0  ^  £,.  r/, 

Proof.  We  prove  the  result  by  induction  on  m. 
Case  m  =  -  1  :  r  is  a  tree  in  this  case.  Let  a/  = 


L.H.s.of(ss)    =    Eingi-Ztn- 
=    (s-^Engf/g 

If  0  56  £  ^  and  rf-  =  0  for  some  i,  then  r,0  =  0  and 


( 


280  Usha  N  Bhosle 

that  for  all  r  and  £  =  (q\  ,  .  .  .  ,  5i,  .  .  .  ,  qj)  satisfying  the  given  conditions,  one  has 


L.H.S.of(SS)     = 


7=1 


where  &,-  =  £[ifi  ^  f(£),^)and&;  =  £><  +  £  if  i  =  i(€)orf(£).  Take  a/  =&i/(g+m)for 
all  /,  then  (SS)  holds.  The  assertion  about  strict  inequality  follows  by  induction  similarly. 

Remark2.l4.  (1  )  Note  that  if  both  a,  a1  satisfy  (SS)  then  for  0  <  t  <  I,  a*  =  ta  +  (\-t}af 
also  satisfies  (SS).  Thus  the  set  of  solutions  a  of  (SS)  is  a  convex  set. 
(2)  Given  /i,  1*2,  ,  1  <  M,  *2  <  /,takea,  =  (&—  2)/fe-l)fori  =  z'i,  i2andaj  =  gi/(g-l) 
for  i  j£i\,i  ^  i^.  Then  in  case  F  is  a  tree  (i.e.  m  =  —  1)  the  inequality  (SS)  holds  (though 
the  strict  inequality  may  not  be  true  eg.  for  r^  =  r/2  =  0).  For  K  —  I  >  0,  the  inductive 
proof  of  Proposition  2.13  then  gives  new  o!  =  (a[,  .  .  .  ,  a'j)  satisfying  the  inequality  (SS). 
It  follows  that  the  inequality  holds  for  a1  ,  0  <  t  <  1  . 

PROPOSITION  2.  15 

Theorem  I  is  true  for  G  =  GL(r). 

(1)  If  pi  \  7t\(C{)  are  unitary  representations  for  all  z,  then  the  vector  bundle  F  on  Y 
associated  to  p  is  a-semistable. 

(2)  If  pi  \  7T\(Ci)  are  irreducible  unitary  for  all  i,  then  F  is  a-stable. 

Proof. 

(1)  As  in  Propositions  3.9  and  3.7(3)  of  [U2],  it  can  be  seen  that  the  vector  bundle  F  on 
Y  corresponds  to  a  QPG  j£  =  (E,  Fj(E))  on  JJ  F/  and  F  is  a-semistable  (resp.  a-stable) 
if  and  only  if  E_  is  (a,  l)-semistable  (resp.  (a,  l)-stable).  Note  that  E  =  [JE/  is  the 
pull-back  of  F  to  [J  YI  -  By  Theorem  2,  [U3],  the  vector  bundles  EI  on  Y-t  associated  to  pi 
are  semistable  for  all  i  .  Hence,  for  any  subsheaf  A//  of  Ej  ,  one  has  x  W'  )  <  fi  (  1  —  £/  )  »  n  = 
(note  that  degree  (£/)  =  0).  Thus 


E 


where  the  summation  over  j  is  taken  for  1  <  j  <  K.  For  the  choice  of  {at-},  made  in 
Proposition  2.13,  we  get 

SN<l-K-Y,Si  =  \Y,K(Ei)-rK\    Ir. 
Thus  (E,  Fj(E))  is  (a,  l)-semistable  and  hence  F  is  a-semistable. 


Principal  G  -bundles  on  nodal  curves  28  1 

(2)  We  need  to  consider  two  cases.  With  the  notations  in  the  proof  of  (1)  if  r/  =  0  for  some 
/  then  by  Proposition  2.  13,  we  have  SN  <  /  —  -&"  —  E  ft  •  Ifn  ^  0  for  all  z  ,  then  there  exists 
an  z'o  such  that  0  ^  r/()  ^  r.  Since  E-l(]  is  stable  by  Theorem  2  [U3],  we  have  x  W-D)  < 
r/0d  -  ft0).  Therefore,  SN  <  E;  r/(l  -  #)  -  E;  *'  (#)/  E;  ^  <  /  -  K  -  E  ft  (by 
Proposition  2.13).  Thus  (E,  F/(E))  is  (a,  l)-stable  and  so  F  is  0-stable. 

Remark  2.16.  The  proof  of  Proposition  2.13  shows  that  there  exist  curves  Ym,m  = 
0,  .  .  .  ,  n  +  1  such  that  (1)7°  =  F,  (2)  Yn+  1  is  a  curve  with  ordinary  nodes  such  that  the  dual 
graph  of  F'14"1  is  a  tree  after  omitting  loops  (3)  Ym+  [  is  obtained  from  Ym  by  blowing  up  a 
node  which  lies  on  two  different  components.  For  m  =  0,  .  .  .  ,  n,  let  (pm  =  ym+!  ~>  ym 
be  the  natural  surjective  maps.  Let  F  denote  a  unitary  (resp.  irreducible  unitary)  vector 
bundle  on  Y.  The  proofs  of  Propositions  2.13  and  2.15  together  show  how  the  'polariza- 
tion' a  =  (a\9  .  .  .  ,  a/)  for  which  the  vector  bundles  (p*TF  are  a-semistable  (resp.  ^-stable) 
varies  as  we  go  down  the  tower  of  curves  {Ym}. 

Proof  of  Theorem  1. 

(1)  If  Pd  is  unitary,  so  is  Ad  o  pc-t  =  (Ad  o  p)q  .  Therefore  there  exist  positive  rational 
numbers  a\,  .  .  .  ,  <2/  with  E  #/  =  1  (depending  only  on  P  and  #/)  such  that  the  vector 
bundle  ^Adop  ^  Ad  T  associated  to  Ad  o  p  is  tf-semistable  (Proposition  2.15).  Hence 
T  is  a-semistable. 

(2)  By  Proposition  2.6,  the  principal  G-bundles  E(  on  YI  associated  to  pi  are  stable  for 
all  i.  We  claim  that  for  the  choices  of  {#/},-  as  in  the  proof  of  (1),  the  QPG  (£,  (cr/)) 
corresponding  to  .T7  is  (a,  l)-stable.   The  result  follows  from  the  claim  in  view  of 
Proposition  2.1  1.  To  prove  the  claim  we  check  that  the  condition  (*2')  of  Lemma  1.9 
is  satisfied  for  any  reduction  s  of  the  structure  group  to  P.  Let  r/  be  the  rank  of  5^.  . 
Since  Ss  is  a  proper  subsheaf  of  £(g),  Er;  i=-nl.  Since  E/  are  stable,  by  Lemma  2.  3, 
X(Sst)  <  n(l  —  gi)  and  the  inequality  is  strict  if  0  <  r,-  <  n.  By  Proposition  2.13,  for 
the  choices  of  {#/}  as  in  (1),  one  has 


if  r/o  =  0  for  some  IQ  and  0  ^  E  n  •  If  n  ^  0  for  all  i  ,  since  E  n  i^nl,  there  exists 
an  z'o  such  that  0  <  r/0  <  n.  Then  x(55)  <  ^7/0  —  ^i)  and  so 


<     (I  -  K  -  E  ft)  (    JW    »  by  Proposition  2.13, 


This  proves  the  claim. 


282  Usha  N  Bhosle 

3.  The  Picard  group  of  the  stack  of  QPGs 

In  this  section  Y  denotes  a  reduced  connected  projective  curve  with  ordinary  nodes 
=  1,  .  .  .  ,  J  as  only  singularities.  Let  [Yi),i  =  1,  .  .  .  ,  /  be  the  irreducible  components  of 
Y  and  Q  the  desingularization  of  7/.  Let  C  =  {Jt-  Q  be  the  desingularization  of  F.  For 
convenience  of  notation,  we  fix  an  orientation  of  the  dual  graph  of  Y  .  For  I  <  j  <  J,  let 
i  (y),  £(7)  denote  the  initial  and  terminal  points  of  j  in  the  dual  graph.  They  correspond  to 
curves  C/Q-),  Ct(j)  intersecting  at  vy-.  Let  Xj  €  C/(y)  and  z7-  €  C/-  (_/)  be  the  two  points  of 
C  mapping  to  yj  €  Y  and  Dj  =  Xj  +  Zj,  j  =  1,  ...,/.  Let  G  denote  an  affine  simply 
connected  simple  algebraic  group  over  C  (or  an  algebraically  closed  field  of  characteristic 
zero).  For  i  =  1,  ...,/,  fix  points  /?/  €  C/,  /?/  not  mapping  to  a  singular  point  of  Y.  Let 
C*==C/-{/7i},C*=C-U/p/. 

The  results  of  this  section  were  inspired  by  [LS].  If  G  is  semisimple,  then  a  principal 
G-bundle  on  a  smooth  curve  C  is  trivial  on  the  complement  of  a  point  in  C.  This  no  longer 
holds  if  C  is  replaced  by  a  nodal  curve  Y.  The  results  of  [LS]  cannot  be  generalized  directly 
to  G-bundles  on  Y.  Hence  we  work  with  QPGs  on  C.  Though  we  closely  follow  the  ideas 
in  [LS],  the  generalization  to  QPGs  is  not  straightforward.  All  the  functors  involved  have  to 
be  defined  carefully  to  take  care  of  the  additional  structure  (generalized  parabolic  structure). 
Unlike  the  usual  parabolic  structure  which  is  supported  on  isolated  points,  the  generalized 
parabolic  structure  is  supported  on  divisors,  so  one  has  the  action  of  G  x  G  rather  than  G. 


3.1    The  stack  Q*j*  and  the  stack 

Let  Aff/k  be  the  flat  affine  site  over  the  base  field  k  =  C,  i.e.  the  category  of  A>algebras 
equipped  with  fppf  topology.  Let  R  denote  a  fc-algebra,  C-^R  :=  C/x  spec  R  and 
C^  =  C*x  spec  R.  Let  qi  be  a  local  parameter  at  the  point  /?/,  i  =  1,  .  .  .  ,  7.  Let  LGJ 
denote  the  £-group  defined  by  associating  ioR  the  group  G  (/?(#/)).  LetLj  .  (resp.  LG')be 
the  fc-group  defined  by  associating  to  R  the  group  G(R[[qi]])  (resp.  G(r\CfR9  Oc*  ))). 

Define  LG  =  ^LG^L*  =  UiLG  =  \\*LG  •  Let 


The  indgroup  L£  acts  on  QG,C-  For  each  7  ,  the  evaluation  at  Xj  and  z;-  gives  an  evaluation 
map  ej  :  LCG  ->  G  x  G.  G  x  G  acts  on  G  by  (#1  ,  gi)g  =  g"^lgg2-  Thus  we  have  a  natural 
action  of  L£  on  ficfc-  ^et  ^G\2cPc  ^e  the  quotient  stack. 

To  an  object  R  e  Aff/k,  associate  the  groupoid  whose  objects  are  families  of  QPGs 
(£*,  (CT/))  on  C  parametrized  by  spec  R  and  whose  arrows  are  isomorphisms  of  the  families 
of  QPGs  i.e.  isomorphisms  of  E  which  preserve  the  parabolic  structures  (a/).  For  any 
morphism  R  -»  #;  we  have  a  natural  functor  between  the  associated  groupoids.  Thus  we 
get  a  &-stack  of  (generalized)  quasiparabolic  G-bundles  on  C.  We  denote  this  stack  by 
Bung?- 

Theorem  2.  There  exists  a  canonical  isomorphism  of  stacks 
TT      •  7c\r)gpar  J 

tfpar  -  LG  \QGC  ~ 
The  projection  7Tpar  :  fiffc  "^  ^unGPC  ^  l°caMy  trivial  in  etale  topology. 


Principal  G-bundles  on  nodal  curves  283 

Proof.  QG,C  represents  the  functor  which  associates  to  every  fc-aigebra  R  the  set  of  iso- 
morphism classes  of  pairs  (£,  p)  where  E  is  a  G-bundie  over  CR  and  p  is  a  trivialization 
of  E  over  C£  ([LS],  Proposition  3.10).  Hence  Q§^  represents  the  functor  PG  which 
associates  to  R  the  isomorphism  classes  of  triples  (E,  p,s)  with  (E,  p)  as  above  and 
s  €  Y\J  G(£),  s  =  (5-1,  .  .  .  ,  Sj),  Sj  €  G(R)  =  Mor  (Spec  jR,  G),  G  being  the  jth  factor 
in  Y\j  G.  Such  a  triple  gives  a  family  of  QPGs  (£,  (cr/))  parametrized  by  5  =  Spec  R  as 
follows.  Let  5j  :  S  xxj  x  G  ->  S  x  Zj  x  G  be  given  by  J;-  (j,  jt,-  ,  g)  =  ($,  ^  ,  £.?/  (j)) 
for  5-  €  S,  g  e  G.  Define  a;  :  fijsx^  -»  £|SxZy  by  ay  =  pj^.  o  Jy-  o  p{SxXj.  Thus 
we  get  a  universal  QPG  over  Q^™  x  C,  giving  a  map  7rpar  :  gg*"  -*  Bung^.  Being 
L^-invariant,  this  map  induces  a  morphism  of  stacks  7fpar  :  ^£  \2cfc  ->  Bung3^. 

To  define  a  morphism  Bung"  -»  L%\Q%%,  for  each  *  and  (£»  (°V))  £  filing^  (/?) 
we  have  to  give  a  L£  -bundle  T(R)  on  Bun^arc(j?)  together  with  an  Lg-equivariant  map 

T(R)  ->  Qg^GR).  Take(^'  (^O)  e  Bxing^(«).  For  any  JZ-algebra/f7,  let  Spec  /Z7  =  5' 
and  T(R')  =  the  set  of  isomorphism  classes  of  pairs  (PR',  crf)  where  PR'  is  a  trivialization 
of  ER*  overC^,  anda7  =  (orj)y,  orj  :  £|5/XJCj.  «  £|s'xz/  is  the  G-isomorphism  which  is  the 
pull  back  of  Oj  to  /?'.  This  defines  an  7?-space  T  with  the  action  of  the  group  L£  (acting  on 
PR,).  It  is  an  Lg-bundle  ([DS];  also  [LS],  Theorem  3.11).  As  Q*?*  represents  the  functor 
PG  »  to  every  element  (p#/  ,  cr/?/)  ofT(R')  corresponds  an  element  of  Qcfc  (^')  giving  a  ^c~ 
equivariant  map  T  ->  GG^C-  Hence  we  get  a  morphism  of  stacks  Bun|p£  ~>  IG\GG  c 
which  is  clearly  the  inverse  of  7fpar. 

To  check  the  local  triviality  of  7rpar  in  etale  topology,  we  have  to  show  that  for  any 
morphism  /  from  a  scheme  S  to  Bun^P"  the  pull  back  of  the  fibration  7rpar  to  S  is  etale 
locally  trivial  i.e.  admits  local  sections  for  the  etale  topology.  Such  a  morphism  corresponds 
to  a  QPG  (E,  (ay))  over  S  x  C.  For  s  €  S,  we  can  find  an  etale  neighbourhood  U  of  s 
and  a  trivialization  p  ofE\uxc*  ([DS]).  Using  p,  the  G-isomorphism  a/  gives  a  morphism 
Sj  :  U  ->  G.  The  triple  (E,  p,  (sj))  defines  a  morphism  /'  :  U  ->  Q^c  sucri  tnat 
#par  o  /;  =  /;  i.e.  the  section  over  U  of  the  fibration  7Tpar.  This  completes  the  proof  of 
Theorem  2. 

PROPOSITION  3.2 
One  has 

(1)  Kcfic,c« 

(2)  Kc(fig) 


(1)  It  is  known  that  each  QG,C,  is  an  ind-scheme  which  is  an  inductive  limit  of 
reduced  projective  Schubert  varieties  XJ,MJ,  this  ind-scheme  structure  coincides  with  the 
one  by  Kumar  and  Mathieu  ([LS],  Proposition  4.7).  One  has  Hl(XitWt  O)  =  0  ([KN,  M]). 
It  follows  that  Pic  Qctc  **  ®/  Pic  QG,C/  -  It  is  known  that  Pic  (QG.Q)  =  Z^Qc.c,.  (1)  for 
all  i  ([LS],  4.10;  [M];  [NRS],  2.3)  The  first  assertion  follows. 
(2)  Since  J"J  .  G  is  a  simply  connected  affine  algebraic  group  Pic  (FT  .  G)  is  trivial  The  ind- 

scheme  QG,CI  is  the  inductive  limit  of  integral  projective  reduced  (generalized)  Schubert 
varieties  X^w.  with  Hl  (XijWi  ,  O)  =  0.  By  III,  Exer.  12.6  [H]  it  follows  that  Pic  (Xj^,  x 
J~J  .G)  «  Pic(X"i|lfn)  ([H],  III,  Exer.  12.6)  and  therefore  by  induction  on  i  one  sees  that 


284 


Usha  N  Bhosle 


Y      vTT  G  and  the  restriction  OQCtC.  (1)  lx;>J. 
**.«>,  *  {  ^ 


ic  (QSGPC) 


a  reference. 


function  on  G  is  constant. 


. 

group  Ga  are  constant  functions. 


.  Let 


I/  «G.,  this  functio 


fore,inviewoftheclaim 


o, 

-  " 


***-*^- 


3.4.  For  each  i,  there  are  ^morphisrn,  ,  of 

phisms<  :  Pic(Bunc,c,.)  -  P  ^  jf/l 
PWellasitspullbacktoBunG,c,then      L  - 

the  pull  back  of  CWD  to  Qo.c 


>  Bunr;  c-  inducing  isoi 
r  ^  (Bunc. 

([LS,  So.  IT).  Hence  rf  we  di 
have  ff.(L/)  =  Ofl<w 
.  Similar  argument  , 


BU..C. 


we  have  a  commutative  diagram 
Pic(Buno,c)    ^ 

«•; 


Principal  G  -bundles  on  nodal  curves  285 


^)  is  the  set  of  L£  -linearizations  of  the  trivial  line  bundle.  Any 

such  linearization  is  given  by  an  invertible  (regular)  function  h  on  LCG  x  Q^fc  satisfying 
a  cocycle  condition.  £>G,C  being  an  inductive  limit  of  integral  projective  schemes  ([LS], 
4.6)  has  no  non  constant  regular  functions.  Since  G  is  simple,  fj  .  G  has  no  invertible 
nonconstant  regular  functions  (Lemma  3.3).  Hence  h  is  the  pull  back  of  an  invertible 
function  on  L^.  Since  it  satisfies  a  cocycle  condition,  it  is  in  fact  a  character  on  L^. 
By  [LS],  Lemma  5.2,  h  is  trivial.  Thus  the  forgetful  morphism  is  injective.  Hence  the 
composite  7r*par  :  Pic(Bun*P£)  ->  Pic^Cfig^)  ->  Pic(Q$%)  is  injective.  Thus^rp*ar  is 
an  isomorphism.  Similarly  jr*  is  an  isomorphism  and  hence  <I>*  is  also  an  isomorphism. 
We  have  proved  the  following  theorem. 

Theorem  3.  Let  G  be  a  simple  simply  connected  affine  algebraic  group  over  C.  Then  we 
have  the  following  isomorphisms. 

(1)     K 
(2) 

where  Lj  and  LI  are  thepullbacks  of  the  generator  of  Pic  (Bun^.c/  )  t°  Buno,c 
respectively. 

Remark  3.5.  For  G  —  GL(n),  SL(rc),  Sp(2n),  the  moduli  stack  (resp.  moduli  space)  of 
bundles  on  Y  is  isomorphic  to  the  moduli  stack  (resp.  moduli  space)  of  QPGs  on  C  ([Ul, 
U2,  U4]).  Hence  we  have 

Pic(BunG,r)  ^  ®/z 
for  G  =  GL(rc),  SL(n)  or  Sp(2n). 

PROPOSITION  3.6 

Assume  that  C  is  irreducible  and  G  as  in  Theorem  3.  Let  (Bun^)88  denote  the  substack 
corresponding  to  ct-semistable  QPGs.  Then 

Pic(Bun*?£)ss  w  Z. 

Proof.  We  claim  that  a  QPG  (E,  a)  is  a-semistable  (resp.  stable)  for  any  a,  0  <  a  <  1 
if  the  underlying  bundle  E  is  semistable  (resp.  stable).  The  semistability  (resp.  stability) 
of  E  implies  that  deg£(p)  <  (resp.  <)0.  Since  ay  is  an  isomorphism,  the  subspace  a, 
of  E(g)Xj  0  E(g)Zj  maps  isomorphically  onto  E(g)Xj  under  the  projection  map.  Hence 
cry  n  (E(p)Xj  0  ^(p)^)  maps  injectively  into  E(p)Xj  and  hence  has  dim.<  rank  E(p). 
It  follows  that  pardeg  E(p)  <  (resp.  <)aJ  rank£(p).  The  claim  now  follows  from 
Lemma  1.6. 

The  morphism  0  :  Bun^f^  ->  BunG,c  (forgetting  the  quasiparabolic  structure)  is  a 
surjective  morphism  with  isomorphic  fibres.  It  follows  from  the  claim  that  (f)~l(BuriG,c 
-  Bung  c)  D  Bungc  -  (Bun^)88.  Hence  codim.Bun«par  (Bung*"  ~  (Bunf^)88)  > 
codim.Bunc,c(^unc/,c  —  Bung  c).  Since  the  latter  is  >  2  for  g  >  2  ([L-S],  9.3)  the  same 
is  true  for  the  former.  The  result  now  follows  from  Theorem  3. 


286  Usha  N  Bhosle 

3.7.  Results  in  case  G  =  GL(n),  SL(ri)  : 

In  case  of  vector  bundles  we  have  the  following  results  on  Picard  groups  of  moduli  spaces 
([U5,  U6]).  Let  Y  denote  an  irreducible  reduced  curve  over  C  with  at  most  ordinary  nodes 
as  singularities.  Let  £  be  a  line  bundle  on  F.  Let  U'Y(n,d)  (resp.  U'c(n,d))  denote 
the  moduli  space  of  semistable  vector  bundles  of  rank  n  and  degree  d  (resp.  with  fixed 
determinant  C)  on  F.  Let  Uy(n,  d)  (resp.  U£(n,  d))  denote  the  open  subset  of  U'Y(n9  d) 
(resp.  Uf^(n,  d))  consisting  of  stable  vector  bundles.  Letgc  (resp.  gy)  denote  the  geometric 
(resp.  arithmetic)  genus  of  Y. 

I.  Assume  that  gc  >  2.  Then,  except  possibly  for  gc  =  2,  n  =  2,  d  even,  one  has 


2.  Pic  U's  (n,  d)  «  Pic  U'(n,  d)  «  Pic  J  0  Z,  where  J  denotes  the  Jacobian  of  Y. 
II.  Assume  that  gy  =  2,  n  =  2.  Then 
Pic  U'c(2,d)&Z. 

4.  Corapactifications 

In  general,  the  moduli  spaces  MQ  of  principal  G-bundles  on  a  nodal  curve  F  are  not 
complete.  In  case  G  =  GL(ri)  a  compactification  of  MQ  is  given  by  the  moduli  space  of 
torsionfree  sheaves  of  rank  n  (and  fixed  degree)  on  F,  this  compactification  is  not  normal.  A 
normal  compactification  of  MG  is  obtained  as  the  moduli  space  of  (generalized)  parabolic 
bundles  on  the  desingularization  C  of  F  ([Ul,  U2]).  This  can  be  done  for  other  classical 
groups  G  =  O(n),  SO(n),  Sp(2ri)  also,  we  briefly  describe  the  main  result  (Theorem  5). 
The  details  will  appear  elsewhere  [U4].  To  construct  a  normal  compactification  of  MG, 
one  needs  a  good  compactification  of  G  and  hence  a  good  representation  of  G.  In  case 
of  classical  groups  we  use  their  natural  representations.  For  a  general  group  G,  a  natural 
choice  is  the  adjoint  representation.  Unfortunately  it  gives  a  compactification  of  G  only  if 
G  is  of  adjoint  type  ([DP],  §6;  [S];  [D]).  Using  this  compactification  we  give  a  more  general 
definition  of  QPGs  in  case  G  has  trivial  centre.  For  classical  groups  and  adjoint  groups  we 
'compactify'  the  stack  Bun^P  ^  and  also  compute  the  Picard  group  of  the  compactification. 
In  case  of  classical  groups,  the  compactifications  of  moduli  spaces  obtained  are  complete 
normal  varieties  (see  Theorem  5).  We  do  not  prove  that  the  'compactification'  is  a  proper 
stack  in  case  of  adjoint  groups.  It  will  be  useful  to  know  a  natural  (canonically  defined) 
compactification  of  G  in  the  general  case. 

4.1.  Let  the  notations  be  as  in  §3.  We  further  assume  that  G  is  a  semisimple  algebraic 
group  with  trivial  centre.  Let  g  denote  the  Lie  algebra  of  G,n  =  dimg.  GxGactsong0g 
(via  adjoint  representation)  and  hence  on  the  Grassmannian  Gr(n,  g  0  g)  of  n-dimensional 
subspaces  of  g  0  g.  Let  AG  denote  G  embedded  in  G  x  G  diagonally.  Since  G  has  trivial 
centre  the  adjoint  representation  is  faithful.  Hence  G  «  G  x  G/AG  gets  embedded  in 
Gr(n,  g  0  g)  as  G  x  G-orbit  of  Ag  e  Gr(rc,  g  0  g).  Let  F  be  the  closure  of  the  G  x  G 
orbit  of  Ag  in  Gr(n,  g  0  g). 
Given  a  principal  G-bundle  E  and  disjoint  divisors  Dj  =  Xj  -f  Zj  on  C,  define 

EJ  =  EXJ  x  EZJ  =  G  x  G,  JE'(F)  -  E*  x(GxG)  (F),  j  =  1,  .  .  .  ,  J. 


Principal  G  -bundles  on  nodal  curves  287 

A  QPG  (quasiparabolic  G-bundle)  is  a  pair  (£,  (cry))  where  E  is  a  principal  G-bundle  and 


DEFINITION  4.2 

QPGs  (£,  (cry))  and  (£',  (a'.))  are  isomorphic  if  there  is  an  isomorphism  /  :  E  ->  £'  of 
principal  G-bundles  which  maps  ay  to  (cr'.)  i.e.  for  the  isomorphism  /ji  :  Ey(F)  -»  EfJ'(F) 
one  has  //(cry)  =  aj  . 

4.3.  A  family  of  QPGs  (£,  (oy))  -»  C  x  7  is  a  family  of  G-bundles  £  -»  C  x  T  together 
with  a  section  a/  :  T  —  »•  £J'(F). 

Remark  4.4.  (  1  )  The  following  diagram  commutes 


GxG  --V     F  =  Gx  G/AG 

*2  i  I  'l 

GL(g)xGL(g)     4       Gr(n,g0g). 

Here  t\  is  inclusion,  *2  =  product  of  adjoint  representations  of  G  in  g,  /i2(/i,  /2)  = 
subspace  of  g  0  g  generated  by  {(/if,  /zv),  u  e  g}  and  t\  o  /ii  is  the  map  inducing  the 
Demazure  embedding  of  G  (by  identifying  G  with  G  x  G/AG). 

(2)  Recall  that  a  (generalized)  quasiparabolic  structure  (over  Dj  =  zy  +  zy  ,  j  =  1  ,  .  -  •  ,  /) 
on  a  vector  bundle  N  of  rank  n  is  given  by  an  n-dimensional  subspace  ofNXJ  ©  A^Zj.  ,  j  e  J 
i.e.  by  an  element  of  HyGr(n,  NXj  0  A^.)  [Ul].  Given  a  family  of  QPGs  £  ->  C  x  7,  let 
£(g)  be  the  family  of  vector  bundles  of  rank  n  associated  to  S  via  the  adjoint  representation 
of  G  in  g.  It  follows  from  the  above  commutative  diagram  that  a  composed  with  the 
injection  Tlj£J(F)  ->•  (n;-£J(Gr(n,  g  0  g)))  gives  a  quasi  parabolic  structure  on  £(g). 


4.5     77z*  5-tad:  fi        aAifl?  r/ze  stack  Bung*" 

Let  the  notations  be  as  in  3.1.  Let  <2c!c  =  2o,c  x  Fl;  ^-  ^e  ind-scheme  QG,C  is  ind- 
proper,  so  is  <2GPc-  ^e  indgroup  L$  acts  on  Qc.c-  For  each  jf,  the  evaluation  at  Xj  and 
zy  gives  an  evaluation  map  ej  :  LQ  ->  G  x  G.  G  x  G  acts  on  F  naturally.  Thus  we  have 
a  natural  action  of  L£  on  "fig^-  Let  L^YOg/;  ^e  the  quotient  stack. 

As  in  3.1,  we  define  the  &-stack  of  (generalized)  quasiparabolic  G-bundles  on  C  (with 
extended  definition  of  the  parabolic  structure  using  F).  We  denote  this  stack  by 
It  contains  Bung*"  as  an  open  substack. 

Theorem  4.  (1)  There  exists  a  canonical  isomorphism  of  stacks 


Moreover  the  projection  map  fig^c  ->  Bung3"  w  locally  trivial  for  etale  topology. 

(2)  Lef  G  be  a  simple,  simply  connected  affine  algebraic  group  over  C.  Then  there  exists 

an  isomorphism 


288  Usha  N  Bhosle 

Pic  (Bunfacr)  ^  0/ZL/  0  0yPicjP, 
where  LI  are  line  bundles  coming  from  BuriG,q- 

Proof.  The  proof  is  on  similar  lines  as  that  of  Theorem  2  and  Theorem  3,  we  omit  some 
details  to  avoid  repetition. 

(1)  QGX;  ^presents  the  functor  PC  which  associates  to  every  ^-algebra  R  the  set  of 
isomorphism  classes  of  triples  (£",  p,  s)  where  E  is  a  principal  G-bundle  on  C/e,  /?  is  a 
trivialization  of  E  over  C£  and  s  €  Mor  (Spec  #,  f];  F).  Then  s  =  (s\,  .  .  .  ,  sj),  Sj  e 
Mor  (Spec  R,  F)  for  all  j.  We  can  associate  to  such  a  triple  a  QPG  (E,  (ay))  on  C#. 
We  only  need  to  define  for  each  7,  morphism  cry  :  S  ~>  E-i(F),  S  =  Spec  7?.  The 
restriction  of/)"1  gives  isomorphisms  5  x  Xj  x  G  ^  E\sxXj,  S  x  Zj  x  G  ^  E\sxZj  and 
hence  an  isomorphism  of  G  x  G-bundles  S  x  G  x  G  ==  (5  x  Xj  x  G)  x$  (S  x  zy  x 
G)  «  EISXJC/  X5  £|Sxzj  =  EJ.  Therefore  we  have  an  isomorphism  of  associated  fibre 

bundles  pj(F)  :  S  x  F  4-  Ej(F).  Define  a/  by  0j(s)  =  pj(F)(s,Sj(s)).  It  follows 
that  Q^Q^C  x  £•  has  a  umversal  QPG  and  we  have  an  L^-equi  variant  morphism  of  stacks 
7rpar  :  QQ  c  "~^  ^uncPC*  ^^s  inc^uces  trie  morphism  jf  par  on  the  quotient  stack. 

To  define  the  inverse  of  jfpar5  let  (E,  (a/))  6  Bun^(tf).  Let  Rf  be  an  ^-algebra,  5;  = 
Spec  Rf.  Let  T(Rf)  be  the  set  of  pairs  (p/?/,  a1)  where  PR>  is  a  trivialization  of  £V,  a7  = 
(cr[,  .  .  .  ,  o'j)  where  GJ  is  a  pull  back  of  ayVj.  This  defines  a  T-space  with  an  action  of 

L£  (via  p^/)»  ^  is  an  L^-bundle  [DS].  We  now  define  a  L^-equivariant  map  T  ->  GG^C- 
Given  (p#,  a1)  e  T(R'),  we  define  jj  :  57  ->  F  by  s'j  =  /7rF  o  ((PR')J(F))~[  ocrj.  Then 


/,  (/•))  e  PG(^O-  Since  2c?Pc  represents  the  functor  PG,  this  defines  a  map 


— -gpar    .    .     _.  r  .        .          rr.1      ._  c  m 

a  :  T  -»  2G,c» *t  1S  ^G-equivanant.  The  L^-bundle  T  together  with  a  give  a  morphism 
of  stacks  from  Bun^f  ^  to  the  quotient  stack  L£  \  GGX;  which  is  easily  seen  to  be  the  inverse 
Of7fpar. 
The  assertions  about  local  triviality  of  7fpar  follow  similarly  as  in  Theorem  2. 

(2)  Using  the  facts  that  each  <2G,c/  is  an  inductive  limit  of  reduced  projective  varieties 
XitW  with  Hl(Xt,w,  O)  •=  0  and  F  is  a  projective  variety  with  Hl(F,  O)  =  0,  it  can  be 
proved  that  Pic  "gf?"  w  ®/ZO(2GiC.(l)  ©  0;-PicF  (similarly  as  Proposition  3.2).  The 
injectivity  of  TT*  follows  exactly  as  in  Theorem  3.  Note  that  F  being  a  projective  variety 

j2c,c  is  an  inductive  limit  of  integral  projective  schemes  and  hence  has  no  nonconstant 
regular  functions. 
We  now  check  the  surjectivity  of  n*.  We  have  a  commutative  diagram 

Pic(BunG,c)     ^> 
<P*± 

Pic(BungGpacr)     ^ 

with  ^  the  forgetful  morphism  and  the  right  vertical  arrow  is  the  inclusion  as  direct  sum- 
mand.  Hence  one  has  7r*ar(<p*L^)  =  OQG  c  (1),  L\  being  the  pull  back  of  the  generator 
of  Pic  (Bunc.c/)  to  Pic  (BunG,c)-  Thus  for  the  surjectivity  of  ^*ar  it  suffices  to  show  that 
there  exist  line  bundles  {Z^  .}  on  Bun^p^  which  pullback  to  the  generators  of  @j  Pic  F. 


Principal  G  -bundles  on  nodal  curves  2go 

From  the  construction  and  results  in  [S],  it  follows  that  Pic  F  is  a  lattice  of  rank  r  generated 
by  Lj,  i  =  1,  .  .  .  ,  r,  r  =  rank  of  G.  For  each  z,  there  exists  a  G  x  O  module  Wt  and 
a  G  x  G  equivariant  embedding  F  -*  P(W/)  such  that  OP(Wi)(\)  restricts  to  L'  on  F. 
Given  a  family  of  QPGs  (E,  (cr/))  on  C  x  Spec  R  one  has  £V(F)  C  E->  (P(W,-)).  Let  L' 
denote  the  line  bundle  on  Ej  (P(W,-))  (and  also  its  restriction  to  EJ(F))  which  restricts  to 
O/>(W,)0)  on  each  fibre.  The  pull-back  of  L'.j  by  cr/:  Spec  #  ->  £->'(F)  is  a  line  bundle 
L;  •  £  on  Spec  R.  This  construction  can  be  done  for  any  R.  Hence  {L\  .  R]  define  a  line 
bundle  L'y  on  the  stack  Bung^.  By  construction,  J^^L^j)  is  the  generator  of  the  jth 
factor  Pic  F  in  Pic 


CflStf  of  classical  groups 

For  the  simple  and  simply  connected  classical  groups  SL(n)  and  Sp(2ri)  the  compactifi- 
cations  F  of  G  are  defined  using  natural  representations  (described  below).  We  claim  that 
Theorem  4  holds  in  these  cases  also.  The  existence  of  the  isomorphism  7fpar  and  injectivity 
of  TT*  can  be  seen  exactly  as  in  the  proof  of  the  Theorem  4.  We  only  need  to  check  the 
surjectivity  of  7f*ar,  this  is  done  below. 

4.6.  Case  G  =  SL(n).  For  G  =  5L(n),  the  compactification  F  of  G  using  natural 
representation  of  G  ([Ul,  U4])  can  be  described  as  follows.  SL(ri)  x  SL(ji)  is  embedded 
diagonally  in  SL(2ri)  C  GL(2n).  Let  G  ~>  GL(V)  be  the  natural  representation.  Let 
P  c  SL(2n)  be  the  stabilizer  of  the  diagonal  in  V  0  V\  P  is  a  maximum  parabolic 
subgroup.  The  Grassmannian  Gr  =  SL(2n)/P  is  embedded  in  P(A  (V  0  V))  by  Pliicker 
embedding.  Let  {P/t,  ...,/„}  denote  the  Pliicker  coordinates.  Let  F  be  the  hyperplane  sec- 
tion of  Gr  defined  by  PI,  ...  )W  =  P/i+i,  ,2/1-  Then  F  can  be  regarded  as  a  compacti- 
fication of  SL(n)  with  SL(n)  identified  with  the  subset  of  F  defined  by  PI,  .  .  .  .,n  ^  0. 
The  generator  of  Pic  Gr  &  Z  is  the  line  bundle  associated  to  the  character  vun  on  P  and  its 
restriction  to  F  is  the  generator  L'  of  Pic  F  ^  Z.  F  —  SL(ri)  is  a  divisor  Dr  in  F  to  which  L1 
is  associated.  Given  afamily  of  QPGs  (E,  (a/))  on  Cx  Spec  R,  one  has  £•>  (F)  C  £V(Gr). 
Let  Z/;.  be  the  line  bundle  on  F^Gr)  associated  to  the  P-bundle  E*(SL(2n))  -+  £V(Gr) 
via  the  character  wn.  The  pull  back  of  Z/.  by  a;-  :  Spec  R  ->  £7  (F)  C  E-i  (Gr)  is  a  line 
bundle  l!j  R  on  Spec  /?,  L'.  ^  define  a  line  bundle  U-  on  the  stalk  Bun^F^.  By  construction, 

TTparCLy)  is  the  generator  of  the  jth  factor  Pic  F  in  Pic  £>G,C-  Hence  the  niorphism  in 
Theorem  4(2)  is  a  surjection  and  thus  an  isomorphism  for  G  =  SL(n)  and  ^P  as  above. 


4.7.  Case  G  =  Sp(2n).  In  case  G  =  Sp(2n)  also  one  can  use  the  natural  representation  of 
GtodefineF(§5,[U4]).  LetG  ~>  GL(V)  be  the  natural  representation.  We  regard  Sp(2n) 
as  the  group  Sp  (q,  V)  of  automorphisms  of  V  preserving  a  symplectic  form  (nondegenerate 
alternating  form)  q  on  V.  Then  F  is  the  variety  of  maximum  isotropic  subspaces  for 
q  ©  (-q)  on  V  0  V.  The  group  Sp(2n)  x  Sp(2n)  =  Sp(<?,  V)  x  Sp(-$,  V)  is  embedded 
in.Sp(0®(-0),  V0V)  =Sp(4n)  diagonally.  ThenF  «  Sp(4n)/P,P  being  the  maximum 
parabolic  subgroup  of  Sp  (4ri)  which  is  the  stabilizer  of  the  maximum  isotropic  subspace  A  y 
of  V  0  V ,  Pic  F  =  ZL' ,  L'  being  the  line  bundle  associated  to  the  fundamental  weight  w2n  - 
Given  a  family  of  QPGs  (F,  (cr/))  on  C  parametrized  by  5  =  Spec/?, cry  :  5  -*  £'(F),one 
hasF^'(F)  =  EJ(Sp(4n)/P)  and£J(Sp(4n))  ->  ^'(F)  isaP-bundle.  Let  ^  denote  the 


290  Usha  N  Bhosle 

line  bundle  on  E-i(F)  associated  to  this  P-bundle  via  the  character  W2n>  Let  L'>  R  denote 
the  line  bundle  on  S  which  is  the  pullback  of  this  line  bundle  by  a/.  This  constructior 
being  valid  for  any  R,  it  defines  a  line  bundle  Z/;.  on  the  stack  Bun^.  Clearly,  n*(L'j] 
is  the  generator  of  Pic  F,  the  j'th  factor.  It  follows  that  the  injection  in  Theorem  4(2)  is  ar 
isomorphism  for  G  =  Sp(2n)  with  F  defined  as  above. 

The  following  definitions  and  results  are  stated  for  O(n)-bundles,  they  hold  for  Sp(2n)- 
bundles  also  with  orthogonal  replaced  by  symplectic  and  n  replaced  by  2n . 

DEFINITION  4.8 

An  orthogonal  bundle  (£,  q)  on  C  is  an  /-tuple  of  vector  bundles  E  =  (E\, . . . ,  £/),  £/  - 
a  vector  bundle  on  F/  with  a  nondegenerate  quadratic  form  q-{  and  q  =  (#!,...,#/).  W< 
assume  that  rank  E\  =  n  for  all  /,  we  call  n  the  rank  of  E.  For  a  closed  point  x  e  C,  let  qx 
denote  the  induced  quadratic  form  on  the  fibre  Ex. 

DEFINITION  4.9 

A  generalized  quasiparabolic  orthogonal  bundle  (orthogonal  QPB  in  short)  on  C  is  ar 
orthogonal  bundle  (E,  q)  of  rank  n  together  with  n-dimensional  vector  subspaces  Ff(E] 
of  EXj  ©  EZj  which  are  totally  isotropic  for  qx.  ®  (— qzj). 

Theorem  5.  Assume  further  that  Y  is  irreducible.  Then  there  is  a  coarse  moduli  space  M 
for  ct-semistable  orthogonal  QPBs  of  rank  n,  a  e  (0,  1)  being  rational.  M  is  normal  ant 
complete. 

Let  U  be  the  moduli  space  of  orthogonal  sheaves  of  rank  n  on  7.  Assume  that  0  <  a  < 
I ,  a.  is  close  to  1 .  Then 

(1)  there  exists  a  morphism  /  :  M  ->  U. 

(2)  Let  UK  be  the  subset  of  U  corresponding  to  stable  orthogonal  bundles.    Then  the 
restriction  of  /  to  f~l(^)  is  an  isomorphism  onto  Ufr 

Acknowledgement 

I  would  like  to  thank  Tomas  Gomez,  S  Kumar,  R  V  Gurjar  and  N  M  Singhi  for  usefu 
discussions  during  the  preparation  of  this  paper. 

References 

[BL]  Beauville  A  and  Laszlo  Y,  Conformal  blocks  and  generalised  theta  functions.  Comm.  Math, 

Phys.  164(1994)385-419 
[D]  Demazure  M,  Limites  de  groupes  orthogonaux  ou  symplectiques,  unpublished  preprini 

(1980) 
[D-P]  De  Concini  C  and  Procesi  C,  Complete  symmetric  varieties.  Invariant  theory,  Proc.  Monte- 

cantini  (Springer)  (1982)  LNM  996 
[D-S]  Drinfeld  V  and  Simpson  C,  B -structures  on  G-bundles  and  local  triviality,  Math.  Res.  Lett, 

2  (1995)  823-829 

[H]  Hartshorne  R,  Algebraic  Geometry  (Springer- Verlag)  (1977) 
[K]  Kumar  S,  Demazure  character  formula  in  arbitrary  Kac-Moody  setting,  Inv,  Math.  89  (1 987^ 

395-423 

[KN]  Kumar  S  and  Narasimhan  M  S,  Picard  group  of  the  moduli  spaces  of  G-bundles,  e-prini 
alg-geom/9511012 


Principal  G -bundles  on  nodal  curves  29 1 

[L-R]  Laumon  G  and  Rapoport  M,  The  Langlands  lemma  and  the  Betti  numbers  of  stacks  of 

G-bundles  on  a  curve.  Int.  J.  Math.  7  (1996)  29-45 
[L-S]  Laszlo  Y  and  Sorger  C,  The  line  bundles  on  the  moduli  of  parabolic  G-bundles  over  curves 

and  their  sections.  Ann.  ScL  de  ENS  30(4)  (1997) 
[M]  Mathieu  O,  Formules  de  caract'eres  pour  les  algebres  de  Kac-Moody  generates  Asterisaue 

(1988)  159-160  ' 

[NRS]  Narasimhan  M  S,  Ramanathan  A  and  Kumar  S,  Infinite  Grassmannian  and  moduli  spaces 

of  G-bundles,  Math.  Ann.  300  ( 1 993)  395-423 
[Rl]  Ramanathan  A,  Stable  principal  bundles  on  a  compact  Riemann  surface,  Math  Ann  ?13 

(1975)  129-152 
[S]  Strickland  E,  A  vanishing  theorem  for  group  compactifications.  Math.  Ann.  277  (1987) 

165-171 
[So]  Sorger  C,  On  moduli  of  G-bundles  of  a  curve  for  exceptional  G,  Ann.  Sci.  E.N.S.  (4)  32(1) 

(1999)  127-133 

[Sp]  Springer  T  A,  Linear  algebraic  groups.  Progress  in  Mathematics  (Birkhauser)  (198  I ) 
[T]  Teleman  C,  Borel-Weii-Bott  theory  on  the  moduli  stack  of  G-bundles  over  a  curve   Invent 

Math.  134(1998)  1-57 
[Ul]  Usha  N  Bhosle,  Generalised  parabolic  bundles  and  applications  to  torsionfree  sheaves  on 

nodal  curves,  Arkiv.  for  Matematik  30(2)  ( 1 992)  1 87-2 1 5 
[U2]  Usha  N  Bhosle,  Vector  bundles  on  curves  with  many  components,  Proc.  London  Math.  Soc. 

79(3)  (1999)  8 1-1 06 
[U3]  Usha  N  Bhosle,  Representations  of  the  fundamental  group  and  vector  bundles,  Math.  Ann. 

302(1995)601-608 
[U4]  Usha  N  Bhosle,  Generalised  parabolic  bundles  and  applications  II,  Proc.  Indian  Acad.  ScL 

(Math.  Sci.)  106(4)  (1996) 

[U5]  Usha  N  Bhosle,  Picard  groups  of  the  moduli  spaces  of  vector  bundles,  314  (1999)  245-263 
[U6]  Usha  N  Bhosle,  Picard  groups  of  moduli  of  semistable  bundles  and  theta  functions,  TIFR 

preprint  (1999) 


Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  Ill,  No.  3,  August 2001,  pp.  293-318. 
©  Printed  in  India 


Uncertainty  principles  on  two  step  nilpotent  Lie  groups 


SKRAY 

Stat-Math  Unit,  Indian  Statistical  Institute,  203  B.T.  Road,  KolkataTOO  035,  India 
Present  Address:  Department  of  Mathematics,  Indian  Institute  of  Technology,  Kanpur 
208  016,  India 
.  E-mail:  res9601@isical.ac.in;  skray@iitk.ac.in 

Abstract.  We  extend  an  uncertainty  principle  due  to  Cowling  and  Price  to  two  step 
nilpotent  Lie  groups,  which  generalizes  a  classical  theorem  of  Hardy.  We  also  prove 
an  analogue  of  Heisenberg  inequality  on  two  step  nilpotent  Lie  groups. 

Keywords.  Uncertainty  principles;  Hardy's  theorem;  two  step  nilpotent  Lie  groups; 
Heisenberg' s  inequality. 

1.  Introduction 

As  a  meta-theorem  in  harmonic  analysis,  the  uncertainty  principles  can  be  summarized  as: 
A  nonzero  function  and  its  Fourier  transform  cannot  both  be  sharply  localized.  When  sharp 
localization  is  interpreted  as  very  rapid  decay,  this  meta-theorem  becomes  the  following 
theorem  due  to  Hardy  ([4]). 

Theorem  1.1  (Hardy).  Let  f  :  R  ->  C  be  measurable  and  for  allx,  y 
(0  1/001  <  Ce-«**2, 
(")  1/001  <  Ct~b*y\ 

where  C,a,b>O.Ifab>l  then  f  =  0  almost  everywhere.  Ifab  =  l  then  f(x)  = 
CoTanx  .  Ifab  <  I  then  there  exist  infinitely  many  linearly  independent  functions  satisfying 
(i)  and  (ii). 

Considerable  attention  has  recently  been  paid  to  discover  analogues  of  Hardy 's  theorem  in 
the  context  of  Lie  groups  ([28, 27, 5, 1, 16,25,24, 12,8,22]).  Coming  back  to  R,  we  see  that 
the  decay  conditions  can  be  stated  as  |kfljr/llL°°(R)  <  °°  an(*  Ik&r /llL°°(R)  <  °°»  where 
e^(x)  =  e**  .  So  one  reasonable  question  is  to  ask:  what  happens  if  lk<ur/lk/>(R)  <  °° 
and  |kfc;r/lk<?(R)  <  oo,  where  1  <  p,q  <  oo?  The  answer  is  given  by  the  following 
theorem  due  to  Cowling  and  Price  ([6]). 

Theorem  1.2  (Cowling  and  Price).  Let  f  :  R  -*  C  be  measurable  and 

(0    \\e<urf\\LP(R)  <  00, 
(ii)    |kfcr/||£/?(R)  <00, 

where  a,  b  >  0  andmin(p,  q)  <  oo.  Ifab  >  I  then  f  =  0  almost  everywhere.  Ifab  <  1 
then  there  exist  infinitely  many  linearly  independent  functions  satisfying  (i)  and  (ii). 

The  proof  of  the  above  theorem  uses  the  following  result  (see  [6]). 
Lemma  1.1.  Ifg  :C~*  Cis  an  entire  function  and  for  1  <  p  <  oo 


294  5  K  Ray 

(i)  \g(x  + 
(ii)  (fR\g 
then  g  =  0. 

The  importance  of  Theorem  1  .2  is  that  even  ifthepoinrwise  decay  is  replaced  by  averagt 
decay,  Hardy's  uncertainty  principle  continues  to  be  true.  As  expected,  the  case  ab  >  '. 
of  Hardy's  theorem  follows  trivially  from  that  of  Cowling  and  Price.  Actually,  if  we  droj 
the  case  ab  =  1,  then,  on  the  real  line  (more  generally  on  Rn),  the  above  theorems  an 
equivalent  (see  [3]). 

The  following  theorem,  which  follows  as  a  corollary  from  a  deep  theorem  of  Beurlin^ 
([14]),  also  suggests  another  generalization  of  Hardy's  theorem. 

Theorem  1.3.  Let  f  :  R  -»  C  be  measurable  and  for  allx,  y 
(i)  l/00|<Ce-fl*W'f 
(ii)  l/OOIrSCe-^l' 

where  C,  a,  b  >  0,  p~l  +  q~l  =  1,  1  <  p,q  <  oo.If(ap)l^(bq)1^  >  2,  thenf  =  (. 
almost  everywhere. 

In  this  paper  our  aim  is  to  get  analogues  of  Theorems  1.2,  1.3  on  connected,  simpl} 
connected,  two  step  nilpotent  Lie  groups  (see  [9,  3]  for  analogues  of  Theorem  1.2  on  othei 
groups).  We  also  prove  an  analogue  of  Heisenberg's  inequality  on  two  step  nilpotent  Lie 
groups  which  was  previously  known  only  in  the  case  of  Heisenberg  groups  (see  [29]). 

This  paper  is  organized  as  follows:  in  §2  we  fix  notation  and  describe  some  backgrounc 
material  leading  to  a  proof  of  the  Plancherel  theorem  via  the  description  of  the  Hilbert- 
Schmidt  norm  of  the  group  Fourier  transform,  and  in  §3  we  prove  the  proposed  analogue 
of  Theorem  1.2  and  indicate  a  proof  of  Theorem  1.3.  In  §4  we  prove  an  analogue  ol 
Heisenberg's  inequality. 

Finally  we  would  like  to  point  out  that  all  the  results  except  Theorem  3.2  are  from  the 
author's  1999  Ph.D.  thesis  of  the  Indian  Statistical  Institute. 

2.  Notation  and  background  material 

For  a  Lie  algebra  g  (we  will  always  work  with  Lie  algebras  over  R),  we  define  g1  =  [g,  g] 


DEFINITION  2.1 


A  Lie  algebra  g  is  called  two  step  nilpotent  if  g2  =  0  and  g1  ^  0.  The  connected  simply 
connected  Lie  group  G  corresponding  to  such  a  g  is  called  a  two  step  nilpotent  Lie  group. 

We  find  it  more  convenient  to  look  at  a  two  step  nilpotent  Lie  algebra  in  another  way. 
Let  B  :  Rn~m  x  Rn~m  -+  Rm  be  a  nondegenerate,  alternating,  bilinear  map.  Let  g  = 
Rw  0Rn~m,  we  define 


where  z,  zf  €  Rm  and  u,  i/  €  R"~m .  Then  [., .]  is  a  Lie  bracket  and  g  is  a  two  step  nilpotent 
Lie  algebra  with  Rm  as  the  center  of  g.  If  on  G  =  Rm  ©  Rn~m  we  define  the  product 

(z,  u).(z',  v;)  =  (z  +  z'  +  -B(v,  i/),  v  +  1/Y  (2-2) 


i/V 


Uncertainty  principles  295 

then  G  is  a  connected,  simply  connected,  two  step  nilpotent  Lie  group  with  g  as  its  Lie 
algebra  and  exp  :  g  ->  G  is  the  identity  diffeomorphism.  In  this  section  we  will  first 
describe  the  effective  unitary  dual  of  a  connected,  simply  connected,  two  step  nilpotent  Lie 
group  g  following  Kirillov  theory.  Our  notations  are  standard  and  can  be  found  in  [7]. 

Let  g*  be  the  real  dual  of  g .  Then  G  acts  on  g*  by  the  coadjoint  action,  that  is  G  x  g*  -»  g* , 
(£.0  -*  8-1  is  given  by 


=     /(Ad(exp-y)(X)), 

=     /(ead~-v(X)) 


Let  /  G  g*.  Then  we  denote  O/  =  the  coadjoint  orbit  of  /.  BI  =  the  skew  symmetric  matrix 
corresponding  to  /,  that  is,  given  a  basis  {Xi , . . . ,  XOT,  Xw+i, . . . ,  Xn]  of  g  through  the 
center  (that  is,  X  i , . . . ,  Xm  span  the  centre  of  g,  we  consider  the  matrix  BI  =  (Bi  (7,  /'))  = 
(/([X,- ,  X7-]) .  r/  =  The  radical  of  the  bilinear  form  £/,  that  is, 

r/  =  {Xefl:/([X,F])=0    for  all  7  eg}. 

Clearly  r/  is  an  ideal  of  g  and  g(=  R/n)  C  n.  f\  —  spanR{Xw+i, . . . ,  Xn]  n  r/.  £/  = 
BI  |  R/1~m  x  R"~w  that  is  restriction  of  BI  on  the  complement  of  the  center  of  g. 

It  follows  trivially  for  two  step  nilpotent  Lie  groups  that  all  the  coadjoint  orbits  are 
hyperplanes  ([17,  23]).  In  fact  we  have  from  the  above,  the  following. 

Theorem  2.1.  Let  /eg*.  Then  £>/=/  +  r/-  where  r,1  =  [h  6  g*  :  h  \  n  =  0}. 

In  particular,  /'  e  O\  if  and  only  if  n  ~  r//  and  /  |  ri  =  I1  \  r//. 

Let  g  be  a  two  step  nilpotent  Lie  algebra  such  that  dim  g  =  n  with  the  basis  B  = 
{Xi, . . . ,  Xm,  Xm+i, . . . ,  Xn}  through  the  centre.  Then  BI  is  the  n  x  n  matrix  whose 
(z,  ;)th  entry  is  /([X/,  X/]),  1  <  i,  7  <  n.  Let  B*  =  {X?, . . . ,  X*}  be  the  dual  basis  of 
g*.  This  is  a  Jordan-Holder  basis,  that  is,  g^  =  spanR{X^, . . . ,  X*}  is  Ad*(G)  stable  for 

I  <j  <n. 
Let  /  €  g*  and  X;  €  B. 

DEFINITION  2.2 

The  term  i  is  called  a  ;'wm/7  i/wtoc  for  /  if  the  rank  of  the  i  x  n  submatrix  of  BI  ,  consisting  of 
the  first  i  rows,  is  strictly  greater  than  the  rank  of  the  (i  -  1 )  x  n  submatrix  of  BI  ,  consisting 
of  the  first  (i  —  1)  rows. 

Since  an  alternating  bilinear  form  has  even  rank  the  number  of  jump  indices  must  be 
even.  The  set  of  jump  indices  is  denoted  by  J  —  {ji, . . . ,  jik}>  Notice  that  j\>m  +  1. 
The  subset  of  B  corresponding  to  J  is  then  {X^ , . . . ,  Xhk }.  Notice  that  if  i  is  a  jump  index 
then  rankB/  =  rank£/~  *  +  1,  where  B\  is  the  submatrix  of  BI  consisting  of  first  i  rows. 

Note  2.1.  These  jump  indices  depend  on  /  and  on  the  order  of  the  basis  as  well.  But 
ultimately  we  will  restrict  ourselves  to  'generic  linear  functionals'  and  they  will  have  the 
same  jump  indices. 


296  SKRay 

Now  we  are  going  to  spell  out  what  we  mean  by  generic  linear  functional.  This  is  also 
a  basis  dependent  definition.  We  work  with  the  basis  B  chosen  above.  Let  RL  (Z)  =  rank^ 
and  Ri  =  max{ /£,-(/)  :  /  €  a*}. 

DEFINITION  2.3 

A  linear  functional  /  e  Q*  is  called  generic  if  /?/(/)  =  /?/  for  all  i,  1  <  z  <  n. 

Let  £Y  =  {Z  e  g*  :  Z  is  generic}.  Since  for  any  Z  €  G*,  we  have  g.Z  |  $  =  I  [3  where 
g.l  =  I  o  Adg"1,  we  get  #/(/)  =  #/(#./),  1  <  /  <  n  and  hence, 

(i)  Z^  is  a  G-invariant  Zariski  open  subset  of  cj*.  So  Z^  is  union  of  orbits, 
(ii)  If  j  is  a  jump  index  for  some  /  e  U,  then  j  is  a  jump  index  for  all  Z  e  U. 

(iii)  Let  Z  e  W,  then  the  number  of  jump  indices  for  Z  is  the  same  as  the  dimension  of  Oi 
(as  a  manifold).  For,  the  rank  of  the  matrix  BI  is  equal  to  the  number  of  jump  indices 
(=  2k,  say)  and  the  dimension  of  the  radical  r/  is  the  nullity  of  the  matrix  of  £/,  which 
is  n  —  2k.  Since  g/r/  is  diffeomorphic  to  0/  (see  [7]),  we  have  dim  #/  =  2k. 

(iv)  Every  orbit  in  U  is  of  maximum  dimension  though  not  every  maximum  dimensional 
orbit  may  be  in  U. 

Note  2.2.  If  Z  €  Q*  is  such  that  BI  is  an  invertible  matrix,  then  r/  =  g  and  then  m  + 1 , . . . ,  n 
are  all  jump  indices  and  moreover 

IA  =  {I  €  £*  :  BI  is  an  invertible  matrix}. 

Clearly,  if  the  codimension  of  3  in  g  is  odd  then  this  cannot  happen.  Following  [  1 8]  and  [19], 
we  call,  the  two  step  nilpotent  Lie  algebra  a  M W  algebra,  if  there  exists  Z  e  Q*  such  that  BI 
is  nondegenerate  (or  the  corresponding  matrix  is  invertible).  For  example,  Heisenberg  Lie 
algebras  and  [2/1,2*  the  free  nilpotent  Lie  algebras  of  step  two  are  MW  algebras  (see  [2]). 

Our  aim  is  to  parametrize  the  orbits  in  U.  We  will  see  that  they  constitute  a  set  of  full 
Plancherel  measure.  We  again  describe  some  notation. 

W  =  {!,...,  m,  ni,...,  nr}  c  {!,...,  n} 

is  the  complement  of  7  in  {1, . . . ,  n},  V)  =  .spanR{X/f.  :  1  <  i  <  2Jfc,  jt  €  /},  VN  = 
spanR{Xi,...,Xw,Jm  :  1  <  i  <  r,nf-  €  AT},  Vj  =  spanR{X*, . . . ,  *y,  V*  = 

spanR{Xt,  -  -  - ,  **  f  X*  :  n,  G  TV},  V*  -  span^X*,.  :  1  <  i  <  r}. 

The  following  theorem  shows  that  there  exist  a  vector  subspace  of  Q*  which  intersects 
almost  every  orbit  contained  in  U  at  exactly  one  point  (see  [7]).  In  the  two  step  case  one 
can  easily  prove  it  using  Theorem  2.1  (see  [23]). 

Theorem  2.2.  (i)  V£  intersects  every  orbit  in  U  at  a  unique  point  (ii)  There  exist  a 
birational  homeomorphism  W  :  (V^  n  U)  x  Vj  -»  U. 

Note  2.3.  For  each  coadjoint  orbit  in  U,  we  choose  their  representatives  from  VJJ  flU.  Note 
that  Vff  C\U  can  be  identified  with  the  cartesian  product  of  V^  and  a  Zariski  open  subset 
U1  of  3*,  where  W'  =  {Z  e  3*  :  £/(/)  =  /?/,!</<  m}. 

We  begin  with  a  brief  discussion  of  Kirillov  theory,  for  details  see  [7].  Let  G  be  a 
connected,  simply  connected  nilpotent  Lie  group  with  Lie  algebra  g.  G  acts  on  g*  by  the 


Uncertainty  principles  297 

coadjoint  action.  Given  any  /'  <E  $*  there  exist  a  subalgebra  ()//  of  g  which  is  maximal  with 
respect  to  the  property 

/'(«/',  I)/'])  =  0-  (2.3) 

Thus  we  have  a  character  ///  :  exp(fy/)  ->•  T  given  by 

Xc(expX)  =  e27r"'(X),  X  e  fy. 

Let  ^=ind^p(h//)x/'.  Then 

(1)  Tii'  is  an  irreducible  unitary  representation  of  G. 

(2)  If  fy  is  another  subalgebra  maximal  with  respect  to  the  property  /'([I)7,  1}'])  =  0,  then 


(3)  JT/J  =  7T/2  if  and  only  if  /i  and  /2  belong  to  the  same  coadjoint  orbit. 

(4)  Any  irreducible  unitary  representation  n  of  G  is  equivalent  to  jr/  for  some  /  €  $*. 

So  we  have  a  map  /<:  :  g*/Ad*(G)  —  >  G,  which  is  a  bijection.  A  subalgebra  correspond- 
ing to  /  €  cj*,  maximal  with  respect  to  (2.3)  is  called  a  polarization.  It  is  known  that  the 
maximally  of  I)  with  respect  to  (2.3)  is  equivalent  to  the  following  dimension  condition 

dim  I)  =  -(dim  g  +  dimr/). 

Now  suppose  g  is  a  two  step  nilpotent  Lie  algebra  and  /  €  g*.  The  following  technique  for 
construction  of  a  polarization  corresponding  to  /,  seems  to  be  standard:  we  consider  the 
bilinear  form  #/  on  the  complement  of  the  center,  we  restrict  J9/  on  its  nondegenerate  sub- 
space,  then  on  that  subspace  we  can  choose  a  basis  with  respect  to  which  BI  is  the  canonical 
symplectic  form.  With  a  little  modification  the  basis  can  be  chosen  to  be  orthononnal  as 
well.  This  is  essentially  what  was  done  to  obtain  a  canonical  polarization  in  [19,  2,  26,  21  ]. 
We  will  set  down  the  basis  change  explicitly;  our  main  ingredient  for  that  is  the  following 
lemma. 

Lemma  2.1.  Let  B  :  Rn  x  R'1  ->  R  be  a  nondegenerate  f  alternating,  bilinear  form.  Then 
there  exists  an  orthonormal  basis  {#/,  7,-  :  1  <  i  <  k}  ofRn  such  that  B(Xi,  Yj)  = 
Sijkj(B),  B(Xi,  Xj)  =  B(Yi,  Yj)  =  0,  1  <  /,  j  <  k,  n  =  2k  where  ±ikj(B)  are  the 
eigenvalues  of  the  matrix  ofB. 

As  a  consequence  we  have  the  following. 

COROLLARY  2.2.1 

Let  I  €  g*.  Then  there  exist  an  orthononnal  basis 

[Xi,  .  .  .  ,  Xm,  Zi(/),  .  .  .  ,  Zr(/),  Wi(l),  .  .  .  ,  Wk(l),  Fi(/),  .  .  .  ,  y*(/)}        (2.4) 


o/Q  such  that 

(a)  n  =  spanR{Xi,  ...,X 

(b)  l([Wt(D,  Yj(l)])  =  Sij^jd),  l<ij<k  and 


=  l([Yi(l),  Yj(l)])  =  0,  1  <  f,  j  <  k. 
(c)  spanR{Xi,  •  •  -  ,  Xw,  Zi(Z),  .  .  .  ,  Zr(/),  Wi(Z),  .  .  .  ,  Wi(/)}  =  I)  is  a  polarization  for  L 


298  S  K  Ray 

For  a  proof  see  [23].  We  call  the  above  basis  an  almost  symplectic  basis.  Given  X  e  ft 
and  a  basis  (2.4)  we  write 

m  r  k  k 

X  =  ^XjXj(l)  +  ^ZjZj(l)  +  ]T  WjWj(l)  +  ^yjYj(l)  =  (xt  z,  u;,  v). 

./=!  7  =  1  7  =  1  ./  =  ! 

Since  we  are  going  to  use  induced  representations  we  need  to  describe  nice  sections  of 
G/H  and  a  G-invariant  measure  on  G/H.  In  our  situation  H  will  always  be  a  normal 
subgroup  of  G.  We  identify  G  and  3  via  the  exponential  map.  Let  I)  be  an  ideal  of  ft 
containing  -5  and  H  —  exp  I). 

We  take  {X\,  .  .  .  ,  Xm  ,  Xm+i  ,  .  .  .  ,  Xm+k,  .  .  .  ,  Xn  }  a  basis  of  ft  such  that 

3  =  spanR{Xi,  .  .  .  ,  Xm},      ()  =  spanR{Xi,  .  .  .  ,  X,n,  Xm+i,  .  .  •  ,  Xm+k}. 

If  L?(.x)  =  g"1*  and  Rg(x)  =  ,xg,  *,  £  e  G,  then  it  is  clear  from  the  group  multiplication 
that  the  Jacobian  matrix  for  either  of  the  transformations  is  upper  triangular  with  diagonal 
entries  1.  Thus  we  have  the  following  lemma  whose  proof  can  be  found  in  [7]. 

Lemma  2.2.  Let  ft,  I"),  {X\  ,  .  .  .  ,  Xm  ,  Xm+\  ,  .  .  .  ,  Xm+k,  .  .  .  ,  Xn  }  be  as  before.  Then 
(i)  dx\  .  .  .  ^A'/2  /$  a  left  and  right  invariant  measure  on  G. 
(ii)  a  :  G/H  —  >  G  gfven  /ry 


H  I  =  exp 


is  a  sect  ion  for  G/H. 
(iii)  d/A'/H+jk+i  .  .  .  dxn  is  a  G-invariant  measure  on  G/H. 


Now  we  come  to  the  construction  of  representations  corresponding  to  /  6  V^  Pi  U.  Let 
dim  i't  =  777. 4- J"  and  dim  0\  =  Ik  so  m +r +2k  =  n.  We  choose  an  almost  symplectic  basis 
(2.4)  of  ft  corresponding  to  /  and  get  hold  of  I)/  as  in  Corollary  2.2.1,  c).  On  HI  =  exp(I)/) 
we  have  the  character  //:///->  T.  Let  ni  =  ind^  xi-  We  do  not  use  the  standard  model 
for  the  induced  representation  as  given  in  chapter  2  of  [7],  rather  using  the  continuous 
section  a  given  in  Lemma  2.2.2  and  computing  the  unique  splitting  of  a  typical  group 
element 

(x,  z,  w,  >')  =  (0,  0,  0,  y)  (x  -  ~[(0,  0,  0,  y),  (0,  z,  w,  0)1  z,  w,  0  j  , 

corresponding  to  cr,  the  representation  TCI  is  realized  on  L2(R*)  and  is  given  by 
(TT/CX,  z,  iu,  y)/)(y)         /  €  L2(R*), 

for  y  €  R*. 


Uncertainty  principles 

DEFINITION  2.4 
For  /  €  g*  we  define 


299 


called  the  Pfaffian  of/,  where  (£/);,  =  /([*/,.,  Xjt]),  Xji,Xjs  e  Vj. 

Note  2.4.  If  J  is  the  set  of  jump  indices  for  /,  then  B[  is  nondegenerate  on  V>  and  then 
P/(/)  is  the  Pfaffian  of  B\  (see  [15]).  It  is  easy  to  show  that 

(a)  Aei((B[)is)  is  always  a  square  of  a  polynomial  and  hence  Pf(l)  is  a  homogeneous 
polynomial  in  /  |  3. 

(b)  Pf(l)^OifleU  and  is  Ad*G  invariant. 

We  restrict  our  attention  to  the  representations  TT/  for  I  e  V*  n  U  and,  motivated  by 
the  example  of  the  Heisenberg  groups  ask  the  following  question:  suppose  /  e  Ll(G)  fl 
L2(G).  What  is  the  relation  between  /(TT/)  and  f\f(l  \  5,  u)?  Here  /  is  the  operator 
valued  group  Fourier  transform,  (z,  u)  are  elements  of  the  group  with  z  e  3  and  u  € 
SpanR{Xm+.i,  .  .  .  ,  Xn]  and  ^"i/(/  |  3,  u)  means  the  partial  (Euclidean)  Fourier  transform 
of  /  in  the  central  variables  at  the  point  /  |  g. 

In  the  case  of  the  Heisenberg  groups  Hn  ,  with  Lie  algebra 


with  the  only  nontrivial  Lie  brackets  [Wit  F/]  =  Z,  1  <  i  <  n,  we  have  V^  =  SpanR{Z}, 
Vj  =  SpanR{Wi,  .  .  .  ,  7W}  and  V*  n  W  =  {/  €  1^  :  /(Z)  =  A.  ^  0}.  Then  it  can  be  proved 
easily  (see  [10]),  that  for  /€  L[(Hn)C\  L2(Hn)  mdl  eV*C\U, 

(2.6) 

To  find  an  analogue  of  (2.6),  the  most  important  thing  is  to  find  the  Jacobian  of  a 
transformation  which  we  are  now  going  to  describe. 

Let  I  e  V%  =  SpanR{X*1?  .  .  .  ,  X*r}.  Notice  that  for  Hn  and  /y2,  where  n  is  even, 
VN  =  {0},  so  the  transformation  we  are  going  to  describe,  appears  only  for  those  two  step 
nilpotent  Lie  groups  whose  Lie  algebras  are  not  MW.  Suppose  ln.  =  l(Xni),  1  <  i  <  r; 
we  also  have  l(X^)  =  0,  1  <  i  <  2k.  From  BI  we  have  constructed  an  orthonormal  basis 
{Zi(0,  •  •  •  ,  Zr(/)f  Wi(/),  .  .  .  ,  Wik(/)f  FK/),  .  .  .  ,  7^(/)}  with  respect  to  which  the  matrix  of 
j5/  is  of  the  following  form 


(2.7) 


where  the  2k  x  2k  matrix  S  is  given  by 
/ 

0 


300  SKRay 

where  A./  (/)  >  0,  1  <  i  <  k'  .  Let  /(Z/(/))  =  /),  1  <  z  <  r.  We  consider  the  map 

0:  V^(=Rr)-»r 

0(/np...,/nr)  =  &,...,£).  (2.9) 

Lemma  2.3.  7/i£  modulus  of  the  Jacobian  determinant  of<p  is  given  by 


where  J<p  is  the  Jacobian  matrix  ofcf). 

Proof.  First  we  systematically  describe  the  transformations  which  gave  the  almost  sym- 
plectic  basis.  We  restrict  ourselves  only  to  the  complement  of  the  center,  because  it  is  there 
that  the  change  of  basis  takes  place. 

A  1    : 
A2    '•  [Xni  , 

A3  :{Xnit. 


where  Xn/  =  Xn/  -  Y=i  4(^)^y*»  ^  -  z  ~  r»  so  that  each  ^«/  €  r/-  ^1  is  Just  a 
rearrangement  of  basis  and  hence  is  given  by  an  orthogonal  matrix.  A.I  is  clearly  given  by 
a  lower  triangular  matrix  with  diagonal  entries  equal  to  one.  The  matrix  of  A^  looks  like 


(A'     C'\ 
\   0     D'  ) 


where  A!  is  a  r  x  r  matrix,  C'  is  a  r  x  2k  matrix  and  Df  is  a  2k  x  2&  matrix,  because  A  3 
is  obtained  from  the  following  operations: 

(i)  Gram-Schmidt  orthogonalization  of  {Xn.  :  1  <  /  <  r}. 

(ii)  Finding  the  orthogonal  complement  of  the  span  of  {Xm  :  1  <  i  <  r}. 
(iii)  Choosing  an  almost  symplectic  basis  on  the  nondegenerate  subspace  of  BI. 

Notice  that  for  /  6  VN,  /(X/.)  =  0,  1  <  i  <  2fc;  thus  l(Xnj)  =  l(Xni),  I  <  i  <  r. 
Hence 


Since  |  detAi.detAi.det  A$\  =  1,  we  have  [detAsJ  =  1.  But 
|detA3|  = 


So  |  det  70  1  =  |  det  Df\~l  .    If  we  write  BI  in  terms  of  the  basis  {Xni  ,  .  .  .  ,  Xnr  ,  X/,  , 
.  .  .  ,  Xy^  },  then  the  matrix  of  5/  looks  like 


where  (B/),^  =  l([Xh ,  XJs]).  Thus  clearly  |  det  B[\  =  \Pf(l)\2.  Because  of  A3  the  above 
matrix  changes  to 


0     D'B'^DJ 


Uncertainty  principles  30  1 

which  is  nothing  but  the  matrix  in  (2.7).  So 


Thus  |  det  70  1  =  —  —  -  —  —  as  claimed. 


Now  we  come  to  the  analogue  of  (2.6).  Given  /  e  Ll(G)  n  L2(G)  and  TT  e  G  the  so 
called  group  Fourier  transform  at  n  is  the  bounded  linear  transformation  (realized  on  the 
Hilbert  space  7i.n  )  given  by 


We  recall,  for  /  €  V£  fW,  the  almost  symplectic  basis  (2.4)  and  because  of  the  orthonormal 
basis  change,  d*dzdiyd>'  is  the  normalized  Haar  measure  on  G  we  started  with,  where 


m 
c,z,  w,  y)  = 


The  representation  TT/  corresponding  to  /  is  now  given  by  (2.5).    Let  dlni  ..  .dlnr  de- 
note the  usual  Lebesgue  measure  on  V£  (after  we  identify  V£  with  Rr  through  the  basis 


Theorem  2.3.  Let  f  6  Ll(G)  n  L2(G). 


=  I        \Fif(h,...Jm,xni,...,xnr,u,v)\LAxni...to  (2.10) 

Jw+u 

for  almost  every  I  eV^(~\Uf  where 

F\f(l\,  ...,lm,xni,...,xnr,u,v) 

=  [    f(xi,...,xm9Xnl,...,xn,,u,v) 

Jftm 

andl(Xj)  =1J9  1  <;  <  m. 

/.  Let  0  e  L2(R^).  Then  from  (2.5), 


,  z,  u;,  y)(7r/(-^,  -z,  ~iu,  - 

,  Z,  U),  y)e2ff't-;W-/W-E 


/  /(jc,  z,  u;,  y  -  55)e2^/[~/w-/(z)-Sy=i^^^(0-(i/2)Sj.=lWy(v7^ 

Jf(r+m+2k 

X(/>(y)dxdzdwdy        (by  the  change  of  variable  yr  —  y  -f  ;y) 


302 


~  L 


SKRay 
f(x,  z,w,y- 


Rr+m+2/t 

xcj)(y)dxdzdwdy 


X(j>(y)dxdzdwdy. 


Let 


K/(y,  y)  = 


,  z,  iu,  y  -  3?) 


Since  /  e  Ll(G)  n  L2(G),  it  follows  that  K{  e  L2(Rk  x  R*)  for  almost  every 
Z  e  Vfi  0  W.  Let  Z  |  a  =  (Zi,  -  -  - ,  «  and  /  |  spanR{Zi(Z), . . . ,  Zr(Z)}  -  (Z~i, . . . ,  Zr). 
Then 

7  r      r  >i +  y*  i  n\      y* +  yk .  n\      -^ 

l , . . . ,  Zm,  Zi , . . . ,  Zr,  — - — A.i(Z), . . . ,  — - — Xjfe(Z),  y  -  >•    , 


2        1V/7       '       2 
where  ^"123  stands  for  the  partial  Fourier  (Euclidean)  transform  in  the  variables  x,z,  w. 


Thus    Or/)  is  a  Hilbert-Schmidt  operator  on  L2(R*)  with  the  kernel  K   .  Hence 


"     ^ 

=  L 


y\ 


+ 


If  we  do  the  change  of  variables 

v/  + 


M/       = 


,•(/),     1  <  j  <  Jfc, 


then  the  modulus  of  the  Jacobian  determinant  is  |Aj(Z)  .  .  . 
reduces  to 


and  the  above  integral 


-l 


where  M  =  (MI,  .  .  .  ,  M*)  and  i;  =  (i>i, 
theorem  in  the  variable  u  we  get 


.  .  ,  VK).  By  applying  the  Euclidean  Plancherel 


If  we  integrate  both  sides  of  the  above  equation  on  Vfi  with  respect  to  the  usual  Lebesgue 
measure  and  use  change  of  variables  by  the  map  <p  defined  in  (2.9),  we  get 


Jftr+2k 


Uncertainty  principles  303 

Then  by  applying  the  Euclidean  Plancherel  theorem  on  the  variables  (lni, . . . ,  lnr)  6  Rr 
we  get 

\Pfd}\    t      \\f(Kl)\\2HSdlni...dlnr 
JV 

\f(h,  •  •  • ,  lm,xni  ,...,xnr,u,  v)\2dxni  . . .  dxnr 
This  completes  the  proof. 

Theorem  2.4  (Plancherel  theorem).  For  f  <=  Ll(G)  H  L2(G) 


where  dl  is  the  standard  Lebesgue  measure  on  V^(~  Rw+r)  with  respect  to  the  basis 
{X*,  .  .  .  ,  X£,  X*j,  .  .  .  ,  X*r}. 


/.  Regarding  V$  O  ZY  as  the  Cartesian  product  of  U'  and  Rr  as  in  Note  2.3,  we  integrate 
both  sides  of  (2.  1  0)  with  respect  to  the  standard  Lebesgue  measure  on  3*  (upon  identification 
with  Rm  via  the  basis  {X*,  .  .  .  ,  X*  })  to  get 


L 


(by  (2.  10) 

=    / 

jR/ 


by  using  the  Euclidean  Plancherel  theorem  in  the  outer  integral  (Uf  is  a  set  of  full  Lebesgue 
measure  in  3*).  The  last  integral  is,  of  course,  II/II?2/G\  and  tne  proof  is  complete. 


2.5.  The  situation  is  simpler  if  we  consider  the  case  of  MW  groups.  In  this  case 
fl  W  c  g*  is  Zariski  open  and  for  /  6  U  c  3*,  the  representation  TT/  is  given  by 


where  y  €  R^,  /  €  L2(R^)  and  dim  $/?$•  =  2A;.  Then  it  follows  from  the  calculations  done 
in  theorem  (2.3)  that 


Clearly  |li(/)  .  .  .^(/)|  =  |P/(/)|,  since  5/  is  nondegenerate.  The  Plancherel  theorem 
again  follows  from  integrating  both  sides  on  U  c  3*.  5*0  ?/z^  change  of  variables  through 
the  map  $  is  not  needed  for  MW  groups. 


304  S  K  Ray 

Let  g  be  a  two  step  nilpotent  Lie  algebra  with  a  basis  B  as  before.  Now  we  consider 
elements  of  g  as  left  invariant  differential  operators  acting  on  C°°(G)  where  the  action  is 
given  by 


We  define 

n—m 

«+/  (2.11) 


1=1 

and  as  on  the  Heisenberg  groups,  call  it  the  sub-Laplacian  of  G. 

Given  an  irreducible,  unitary  representation  jr  of  G,  we  look  at  the  matrix  functions  of 
jr  given  by 

0£w  :  G  -»  C,         u,  v  €  Hn 

<«(*)  =  frfeX").  (2.J2) 

Our  aim  is  to  find:  which  matrix  functions  of  representations  are  joint  eigenfunctions  of 
£  and  {X-L  :  1  <  i  <  m}l 

Given  n  e  G  and  Jf  E  g,  we  have 


r=0 


(2.13) 


where  w,  u  are  C°°  vectors  for  jr.  If  A  €  L/(g)  then  it  follows  that 
A(n(g)u,  v]  =  (7r(g)d7r(A)w,  u) 

(see  [7]).  Thus  if  u  is  an  eigenvector  for  djr(A)  then  0£v  is  an  eigenfunction  for  A.  Since 
for  1  <i  <m,  Xi  e  2  (Ufa)),  the  center  of  the  universal  enveloping  algebra,  then  dn(Xi) 
acts  as  a  scalar  (see  [7])  and  hence  <p^v  is  an  eigenfunction  for  Xi  for  any  u,  v.  Thus  our 
job  reduces  to  finding  the  eigenfunctions  of  dn(£)  which  are  also  matrix  functions  of  TT. 
Looking  at  the  case  of  the  Heisenberg  groups  and  the  group  F^,2  (see  [26])  it  is  reasonable 
to  expect  that  dir  (£)  is  closely  related  to  the  Hermite  operator  and,  indeed,  that  is  the  case. 
We  use  on  G  the  exponential  coordinates  given  by  the  above  chosen  basis.  Given 
x  =  E?=l  *iXi  and  x/  =  E?=l  xiXi  denote 


where  (.,.)  is  the  Euclidean  inner  product  on  g  for  which  {X/  :  1  <i  <  njisanortliononnal 
basis.  Then  it  follows  that,  for  1  <  i  <  m, 


=      -(x),  (2-14) 

dxi 

and  form  +  1  <  z  <  n, 


+       ^^^-  (2'15) 


Uncertainty  principles  305 

Now  we  start  with  a  representation  JT/  €  G  such  that  /  |  3  ^  0.  We  get  hold  of  an  almost 
symplectic  basis  (2.4)  with  dimr/  =  ra  +  r  and  dim  O\  =  2&,  so  n  =  2fc  -f  m  -f  r. 
The  representations  717  are  realized  on  L2(R*)  and  are  given  by  (2.5).  Using  the  explicit 
description  (2.5),  it  is  easy  to  see  that  C°°(7r/)  =  S(R*),  the  Schwartz  class  functions  on 
R*.  By  direct  calculation  we  find  the  effect  of  applying  dir/  on  the  elements  of  the  almost 
symplectic  basis,  which  in  turn  describes  d;r/  (£)  . 


Lemma  2.4.  For^  e  S(R) 

(i) 

(ii) 
(iii) 
(iv)  djr/(£)0(?)  = 


*  Because  of  (iv)  now  it  is  easy  to  describe  the  eigenfunctions  of  d7r/(>C).  Let 

4jr2  X^=i  ^-  Then  djr/(£)  =  /z  (/)-}-  L/,  and  /^(/)  >  0.  If  0  is  an  eigenfunction  of  L/  with 
eigenvalue  <:(/),  then  ^  is  an  eigenfunction  of  d;r/(£)  with  eigenvalue  c(l)  +  /x(/).  Again, 
if  0y  is  an  eigenfunction  of  —  -^  -|-  4jt2Xj(l)2x2  on  R,  then  clearly 


is  an  eigenfunction  of  L/.  Since  for  5  e  N,  the  5th  normalized  Hermite  function  hs  is  an 
eigenfunction  of  —  —^  +  x2  with  eigenvalue  2s  +  1,  it  is  clear  that 


is  an  eigenfunction  of  —  JT  +  47r2Aj(/)2jc2  with  eigenvalue  2nXj(l)(2s  +  1)  and  also 
||/4  1|  2  =  1.  So  for  («!,  .  .  .*<**)  €  N*  we  define 

htei,...,$k)  =  nkj=lhlaj(§j),  (2.16) 

where 


Then 

/*  \ 

L'(^)  =     E  27r^(0(^  +  1)    h'u.  (2.17) 

\^=»  / 

Thus 

/  *  \ 

(2.18) 


Now  we  state  a  mild  generalization  of  Theorem  1.2,  which  follows  from  Lemma  2.3  of 
[20]. 


306  5  K  Ray 

Theorem  2.5.  Suppose  f  :  R"  —  >  C  be  a  measurable  function  such  that 
(i)  /Rne"a3rW|2|/(.x)ipd;t  <oo, 

(ii)  /Rn  t*teWVOOI?IGOOrdy  <  oo, 

where  a,  b  >  0,  Q  is  a  polynomial  and  r  >  0  is  any  real  number.  Ifab  >  1  then  f  =  0. 

3.  Extensions  of  Hardy's  theorem 

The  principal  result  in  this  section  is  the  analogue  of  the  Theorem  1  .2  for  two  step  nilpotent 
Lie  groups.  Along  the  way  we  also  talk  about  the  analogue  of  Theorem  1,3.  Hardy's 
theorem  for  Heisenberg  groups  was  proved  in  [28]  and  its  Lp  -analogue  (Theorem  1.2)  and 
the  analogue  of  Theorem  1.3  was  proved  in  [3].  An  analogue  of  Hardy's  theorem  on  two 
step  nilpotent  Lie  groups  was  proved  in  [1]. 

Remark  3.1.  Our  treatment  in  this  section  tacitly  assumes  that  G  is  not  MW.  For  the  case 
of  MW  groups  the  treatment  needs  only  obvious  modifications  using  the  description  of 
il/0r/)ll//s  given  in  Note  2.5. 

In  the  case  of  Heisenberg  groups,  Hardy's  theorem  and  Cowling-Price  theorem  actually 
reduce  to  the  corresponding  problems  on  the  center  of  the  group  by  an  application  of 
(2.6).  Two  step  nilpotent  Lie  groups  having  reasonable  analogue  of  (2.6)  in  Theorem 
2.3,  it  is  expected  that  the  same  technique  may  work  here  also;  and  it  does,  as  we  shall 
show  presently.  Since  we  are  going  to  talk  about  exponential  decay  of  the  group  Fourier 
transform,  we  need  a  growth  parameter  on  the  dual,  where  usual  exponential  makes  sense, 
but  that  has  been  addressed  in  §  1  .  In  our  parametrization  the  dual  is  essentially  a  vector 
subspace  (actually  a  Zariski  open  subset  of  that  subspace)  of  g*,  which  is  good  enough  for 
us. 

Let  g  be  a  two  step  nilpotent  Lie  algebra  with  basis  B  as  before.  G  is  the  corresponding 
connected,  simply  connected,  Lie  group.  We  write  elements  of  g  (as  well  as  G)  by  (;c,  v)  = 
]C/li  xiXi  +  Y!t=\  viXm+i-  The  set  V^  nil  serves  as  the  effective  dual  (that  is,  it  is  a 
set  of  full  Plancherel  measure  in  G)  of  G  and  we  put  Euclidean  norm  there  such  that 
{X*,  .  .  .  ,  X*n  ,  X*.  :  1  <  i  <  r}  is  an  orthonormal  basis.  We  write  elements  of  Vfi  as 


To  prove  an  analogue  of  Theorem  1.2,  we  need  the  following  trivial  lemma. 

Lemma  3.1.  Let  G  be  a  two  step  nilpotent  Lie  group.  Then  there  exists  a  constant  C  such 
that 


IK*,  v)-(*i»  ^i)     II  >  II (*,  u)ll  —  II  (*i,  vi)ll  —  C||(jc,  u)||||(xi,  i>i)||,       (3.1) 
for  all  (;c,  D),  (jci,  ui)  €  G. 

Now  we  come  to  the  proposed  analogue  of  Theorem  1.2. 
Theorem  3.1.  Letf  e  L](G)  0  L2(G)  satisfy 

(ii\     I       pfl DTi  I! (.A-iY)  II    ||  fdr-i      ^  II        I  P  /* ^  1  ^  !H  1  Hi/  ^^  /^^\ 
'     /  T/*  t<  II  y  vy^A  v/ II  we  I     ./  \    /  l^-i'^-vji/    ^»  ^~? 

vv/z^re  I  <  p  <  oo  a/zJ2  <  q  <  oo.  T^aZ?  >  I,  f/ien  /  =  0  almost  everywhere. 


Uncertainty  principles  307 

"oof.  We  first  prove  the  case  p  =  oo  and  later,  use  this  result  for  the  case  1  <  p  <  oo. 

ise  1.  p  =  oo.  In  this  case  we  interpret  (i)  as 

|/(jc,u)|  <  Ae-fl3r|l(*'w)l12.  (3.2) 

e define 


=  /       ( 
JR«-« 


(3.3) 


lere  /„(*)  =  /(*,  u)  and  *  is  the  convolution  on  Rm.  Since  f  €  Ll(G)9  h  e  Ll(Rm) 
d  the  Euclidean  Fourier  transform  of  h  is  given  by 


-      f    /z 

jRm 

=      f 

Jf\n-m 


=    \PfW\f_   \\f(^,y)\\HsdY       (by  (2.10)). 

JVH 


(3.4) 


3W  writing  e*  = 
h(x)\     < 


f 

JR 


<     A          exp(~«7r[2||i;||2 


(3.5) 

lere  a'  <  a  with  a'b  >  i  (the  integral  in  the  last  line  but  one  being  a  polynomial  in  ||*  || ). 
loosing  b'  <  b  such  that  a'b'  >  1  we  have,  on  the  other  hand, 


(^  exp  (|2^||X||2) 


1A/2 


(by  Holder's  inequality,  where  2/q  +  I/a  =  1) 


308  5  K  Ray 


x{exp((*7  -  b)n ||(A.,  y)||2)|P/(A.)|^/2}dXdy  <  oo     (by  (ii)).        (3.6)  . 

Since  (af /2)2b'  =  a'b'  >  1,  by  3.5,  3.6  and  Theorem  1.2  for  the  case  p  =  oo  and  q/2 
(which  is  >  las  g  >  2)  we  get  that  A  =  0  almost  every  where.  Thus  ||/(X,  y  )!!//£  =  0  for 
almost  every  (A,  y)  and  hence  /  =  0  almost  everywhere  by  the  Plancherel  theorem. 

Case  2.  p  <  oo.  Let  **(*,  u)  =  e*ll(*'u)l12  for  k  e  R+.  Suppose  g  e  CC(G)  is  such 
that  supp  g  C  {(-xi,  i>0  :  ||(;ci,  v\)\\  <  ~},  where  m  €  N.  We  choose  (^,  v)  e  G  with 
H(JC,V)||  >  1.  Thus,  if  (;q,  ui)  6  supp^  we  have  \\(x\,  v\)\\  <  \\(x,  v)\\/m  and  hence  by 
Lemma  3.1, 


||(*,  v)(x},  u,)-1!!  >  ||(jc,  u)||     1  -  -    ,  (3.7) 

V      mJ 

where  c?  =  1  +  C.  Thus  for  (jc,  v)  e  G  with  ||(jc,  u)||  >  1  we  have 


=   I 

Jsuppg 


By  (i)  we  have  that  ean  \  f\  is  a  Lp  function  (p  <  oo)onGand#  €  Cc(G),thus<?fl7r|/|*|£| 
is  a  continuous  function  vanishing  at  infinity.  Thus  from  (3.8)  we  have  that 


K/  *  8)(x,  v)\  <  j8e"fljr( 
for  all  (x,v)  e  G  with  Euclidean  norm  greater  than  1  .  By  continuity  of  /  *  g  we  have 

!(/**)(*,  u)l  <  ^e-^^-^^211^'^!'2,  (3.9) 

for  all  (*,  v)  e  G  (possibly  with  a  different  constant).  Since 


from  (ii)  we  get  that 

<  oo.  (3.10) 


We  choose  m  so  large  that  ab(\  -  (d/m))2  >  1.  Then  by  (3.9)  and  (3.10)  we  are  reduced 
to  case  1.  Hence  /  *  g  =  0  almost  everywhere.  Now  by  choosing  g  from  an  approximate 
identity  we  get  /  =  0  almost  everywhere.  This  completes  the  proof. 

Note  3.1.  For  general  two  step  nilpotent  Lie  groups  we  are  unable  to  answer  the  case  q  <  2. 
But  if  G  is  a  MW  group  then  we  have  a  complete  answer,  as  is  shown  in  the  following 
theorem. 

Theorem  3.2.  Let  G  be  a  connected,  simply  connected,  two  step  nilpotent  Lie  group  which 
is  MW.  Let  f  e  L*(G)  n  L2(G).  Suppose  that  for  a,  b  >  0  andmn(p,  q)  <  oo 


Uncertainty  principles  309 


(i)  fcePa**<vW\f(z,v)\Pdzdv  <  oo, 
(ii)  /v^^H/V/JII^I^/COId/  <  oo. 


Then 

(a)  Ifq  >  2,  then  f  =  Oforab  >  1. 

(b)  Ifl  <q  <2  then  f  =  0  r/flfr  >  1. 

Proof.  Part  (a)  is  essentially  in  Theorem  3.1.  So  we  prove  (b).    In  this  case,  for  /  € 
L1(G)nL2(G)wehave 

\\f(*i)\\Hs  =  \pfwr1  f 

•/R2" 
(see  Note  2.5).  Starting  from  (ii)  we  have 


=    f    e«b*n2  (\Pf(l)\-1  f 
JVJ,  V  ./R2" 

=    f    (f     g(l,v)l 
Jv*  \JR^ 


|P/(/)|d/ 


(where  g(l,  v)  = 


\  g      \ 
\  *       \ 

)\    dv  I 

/          J 


g(l,  v)dfji(l)  I    dv  I  (by  Minkowski's  inequality). 

'         I 
Thus  for  almost  every  u,  /v*  #(/,  u)d/x(/)  <  oo,  that  is 

/     e9to||/l|2|^ri/(/,i;)^|P/(/)|(1"i)d/<oo.  (3.11) 

•fy}   . 

But  from  (i)  it  follows  that  for  almost  every  v, 

:oo.  (3.12) 


Thus  for  almost  every  v,  the  function  /(.,  v)  satisfies  the  condition  of  Theorem  2.5  and 
hence  for  ab  >  1,  /  =  0  after  all. 

Going  back  to  connected,  simply  connected,  two  step  nilpotent  Lie  groups  G,  we  observe 
that  the  same  technique  using  the  functions  /  and  h ,  as  in  Theorem  3. 1 ,  yields  the  following 
theorem. 

Theorem  3.3.  Let  f  :  G  ->  C  be  a  measurable  function.  Suppose 

0)  \f(x, 
(ii)  ii/Vx 

where  C  >  0,  p  >  2,  \fp-\-\jq  =  I  and  g,  h  are  nonnegative  functions  with  g  6 
L^R""-""1)  n  L2(Rn~m)  andh  e  Ll(Rr)  Pi  L2(Rr).  If(ap)l/p(bq){fc*  >  2,  then  f  =  0 
almost  everywhere. 


310  SKRay 

4.  Heisenberg's  inequality 

The  classical  inequality  of  Heisenberg  for  L2  functions  on  R  says  that 

a\  1/2   /    /•  \  1/2 

W2l/WI2d*        I      \y\2\f(y)\2dy)      >C||/||2,  (4.1) 

\  /        V./R  / 

where  /  is  defined  by 


-  f 

JR 


and  C  is  a  constant  independent  of  /. 

In  this  section  our  aim  is  to  extend  the  version  of  Heisenberg's  inequality  proved  in 
[29]  for  the  Heisenberg  groups  to  all  connected,  simply  connected,  step  two  nilpotent  Lie 
groups.  Two  other  variants  of  Heisenberg's  inequality  on  Heisenberg  groups  are  available 
in  [13]  and  [28],  but  since  these  results  use  the  existence  of  rotations  on  Heisenberg  groups, 
it  is  not  clear  how,  without  the  notion  of  rotation,  one  should  proceed  to  extend  them  to  a 
general  two  step  nilpotent  Lie  group  (see  [2]). 

We  state  (4.1)  in  a  slightly  different  way.    Let  A  =  —  £?=i  —^  be  the  Laplacian 

on  Rw.  Then  (A/)(}>)  =  4jr2||;y||2/(>')  for  any  Schwartz  class  function  on  Rn.  We 
may  relate  A  to  the  character  yy(x)  =  Q2niy-x  of  R"  by  dyy  (  -^-  j  =  2;nv7-,  and  hence 
<tyy(A)  =  47T2||>'||2.  Thus  we  have 


Since  dy),(A)  is  a  positive,  self  adjoint  operator,  it  has  a  (visible)  square  root,  which  is 
multiplication  by  2n  \\y\\.  Thus  we  define 


-  2x\\y\\f(y)  -  (d 

for  all  Schwartz  class  functions  on  Rn.  Since  the  Fourier  transform  is  an  isomorphism  on 
Schwartz  class  functions,  the  operator  (A)  2  is  defined  completely.  Then  we  can  restate 
(4.1)  as 


a 
^ 


(4.2) 


for  all  /  of  Schwartz  class  on  Rn  ,  where  C  is  a  constant  independent  of  /.  It  is  (4.2)  ,  whose 
analogue  on  connected,  simply  connected,  two  step  nilpotent  Lie  groups  we  are  looking 
for.  As  in  the  case  of  Heisenberg  groups,  here  also  the  proof,  in  principle,  is  close  to  the 
proof  on  R"  (see  [11])  having  the  same  basic  ingredients,  namely,  integration  by  parts, 
Cauchy-Schwartz  inequality  and  the  Plancherel  theorem. 

We  call  a  function  f  on  G  a  Schwartz  class  function  iff  o  exp  is  a  Schwartz  class  function 
on  Q.  We  denote  the  Schwartz  class  functions  by  S(G). 

Replacing  A  2  by  &.,  the  main  result  of  this  section  is  as  follows. 

Theorem  4.1.  Let  G  be  a  connected,  simply  connected,  step  two  nilpotent  Lie  group  and 
f  €  S(G).  Then 


Uncertainty  principles  3 1 1 

\i/2 
v\\2\f(x9v)\2dxdv\ 


(4.3) 
where  C  is  a  constant  independent  of  f  and  C  =  ~  ^frf1^  +/  w  r/ze  sub-Laplacian. 

Let  us  explain  the  meaning  of  (£5  /)(TT/).  We  view  X  G  g  as  a  left  invariant  differential 
operator  on  C°°  (G).  Then  in  view  of  our  definition  of  the  group  Fourier  transform,  we  have 
for  /  e  S(G) 

(Xf)(nt)  =  dni(X)  o  /Or/),  (4.4) 

where  djr/(X)  is  given  by  (2.13).  We  view  the  universal  enveloping  algebra  U(o()  as  the 
algebra  of  all  left  invariant  differential  operators  on  C°°(G).  Since  dni  is  a  representation 
of  g,  it  extends  to  a  representation  of  ZY(g)  realized  on  C°°(7r/).  By  (4.4)  we  have 


as  £  €  £^(&).  In  §2  we  have  seen  that  the  eigenfunctions  of  dir/(£)  are  parametrized  by 
N*  and  are  given  by  (2.16).  Let  {/,-(/)  >  0  :  /  =  0,  .  .  .}  be  an  enumeration  of  those  real 
numbers  such  that  there  exist  a  G  Nk  with 

j  +  1),  (4.5) 

as  cc  varies  over  N*.  Let  £/(/)  =  spanc{/i^  :  djr/(£)(/i^)  =  r/(/)/i/a},  that  is,  £/(/)  is  the 
eigenspace  corresponding  to  the  eigenvalue  ?/(/),  which  is  clearly  finite  dimensional.  If 
Pi  (I)  :  L2(Rk)  -»  £/(/)  is  the  projection,  we  have 

(4.6) 
;=o 

Thus  we  define 


and 


7=0 

Analogous  to  the  Euclidean  spaces,  we  define 

(£*/)(*/)  =  d7T/(£)2  o  /(*/),  (4.9) 

for  all  /  e  <S(G)  and  /  E  V^  n  W.  Thus  the  statement  in  theorem  4.1  makes  sense. 

It  follows  from  (4.5)  that  the  eigenvalues  of  d7r/(£)~~2  are  bounded  by  A,o(/)~5  where 
A.Q(/)  =  min{A.7-(/)  :  1  <  j  <  k}.  As  a  consequence  we  get  the  following  Lemma. 


312  SKRay 

Lemma  4.1.  The  operator  djr/(£)~5  is  bounded  on  L2(Rk). 
Let  us  consider  the  following  elements  of  gc,  the  complex!  fication  of  g, 

Dj(l)  =  Yj(l)  -  iWj(l),         l<j<k,  (4.10) 

Dj(l)  =  Yj(l)  +  iWj(l),         l<j<k.  (4.11) 

Because  of  Lemma  2.4  we  have 

(4.12) 


(4.13) 
For  hs  the  5th  normalized  hermite  function  on  R,  we  define  /i£(x)  =  c1/'4A.j(c1/'2jc))  then 


Using  this  with  (4.  12)  and  (4.  13)  we  get  for  a  e  Nk, 

(4.14) 
(4.15) 

where 


Lemma42.  The  operators  cbr/GDy  (/))  od7T/(£)~2  ^£/djr/(D7-(Z))od7T/(£)"~2  arebound- 
ed  operators  on  L2(R^),  1  <  ;  <  Jt. 


/.  We  consider  the  orthonormal  basis  {/i^  :  a  €  N^}  of  L2(R*).  By  (4.8),  (4.14)  and 
(4.15)  we  have 


.  -t-1), 

and 


Since 


Uncertainty  principles  3 1 3 

and 

/ 2^(020; V 


the  operators  djr/(D/(/))  o  djr/(£)~2  and  djr/(Z)/(/))  o  djr/(£)~5  are  bounded  operators 
on  L2(R*).  This  completes  the  proof. 

Suppose  /  €  S(G)  and  let  /  e  VjJ  O  W  be  arbitrary  but  fixed.  So  we  have  an  almost 
symplectic  basis  (2.4)  of  Q.  Let  I  \  3  =  A,.  We  define 

)=:  f    /O,  lOe-^^djc,  (4.16) 

,/Rm 


that  is,  the  partial  Fourier  transform  in  the  central  component.  So  i;  ->  Fcf&>  v)  is  a 
Schwartz  class  function  on  Rn~m  .  On  Euclidean  spaces,  differentiation  and  multiplication 
are  intertwined  by  the  Fourier  transform.  On  two  step  groups,  as  analogues  of  differentiation 
we  consider  the  operators  D\  (I)  and  Dj  (!)  and  as  analogue  of  Fourier  transform  we  consider 
the  partial  Fourier  transform  defined  in  (4.16).  We  want  to  find  what  plays  the  role  of 
multiplication? 

Let  /  e  S(G)  and  Xj  e  B  C  Q,  m  +  1  <  j  <  n.  By  (2.  15)  it  is  clear  that  Xjf  €  5(G), 
and  an  easy  calculation  shows  that 


.f  v). 
Thus  using  the  basis  in  (2.4)  we  have 


(4.17) 


.iw.y)  (4.18) 

1  <  j  <k.  Thus  writing 


=  f  /- 
\dyj 


iu»y),  (4-20) 

wj 

we  have  from  (4.  17)  and  (4.  1  8) 

.w.y),  (4.21) 

,«w,y).  (4-22) 


Thus  Vj(l)  and  V,-  (/)  play  the  role  of  multiplication. 
Now  we  come  to  the  proof  of  Theorem  4.  1  . 


314  SKRay 

Proof  of  theorem  4.1.  Let  /  €  <S(G)  and  /  |  5  =  X.  Now 
f       \fcf(X,z,w,y)\2dzdwdy 

JRn-w 

=     f       ^/(A-,  z,  u),  y)Fcffr,  z,  w,  y)dzdwdy. 

jRn-m 

Since 

a        a 


we  have  from  the  above  equality 

f       \fcf(k,z,w,y) 

jRrc-m 


,,  z,  u», 

o 

2 


(—    •  d  \ 

If.         .     .(  d  B    \ 

~2j  «-m(-V;' +'IU;')  IF~  ~'a«r/ 


,  2,  W,  >') 


,  z,  w;,  j)dzdw;dy         (by  integration  by  parts) 


cf(Xt  z,  w,  y)dzdwdy 
~\  I      (yj  +  iwjWjW  +  xlj(l)(yj  -  iwj))Fcf(^  z,  ti;, 

/  jR/l-m 


.,z,  w;,  >')dzdw;d>;        (by  (4.  19)  and  (4.20)) 


1      /•  _ 

~o  /       (>y  +  iwj)Fc(Dj(l)f)(^  z,  w,  y}Fcf(^  z,  ™,  y)dzdwdy 

L  Jftn-m 

(by  (4.21)  and  (4.22)).  (4.23) 

Let  us  recall,  if/  varies  over  VJJ  C\U  then  /  |  g  =  A.  varies  over  the  Zariski  open  subset^' 
of  a*  (see  Note  2.3).  Hence 


Uncertainty  principles  3  i  5 

(I       \f(x,v)\2dx6v     =      f    I       \Fcf(X,v)\2dXdv 

JzJRn-m  JU'  Jftn-m 


JU' 
(by  Fubinfs  theorem  and  the  orthogonal  basis  change  on  Rn~m  by  7}  :  spanR{Xm+i, 


,  zt  wt  y)dzdwdy 

U' 


(by  (4.23)) 

f    (-4 

JU'  \    2 


I       Tt-l(yj  4-  i 

2  jRH-m 


+ 


(by  change  of  variables) 

f  -i  •> 

/         \Ti     (vy  -f  JW/)|~KC/(^ 

7  JR/J-W 
i  i  f       f  -  2 

I  (  /     /         |^c(D;-(/)/)(X,u)| 2du( 
[  VJw  JR«-W 

f    f  2        \ 

WR»-«          7  '  / 


(by  Cauchy-Schwartz  inequality  and  nonnegativity 
of  the  integral) 


li' 


f 

JR"-^1 


\\v\\2\f(x,v)\2dxdv) 


/W  JR"-m 

/        |J"c(Dy(/)/)(A.,  f)|2dudA )    [ .  (4.24) 

JU1  JR"-^ 

by  the  Euclidean  Plancherel  theorem  on  g. 


316  SKRay 

By  Theorem  (2.3)  we  have 


cf(^,  V)\2dv, 


where/  |  8  =  A.  and/  |  VN  =  y  =  (/„,, . . . ,  /nr).  Thus 
(If       \Fc(D(Dn(^v)\2dvdk 


<' 


x\Pf(l)\dln 


lni...dlnrdX) 


x\Pf(l)\dlni...dlnrdX) 


i 


IIGC'/)(jr/)ll«l  P/Wld/n,  -  -  -  d/Brd/i  .  .  .  d/m 
r% 

(by  Lemma  4.2J. 
Similarly  as  above  we  can  show  that 


II      IFc&j 

Uf  JW-"1  ' 


Htf/X^lll^/COId/n,  -  -  -d/nrd/i  .  .  .d/m 
f'Jv 


Thus  from  (4.24)  we  have 
l/(*,i 


<  C 

where  C  is  a  constant  independent  of  /.  This  completes  the  proof. 


Uncertainty  principles  3 1 7 

Acknowledgement 

I  am  grateful  to  my  teacher  Prof.  S  C  Bagchi  and  my  friend  E  K  Narayanan  for  several 
useful  discussions.  Most  of  this  work  is  contained  in  the  author's  Ph.D.  thesis  submitted 
to  the  Indian  Statistical  Institute  (1999). 

References 

[1  ]  Astengo  F,  Cowling  M  G,  Di  Blasio  B  and  Sundari  M,  Hardy's  uncertainty  principle  on  some 

Lie  groups,  J.  London  Math.  Soc.  (to  appear) 
[2]  Benson  C,  Jenkins  J  and  Ratcliff  G,  On  Gelfand  pairs  associated  to  solvable  Lie  groups,  Trans. 

Am.  Math.  Soc.  321  (1990)  85-1 16 
[3]  Bagchi  S  C  and  Ray  S,  Uncertainty  principles  like  Hardy's  theorem  on  some  Lie  groups,  J. 

Austral.  Math.  Soc.  A65  (1998)  289-302 

[4]  Chandrasekharan  K,  Classical  Fourier  transforms  (Springer  Verlag)  (1989) 
[5]  Cowling  M  G,  Sitaram  A  and  Sundari  M,  Hardy's  uncertainty  principle  on  semisimple  Lie 

groups,  Pacific  J.  Math.  192  (2000)  293-296 
[6]  Cowling  M  G  and  Price  J  F,  Generalizations  of  Heisenberg's  inequality,  in:  Harmonic  analysis 

(eds)  G  Mauceri,  F  Ricci  and  G  Weiss  (1983)  LNM,  no.  992  (Berlin:  Springer)  443-449 
[7]  Corwin  L  J  and  Greenleaf  F  P,  Representations  of  nilpotent  Lie  groups  and  their  applications, 

Part  1  -  Basic  theory  and  examples  (NY:  Cambridge  Univ.  Press,  Cambridge)  (1990) 
[8]  Ebata  M,  Eguchi  M,  Koizumi  S  and  Kumahara  K,  A  generalisation  of  the  Hardy  theorem  to 

semisimple  Lie  groups,  Proc.  Japan  Acad.  Math.  Sci.  A75  (1999)  1 1 3-1 14 
[9]  Ebata  M,  Eguchi  M,  Koizumi  S  and  Kumahara  K,  Lp  version  of  the  Hardy  theorem  for  motion 

groups,  J.  Austral.  Math.  Soc.  A68  (2000)  55-67 

[10]  Folland  G  B,  A  course  in  abstract  harmonic  analysis,  (London:  CRC  Press)  (1995) 
[11]  Folland  G  B,  Lectures  on  partial  differential  equations  (New  Delhi:   Narosa  Pub.  House) 

(1983) 
[12]  Folland  G  B  and  Sitaram  A,  The  uncertainty  principles:  A  mathematical  survey,  J.  Fourier 

Anal.  Appl  3  (1997)  207-238 
[13]  Garofalo  N  and  Lanconeli  E,  Frequency  functions  on  the  Heisenberg  group,  the  uncertainty 

principle  and  unique  continuation,  Ami.  Inst.  Fourier  40  (1990)  313-356 
[14]  Hormander  L,  A  uniqueness  theorem  of  Beurling  for  Fourier  transform  pairs,  Ark.  Math.  29 

(1991)237-240 

[15]  Jacobson  N,  Basic  Algebra,  (New  Delhi:  Hindustan  Publishing  Co.)  (1993)  vol.  1 
[16]  Kaniuth  E  and  Kumar  A,  Hardy's  theorem  for  simply  connected  nilpotent  Lie  groups,  Proc. 

Cambridge  Philos.  Soc.  (to  appear) 
[17]  Lipsman  R  L  and  Rosenberg  J,  The  behavior  of  Fourier  transform  for  nilpotent  Lie  groups, 

Trans.  Am.  Math.  Soc.  348  (1996)  1031-1050 
[18]  Moore  C  C  and  Wolf  J  A,  Square  integrable  representations  of  nilpotent  Lie  groups,  Trans. 

Am.  Math.  Soc.  185  (1973)  445-462 
[19]  Muller  D  and  Ricci  F,  Solvability  for  a  class  of  doubly  characteristic  differential  operators  on 

two  step  nilpotent  Lie  groups,  Ann.  Math.  143  (1996)  1-49 
[20]  Narayanan  E  K  and  Ray  S  K,  Lp  version  of  Hardy's  theorem  on  semisimple  Lie  groups,  Proc. 

Am.  Math.  Soc.  (to  appear) 
[21]  Park  R,  A  Paley- Wiener  theorem  for  all  two-  and  three-step  nilpotent  Lie  groups,  J.  Fund. 

Anal.  133(1995)277-300 
[22]  Pati  V,  Sitaram  A,  Sundari  M  and  Thangavelu  S,  An  uncertainty  principle  for  eigenfunction 

expansions,  J.  Fourier  Anal.  Appl.  5  (1996)  427-433 
[23]  Ray  S  K,  Uncertainty  principles  on  some  Lie  groups,  Ph.D.  Thesis  (Indian  Statistical  Institute) 

(1999) 
[24]  Sundari  M,  Hardy's  theorem  for  the  ft -dimensional  Euclidean  motion  group,  Proc.  Am.  Math. 

Soc.  126(1998)1199-1204 
[25]  Sengupta  J,  An  analogue  of  Hardy's  theorem  for  semi-simple  Lie  groups,  Proc.  Am.  Math. 

Soc.  128  (2000)  2493-2499 


318  SKRay 

[26]  Strichartz  R  F,  Lp  harmonic  analysis  and  radon  transforms  on  the  Heisenberg  groups,  J  Funct. 

Anal.  96(1991)350-406 
[27]  Sitaram  A  and  Sundari  M,  An  analogue  of  Hardy's  theorem  for  very  rapidly  decreasing 

functions  on  semisimple  Lie  groups,  Pacific  J.  Math.  Ill  (1997)  187-200 
[28]  Sitaram  A,  Sundari  M  and  Thangavelu  S,  Uncertainty  principles  on  certain  Lie  groups,  Proc. 

Ind.  Acad.  ScL  (Math.  Sci.)  105  (1995)  135-151 
[29]  Thangavelu  S,  Some  uncertainty  inequalities,  Proc.  Ind.  Acad.  ScL  (Math.  Sci.)  100  (1990) 

137-145 


Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  1 1 1,  No.  3,  August  2001,  pp.  319-336. 
©  Printed  in  India 


Oil  property  (ft)  in  Banach  lattices,,  Calderon-Lozanowskii  and 
Orlicz-Lorentz  spaces 

PAWEL  KOLWICZ 

Institute  of  Mathematics,  University  of  Technology,  ul.  Piotrowo  3a,  60-965  Poznati, 

Poland 

E-mail:  KOLWICZ@math.put.poznan.pl 

MS  received  24  November  1999;  revised  4  December  2000 

Abstract.  The  geometry  of  Calderon-Lozanowskii  spaces,  which  are  strongly  con- 
nected with  the  interpolation  theory,  was  essentially  developing  during  the  last  few 
years  (see  [4,  9,  10,  12,  13,  17]).  On  the  other  hand  many  authors  investigated  prop- 
erty (ft)  in  Banach  spaces  (see  [7,  19,  20,  21,  25,  26]).  The  first  aim  of  this  paper  is 
to  study  property  (ft)  in  Banach  function  lattices.  Namely  a  criterion  for  property  (ft) 
in  Banach  function  lattice  is  presented.  In  particular  we  get  that  in  Banach  function 
lattice  property  (ft)  implies  uniform  monotonicity.  Moreover,  property  (ft)  in  gener- 
alized Calderon-Lozanowskii  function  spaces  is  studied.  Finally,  it  is  shown  that  in 
Orlicz-Lorentz  function  spaces  property  (ft)  and  uniform  convexity  coincide. 

Keywords.  Banach  lattice;  Calderon-Lozanowskii  space;  Orlicz-Lorentz  space; 
property  (ft). 

1.  Introduction 

Let(X,  11-11)  be  a  real  Banach  space,  andletB(X),  S(X)  be  the  closed  unit  ball,  unit  sphere 
of  X,  respectively.  For  any  subset  A  of  X,  we  denote  by  conv(A)  the  convex  hull  of  A. 

We  denote  by  £Q  W  the  characteristic  of  convexity  and  by  $x  (£)  iht  modulus  of  convexity 
of  the  space  X,  i.e. 

$x(s)     =     i 


We  say  thai  X  is  uniformly  convex  (X  e  (UC)  for  short)  if  SQ  (*)  =  0  (see  [22]). 

Define  for  any  x  i  B(X)  the  drop  D  (x,  B(X))  determined  by  x  by  D  (x,  B(X))  = 
conv({;c}U.£(X)). 

Recall  that  for  any  subset  C  of  X,  the  Kuratowski  measure  of  non-compactness  of  C  is 
the  infimum  a(C)  of  those  £  >  0  for  which  there  is  a  covering  of  C  by  a  finite  number  of 
sets  of  diameter  less  than  £. 

Rolewicz  in  [25]  has  proved  that  X  e  (UC)  iff  for  any  s  >  0  there  exists  8  >  0  such  that 
1  <  ||*||  <  1  +  8  implies  diam  (D  (x,  B(X))  \  B(X))  <  e.  In  connection  with  this  he  has 
introduced  in  [26]  the  following  property. 

A  Banach  space  X  has  the  property  (ft)  (X  e  (ft)  for  short)  if  for  any  £  >  0  there  exists 
8  >  0  such  that  a  (D  (x,  B(X))  \  B(X))  <  s  whenever  1  <  ||jc||  <  1  +  3. 

We  say  that  a  sequence  [xn]   c   X  is  e-  separated  for  some  £   >   0  if  sep{*M}   = 


320  Pawel  Kolwicz 

The  following  characterization  of  the  property  (ft)  is  very  useful  (see  [20]). 

A  Banach  space  X  has  property  (ft)  if  and  only  if  for  every  £  >  0  there  exists  <5  >  0  such 
that  for  each  element*  6  B(X)  and  each  sequence  (xn)  in  B(X)  with  sep{*,z }  >  £  there  is 
an  index  k  for  which 


A  Banach  space  is  nearly  uniformly  convex  (X  E  (NUC))  if  for  every  £  >  0  there 
exists  <$  €  (0,  1)  such  that  for  every  sequence  {xn}  c  B(X)  with  sep{xn}  >  e,  we  have 
conv({A"rt})  n  (1  —  S)B(X)  ^  0.  Rolewicz  proved  the  following  implications  (UC)  =>• 
(ft)  =£•  (NUC)  (see  [26]).  Moreover,  the  class  of  Banach  spaces  with  an  equivalent  norm 
with  property  (ft)  coincides  neither  with  that  of  super-reflexive  spaces  ([21])  nor  with  the 
class  of  nearly  uniformly  convexifiable  spaces  ([19]). 

A  Banach  space  X  is  said  to  have  the  Kadec-Klee  property  (X  e  (H)  for  short)  if  every 
weakly  convergent  sequence  on  the  unit  sphere  is  convergent  in  norm.  The  Banach  space 
X  is  called  to  have  uniformly  Kadec-Klee  property  (X  e  (UKK)  for  short)  if  for  every 
s  >  0  there  exists  <$  e  (0,  1)  such  that  ||jc||x  <  \  -  8  whenever  (xn)  c  B(X),  xn  -^  x 
and  stp[xn}x  >  e.  For  any  Banach  space  we  have  (NUC)  =»(UKK)  =»(KK).  Moreover 
X  e  (NUC)  iff  X  e  (UKK)  and  X  is  reflexive  ([15]). 

Denote  by  TV",  71  and  7£+  the  sets  of  natural,  real  and  non-negative  real  numbers,  respec- 
tively. Let  (r,  E,  IJL)  be  a  measure  space  with  a  or -finite,  complete,  non-atomic  measure  ju.. 
By  L°  =  L°(T)  we  denote  the  set  of  all  ^-equivalence  classes  of  real  valued  measurable 
functions  defined  on  T. 

Let  E  =  (E,  <,  ||.||£)  be  a  function  Banach  lattice  over  the  measure  space  (7",  S,  /z), 
where  <  is  the  usual  semi-order  relation  in  the  space  L°  and  (E ,  ||  •  ||  E)  is  a  Banach  function 
space  (i.e.  E  is  a  linear  subspace  of  L°  and  norm  ||-||£  is  complete  in  E).  Let  E  satisfy 
two  conditions: 

(i)  if  x  e  E,  y  e  L°,  |j|  <  |*|  ^-a.e.,  then  >•  e  E  and  \\y\\  E  <  \\x\\  E, 
(ii)  there  exists  function  x  in  E  that  is  positive  on  whole  T  (see  [18]  and  [22]). 
Denote  by  £+,  L^J.  the  positive  cone  of  E,  L°  respectively,  i.e.  L?j_  =  {x  e  L°  :  x  >  0}. 

Recall  that  E  satisfies  the  Fatou  property  (E  e  (FP))  if  x  e  L°  and  (xm)  e  E  are  such 
thatO  <  xm  /  x  and  supm  \\xm\\E  <  oo,  then*  e  E  and  \\x\\ E  =  lim^oo  \\xm\\E  (see 
[18]  and  [22]). 

We  say  that  Banach  lattice  E  is  uniformly  monotone  (E  e  (UM))  if  for  every  q  e  (0,  1) 
there  exists  p  e  (0, 1)  such  that  for  all  0  <  y  <  x  satisfying  \\x\\E  <  1  and  \\y\\ E  >  % 
we  have  \\x  -  y\\E  <  1  -  p.  Then  the  modulus  p  (•)  of  the  uniform  monotonicity  of  E  is 
defined  as  follows: 

p  (q)  =  inf  {1  -  ||*  -  y\\E  :  \\x\\  E  <  1,  \\y\\E  >q,0<y<x}. 

A  Banach  lattice  E  is  called  order  continuous  (E  e  (OQ)  if  for  every  x  e  E  and  every 
sequence  (xm)  e  E  such  that  0  <-  xm  <  |jc|  we  have  ||JC/M  \\E  ->  0  (see  [18]  and  [22]).  It 
is  known  that  if  E  €  (UM),  then  E  e  (OC) . 

A  function  $  :  T  x  7£  — >  [0,  oo)  is  said  to  be  a  Musielak-Orlicz  function  if  $(-,  «) 
is  measurable  for  each  u  e  K,  <$(t,  0)  =  0  and  $>(£,-)  is  convex,  even,  not  identically 
equal  to  zero  for  M-a.e.  t  e  T.  We  denote  (O  o  x)(t)  =  <&(t,  x(t)).  We  will  write  4>  >  0 
if  ^(?,  •)  vanishes  only  at  zero  for  M  -  a.e  t  e  T.  For  every  Musielak-Orlicz-function  4> 


Calderon-Lozanowskii  and  Orlicz-Lorentz  spaces  321 

we  define  complementary  function  in  the  sense  of  Young  <$*  :  T  x  7£  —  >  [0,  oo)  by  the 
formula  $>*(?,  v)  =  sup{u  \v\  -  <t>(r,  u)}  for  every  v  e  U  and  t  e  T  . 

H>0 

We  say  that  Musielak-Orlicz  function  $  satisfies  the  ^-condition  (3>  e  A^)  if  there 
exist  a  constant^  >  2,  asetA  e  £  of  measure  zero  and  a  measurable  non-negative  function 
ft  6  E  such  that 


for  every  t  e  T  \  A  and  every  u  e  7£  (see  [11]  when  E  =  Ll  and  [10]  in  general).  Then 
<£  €  A|  iff  there  exist  a  constant  k  >  2,  a  set  A  e  £  of  measure  zero  and  a  measurable 
non-negative  function  /  €  L+  such  that  $>  o2f  e  E  and 


,  2w)  <  &<£>(?,  M) 

for  every  *  €  T  \  A  and  every  «  >  /(O  (see  [10]). 
Define  on  L°  a  convex  modular  /$  by 


oo  otherwise. 

By  the  function  lattice  E®  we  mean 

#4)  =  {x  €  L°  :  I$(cx)  <  oo  for  somec  >  0}, 
equipped  with  so  called  Luxemburg  norm  defined  as  follows: 


We  will  assume  in  the  paper  that  E  e  (FP),  so  (£$,  ||-||<j>)  is  a  Banach  space  (see  [9]).  The 
space  E®  (when  <!>  >  0)  is  a  special  case  of  the  Calderon-Lozanowskil  construction  of  the 
lattice  (see  [10]).  As  for  theory  of  Calderon-Lozanowskil  space  we  refer  to  [3],  [23]  and 
[24]. 

If  E  =  L1  ,  then  E®  is  the  Musielak-Orlicz  space  equipped  with  the  Luxemburg  norm. 
If  E  is  a  Lorentz  space  Aw,  then  E®  is  the  corresponding  Musielak-Orlicz-Lorentz 
space  (A^)^  equipped  with  the  Luxemburg  norm  (see  [12,  16,  17]).  If  additionally 
<b(t  ,  u)  =  |w|  for  every  f  e  T,  then  the  space  (A^)^  is  the  Lorentz  space  A^.  Recall 
that  the  function  (*)  :  [0,  y)  ->•  R+  with  y  =  /x(J)  is  said  to  be  the  weight  function, 
if  it  is  strictly  positive,  nonincreasing  and  locally  integrable  function  with  the  respect  to 
the  Lebesgue  measure  v.  Then  A^  consists  of  all  functions  x  :  [0,  y)  —  >  7£  measurable 
with  respect  to  v  for  which  ||jc||  =  f£  x*(t)co(t)dt  <  oo,  where  **  is  the  decreasing 
rearrangement  of  x  (see  [2]).  Recall  that  mx  denotes  the  distribution  function  of  x,  i.e. 
mx  (X)  =  y  ({t  €  [0,  y]  :  \x  (r)|  >  X})  for  all  A.  >  0.  The  decreasing  rearrangement  func- 
tion of  jc  is  denoted  by  x*  and  is  defined  by  jc*  (r)  =  inf  {A  >  0  :  mx  (A)  <  f  }.  Denote  by 
5(r)  =  /Or  6>(j)dj.  The  weight  function  is  called  regular  if  inf,>o  5(20/5(0  >  1  in  the 
case  y  =  co  and  info<r<y0  S(2t)/S(t)  >  1  for  some  yo  <  y/2  in  the  case  y  <  oo. 

2.  Results 

It  is  a  natural  question  whether  a  geometric  property  in  Banach  lattices  can  be  equivalently 
considered  only  for  nonnegative  elements  (see  [13,  17]).  In  order  to  consider  that  problem, 
in  case  of  property  (ft)  ,  let  us  introduce  new  notions. 


322  Pawel  Kolwicz 

DEFINITION  1 

We  say  that  a  Banach  lattice  E  has  the  property  (ft+)  if  for  every  s  >  0  there  exists 
8  =  8  (s)  >  0  such  that  for  each  element  x  e  B(E+)  and  each  sequence  (xn)  in 
with  sepfcfl}  >  s  there  is  an  index  k  for  which 


2      ||  -^     "' 
DEFINITION  2 

We  say  that  the  Banach  lattice  (E,  \\-\\ E)  is  orthogonally  uniformly  convex  (E  e  (UC1) 
for  short),  if  for  each  s  >  0  there  is  8  =  8  (e)  >  0  such  that  for  x,  y  6  B(E)  the 
inequality  max  \\xXAxy  \\E  ,  ||>'X^-y  ||£ }  >  e  implies  ||(jr  +  y)  /2||£  <  1  -5,  where  A^v  = 
suppjc-suppy  andA-f-J5  =  (A  \  B)  U  (J5  \A). 

Lemma  1.  (Theorem  1  from  [13]).  L^/  £"  Z?e  any  Banach  function  lattice.  Then  E  €  (UM) 
rj^/or  any  £€(0,1)  //z^re  r\  (e)  >  0  .s-wc/z  that  for  any  x  e  E+  with  \\x  \\  E  ~  1  and  for  any 
A  e  D  McA  rto  ||^XA  HE  >  fi  r/iere  /zoW^  |^Xr\A  |  E  <  1  ~  r\  (s). 

Lemma2.  (Lemma  1.4  w  [14]).  Letx,y  6  X  \  {0} .  Denotex=  x  / \\x\\ x  .  Ifmm[\\x\\x  , 

,     A      A 
and    x  -  y       >e,  then 


\\x  +  y\\x  <l~  i&x  (e))  (Ikb  +  \\y\\x)  - 

Obviously  if  £  6  (UC)  ,  then  E  e  (UC1)  .  It  is  known  that  uniformly  convex  Banach 
function  lattice  is  uniformly  monotone  ([13]).  Moreover,  we  will  prove  the  following: 

Lemma  3.  Let  E  be  any  Banach  function  lattice.  IfEe  (UC1)  ,  then  E  e  (UM)  . 

Proof.  Assume  that  E  £  (UM)  .  By  Lemma  1  we  conclude  that  there  exists  s  >  0  such 
that  for  every  n  €  N  there  exist  xn  e  £+  with  ||jcj|£  =  1  and  a  set  Bn  e  E  such 
that  H-x/iXflJs  -  £  and  \\xnXT\Bn\\E  >  1  -  1/n.  Let  un  =  A:n  and  ^  =  *nXT\Bn- 
Denote  An  =  suppw;i  ^  suppu^.  Then  max  {i|^X/\J|£  ,  IKjuJ^}  >  e.  Moreover, 
since  un  >  vnt  so  we  get  ||«,2  +  vn\\E  >  \\2vn\\E  >  2(1  -  l//j)  .  Hence  E  $  (UC1)  . 

Remark  1.  The  converse  implication  is  not  true.  Let  £  =  L1.  Obviously  L1  e  (UM)  . 
Let*,;y  e  S  (L1)  andsuppx-Hsuppy  =  0.  Then  max  j||^X^v||Li  ,  |bxA,ylLi)  =  l  and 
=  l.ThusL1  ^  (UC-1)  . 


Theorem  1.  Ler  E  fee  a  function  Banach  lattice.  Then  E  e  (ft)  iff  E  e  (fi+)  and  E  e 


Proof  Necessity.  Clearly,  if  £  e  (j8),  then  E  e  (fi+).  Moreover,  taking  into  account  that 
(NUC)  =»  (H)  (see  [15])  and  (H)  =>  (OC)  in  any  Banach  function  lattice  (see  [8]),  we 
get  (ft)  =>  (OC).  Moreover,  by  Lemma  3  we  conclude  that  (UC-1)  =»  (OC)  .  Hence  it  is 


Calderon-Lozanowskii  and  Orlicz-Lorentz  spaces  323 

enough  to  prove  that  if  £  g  (UC1)  and  £  e  (OC),then£  £  (/J) .  Assume  that  E  g  (UC1) 
andE  e  (OC) .  Then  there  exists  s  >  0  s  uch  that  for  every  n  e  N  there  exist  xn,yn  e  B  (E) 

with  max  |||^XA.VnVn  ||£  ,  i^X/i^  Uj  £  £  and  II**  +  >'«IU  >  20  ~  !/>0  -  Denote 
An  =  Aj^y,,.  Divide  the  set  An  into  two  disjoint  subsets  A\  =  suppjc71  \  suppy,,  and 
A^  =  supp)Vz  \  suppj^.  We  divide  the  proof  into  two  parts: 


1 .  Suppose  that 


>  s.  We  claim  that  for  every  n  e  J\f  there  exists  a  set  Bn  c  A 


of  finite  measure  such  that 


x,iXAln\Bn  <  £/2-  Fix  n  e  A/".  Since  the  measure  /z  is 
cr-finite,  there  exists  an  increasing  sequence  of  sets  of  finite  measure  OS*)^  C  A,1, 
such  that  UStLi  $k  =  Aln.  Denote  Wk  =  xnXA},\sk-  Then  Wk  I  0  a.e..  Since  E  is  order 
continuous,  so  \\Wk\\E  ->  0,  what  proves  the  claim.  Then  ||^/iX^  |  E  >  £/2  for  every 
n  6  jV.  We  decompose  each  set  #,2  into  the  family  of  sets  Bn  (7,  2^)  for  7  =  1 ,  2, . . . ,  2^ 
and/:  =  1,2,...,  by  the  following  iteration.  We  divide^  into  two  disjoint  sets  Bn  (I,  2) 
and  £,z  (2,  2)  such  that  IJL  (Bn  (1 ,  2))  =  \JL  (Bn  (2,  2)) .  Suppose  that  for  fixed  k  the  sets 
Bn  (j,  2k)  (1  <  7  <  2*)  are  already  defined.  Toobtainsets  £„  (7,  2*+1)  (1  <  7  <  2*+1) 
we  divide  every  set  Bn  (7,  2*)  (1  <  7  <  2*)  into  two  disjoint  sets  £„  (27  -  1,  2k+{] 
and  £rt  (27,  2*+1)  such  that  //  (/?„  (27  -  1,  2fc+i))  =  IJL  (Bn  (27,  2^1)).  Define  on  the 
set  5n  the  &th  Rademacher  function  by 

II        for  r  e  B^  (7,  2*)  with  odd  7,  1  <  7  <  2*. 
- 1     fort  e  Bn  (7,  2^)  with  even  7,  1  <  7  <  2*, 
0       for  f  g  Brt  (7,  2*) . 


Let 


for  every  n,  /:  E  A/".  We  will  prove  that  for  every  n  €  TV"  we  have  /^  ->  0  as  &  ->  oo  in  E. 
Recall  that  a  A'dY/z^  <afwa/  E'  of  E  is  defined  by 


E  =  |/z  €  L°  :  ||  A  H-/  =  sup]  /  |/i  (/)^(0ld/x  :  g  e  E,  ||^||F  <  1  [  <  oo 
[  *  [JT  \ 

It  is  known  that  E'  is  a  Banach  function  lattice.  Moreover  E*  =  E  iff  E  €  (OC)  (see 
[18]  and  [22]).  Then  for  fixed  n  e  AT  and  every  A*  €  E*  we  get 

lim  A*  (/£)  =  lim    f  //'  (f)  A  (0  d/x  =  lim    /*   r£  (r)  xn  (0  A  (0  d/x  =  0 

since  xw  (f)  A  (r)  is  real  integrable  function.  It  follows  by  the  fact  that  the  set  of  sim- 
ple functions  is  dense  in  Ll  and  for  every  simple  function  b  defined  on  Bn  there  holds 

w 

lim/^oo  fB  r%  (t)  b  (t)  dfji  —  0.  Therefore  for  every  n  e  M  we  have  /^  -*•  0  as  k  ->  oo 
in  E.  Moreover  |/^  |  >  s/2  for  every  n,  k  e  N.  Then,  applying  Hahn-Banach  theorem, 
it  is  easy  to  prove  that  for  every  n  €  N  there  exists  a  subsequence  (§k)k=i  ^  (fk)k=\ 
such  that  sep  [g%}E  >  e/4.  For  every  n  e  A/"  let 

"jt  ~  ^  ~t~  -K/iXsuppjCflXBu  ,  ^  =  1,  2,  .... 


324  Pawel  Kolwicz 

Then  for  every  n  €  A/"  we  get  sep  [hnk  }  E  >  e/4.  On  the  other  hand  for  every  n  eN 

bn  +  hl\\E  =  \\yn  +  xn\\E>2(l-l/n) 
for  all  k  eM.lt  means  that  E  $  (p)  . 
2.  If  I  ynXA2  II  E  -  £s  t^ien  ^e  Pro°f  *s  analogous.  ^ 

Sufficiency.  Takes  >  0.  Let*  6  £(£).  Take  (*„)£!  j  C  B(E)  with  sep{jcn}£  >  £.  Denote 
by  jc+  and  *~  the  positive  and  negative  part  of  ;c,  respectively.  We  will  show  that  there 
exists  a  subsequence  (ZnJJLj  C  O/i)^  such  that  sepfc+}  >  s/2  or  sepfc"}  >  e/2.  For 
every  n  ^  m  we  have  ||*+  -  x+  \\  E  >  e/2  or  |*~  -x~\\E>  e/2. 

1.  Consider  the  element*]  and  the  sequence  (xn)%L2-  Then  there  exists  a  subsequence 

(4°)°°      C(*n)^2SUchthat 

V  /  J7=l 

L-J1"  -  *<1)+      >  £/2  for  every  n  e  A/"  or   L~  -  jc^I}"  II     >  e/2  for  every  n  6  Af  . 

II  £  ||  ||  £  j, 

Denote  y{l)  =  jci  and  >'^j  =  *,(/}  for  every  n  e  A/". 


2.  Consider  the  element  *j(1)  and  the  sequence   x  .  Then  there  exists  a  subsequence 

C  (40)00    such  that 

l         \        /n=2 


\\E 


>  s/2  for  every  n  €  A^  or 


>  fi/2  for  every  n  e 


Denote  vp}  =  *[1}  and  y^  =  42)  for  every  n  e  M. 

Taking  the  next  steps  analogously  we  conclude  that  there  exists  a  sequence  (jk)^L\ 
natural  numbers  and  the  sequence  of  subsequences  (y^V°    ,  k  =  1 , 2, . . .  such  that 


and 
or 

Define  zn  =  y^n  for  every  n  e  N.  The  sequence  (zn)  satisfies  the  required  condition. 
Denote  still  this  subsequence  (zn)  by  (jcn).  Let  sep{jc+ }  >  e/2.  Denote  by  p  (•)  the  modulus 
of  the  uniform  monotonicity  of  E,  by  8E  (•)  the  function  8  (•)  given  in  Definition  1  and  by 
<$£  (•)  the  function  8  (•)  given  in  Definition  2.  We  denote  some  constants 


°>     0  <  a  <  min  f^/8,  e/144} , 

0  <  fc  <  a,  (52  =  3£  (afr)  >  0,  (1) 

0,       p2  =  p  aV/2   >  0. 


Calderon-Lozanowskii  and  Orlicz-Lorentz  spaces  325 

For  every  n  ^  m  we  have  \\\xn\  —  |JC/«|||E  -  ab  or  IIU/zl  ~"  \xm  I  II  E  <  ®b.  Hence,  analo- 
gously as  in  the  previous  part  of  the  proof,  we  may  find  a  subsequence  (v,z)^Lj  C 
such  that 

\\\yn\-\ym\\\E^ab  fora11  n^m  or  \\lyn\-\ym\\\E  «*b  fora11 

Denote  still  (yn)  by  (xn).  We  consider  two  cases: 


I.  Assume  that  \\\xn\  —  Ijc^lll^  >  ctb  for  every  n  ^  m.  Then  sep{|^n|}  >  ab.  Denote 
yn  =  |jcn|  and  y  —  \x\  .  Hence  yn  e  5  (E+)  and  sep  {yn}£  >  ctb.  Basing  on  property 
(j8+)  we  find  a  number  k  €  M  such  that  ||(>»  +  yk)  /2\\E  ~  *  ~"^2,  where  82  is  defined 
in(l).  Consequently  ||jc  4-  xk  II  E  <  lly  +  ^IU  <2(l-^2). 

II.  Suppose  that 

II  \xn  I  -  \xm  \\\E  <ab  for  every  «  ^  m.  (2) 

For  every  n^m  denote 

A[nm     =     suppo:+nsuppjc+  and  A2nm     =     supp-r+  -s-  suppjr+, 

where  A  -f-  B  =  (A  \  B)  U  (B  \  A)  .  Since  sep{jc+}  >  fi/2,  then  for  every  n  ^  m  we  get 
1  W"  -  *,t)  XAL  II  E  >  */*  ™  II  W"  -  ^)  XxL  11  fi  >  fi/4.  Suppose  that  ||  (jc+  -  x+) 
1         —  £/^  ^Or  some  n  ^  m.  Then 


I]   I   „        I  I  ....          I  II  x. 

II  \Xn  I  —  l^mlllfi  > 

which  is  a  contradiction  with  (2).  Hence 

Y     —  r     1  Y  j  •>  >  P  /4  for  i 

*n        Am)  *-A%m     p  -~  &l^  1U1  ' 

Decompose  the  set  A1nm  into  two  disjoint  subsets 


Notice  that  for  every  t  e  A~^n  we  have  sgn  (xn  (t)  xm  (t))  =  0.  First  we  will  show  that 

(x*  —  x*)  XA21        —  s/^  ~"  2ab  for  every  n  ^  m.  (4) 

Ifmax|||*nXA22  ||£,  \XmXAll  IE}  -  ab> then  I  I*"'"  '^HU  -  a^'  which  is  a  contra- 
diction with  (2).  Hence  max  I  |U7/xA22  II  r,  lUmXA^  II  r  \  <  ab-  Suppose  that  (4)  is  not 

[  "  •ri«/j;  ''  tL>     "  "•nm  "  *->  J 

true.  Then,  in  view  of  (3),  for  some  n  ^  m  we  get 

P/4  <  ll/r+  -  r+W   o          <  l/rjc+  -  x+W,o,  II     - 
«/^  ±  I  (*n        ^m  J  XA2m         5  I  (*„        *m  J  XA^  I E 


which  is  a  contradiction,  so  the  inequality  (4)  holds.  Decompose  the  set  A^m  into  two 
disjoint  subsets  A^1  =  suppjc+  \  supp^  and  A^2  =  suppjc^  \  supp^.  Consequently 


ab>  \\\Xn\-  \Xm\\\E  > 


326 


Pawel  Kolwicz 


>  max 


max 


+  -  xm  )  xA2i  1  1  £  ,  I  (4  -  *„  )  XA™   E  } 


nnm  ||  £ 


for  every  n  ^  m.  By  (4)  we  get  max  jp^x^n 

>  g/8  -  aZ?,  then  IjCx^ii       ^  e/8  "  2ab- If 

£  -  II          /»«  HE 

then  JU^XAJJ?  I     ^  £/8  -  2of^-  Hence 


>   */8  -  ab.  If 
>  e8  -  aft, 


(5) 


Denote  r  A  j  =  min  {r,  ^}  and  r  v  5  =  max  {r,  51}  for  r,  5-  €  K.  For  every  n  €  A/"  define 

Bn  =  {t  e  suppjc  :  |^(r)|  A  \xn(t)\  >  b  (I*  (01  v  |jrn(OI)}  ,     Cn  =  suppx  \  Bw, 
^  =  [t  e  Bn  :  sgn(^  (t)xn  (0)  =  -1}  ,  ^  =  Bn  \  5,[, 

C,|  =  {t  e  Cn  :  |jc(OI  =  WO  I  A  MOD  ,  Cl  =  Cn  \  CA1 

and  for  every  k  ^  n  let 


Dnk  = 


f?  :  MOi  A  |^(OI  >  <*b  (\xk(t)\  v  |^ 
^  (0)  =  -1}  , 


ILL  Suppose  that  LrxBi       >  8a  for  some  n  €  M.  Denote  by  <5<ft  (•)  the  modulus  of 
convexity  of  7£.  Note  that  &K  (2)  =  1 .  Applying  Lemma  2  we  get 


+  *« 


Consequently 


\X+Xn\ 


\Xn\ 


'      |JC/|| 


Hence,  applying  the  uniform  monotonicity  of  E,  we  get  \\(x  +  Xn)  /2\\  E  <  1  —  p\,  where 
pi  is  defined  in  (1). 


IL2.  Let 


xXsl 


for  some  n  e  A/". 


(6) 


Note  that  if 


'«»£ 


—  a»  *en 


>  l^Xc^l^  >  ia>  I- Hence 


FXcj 


<  a. 


(7) 


Furthermore   JC71Xc2       <  /?  <  a.  Consequently  if   *&Xc2       —  ^a  f°r  some  k  ^  n,  then 
II  l*ii  I  ~  I**  I II  >a>ab,  but  this  contradicts  (2).  Thus 


(8) 


Calderon-Lozanowskii  and  Otiicz-Lorentz  spaces 
Moreover  we  will  show  that 

*fcXr>2        <  4oeb  for  every  k^n. 

II  nk  II  E 

Suppose  conversely  that  UA-XD^        -  4aZ?  f°r  some  ^  ^  n  anc*  let 


327 


(9) 


. 


If  f  jr*xD2i        >  2ab,  then    *«XD2i       >  2.  But  *„  e  B  (£) .  Hence 

«*  II  £,  II  n*  I!  E 


lab.  But 


nk  II  E 


<  ab.  Consequently  |||^|  -  |.^|||  >  ab,  which  is  a  contradiction 


with  (2),  so  (9)  is  proved.  We  divide  the  proof  into  two  parts: 


>  a  for  some  k  ^  n.  Note  that  for  every  f  €  E{k  we  have 


and  sgn(;r* 


a. 


Since  SK  (2)  =  1 ,  so  applying  Lemma  2  we  get 

(*  + 
Then 


Hence,  similarly  as  in  case  II.  1,  we  conclude  that  \\(x 
defined  in  (1). 


<  1  —  p^  where  ^2  is 


nk  II  E 


a  f°r  everY  k  ^=  n.  Then,  by  (8)  and  (9),  we  get 


+ 


for  every  k  ^  n. 


(10) 


Notice  that  A^  n  £^  =  0  for  every  k  ^  n.  Furthermore  the  inequality  (5)  yields 
>  £/8  —  2ab  for  every  k  ^  n.  Consequently,  by  (1)  and  (10),  we  obtain 


'nk  (I  E 

S  8 

8  ~~  9(*  -  16 

for  every  k  ^  n.  Let  z  =  |jc|  X/?2uc2-  Denote  7*  -  suppz  -4-  suppjt*.  Then,  by  (1 1),  we  get 
||^XrJ|£  v  I^XT/JE  ^  ^/16  for  every  k  e  J\f.  Since  E  e  (UC1) ,  then  llzH-^/rlU  < 
2  (1  -  5i)  for  every  A;  €  A/",  where  ii  is  defined  in  (1).  Thus,  by  (1),  (6)  and  (7),  we  obtain 


9of 


Combining  all  of  the  cases  we  get  ||(jc  +Xk)/2\\E  <   I  —  ^  for  some  k  €  A/",  where 
A  =  min  {#2,  /?i,  pa.  3^i/8} ,  which  finishes  the  proof.  D 


328  Panel  Kolwicz 

An  immediate  consequence  of  Lemma  3  and  Theorem  I  is  the  following. 

COROLLARY  1 

Let  Ebea  Banach  function  lattice.  IfE  e  (ft)  ,  then  E  e  (UM). 

Here  we  will  find  some  necessary  and  sufficient  conditions  for  the  property  (ft)  in  the 
space  £0.  We  will  need  the  following  notion.  We  say  that  Musielak-Orlicz  function  $ 
satisfies  the  uniform  &%  -condition  (we  write  O  e  A2  for  short)  if  there  exist  a  constant 
k>2  and  a  set  A  e  £  of  measure  zero  such  that  O(r,  2u)  <  kQ(t  ,  u)  for  every  t  e  T  \  A 
and  every  u  ell. 

For  t  e  T  the  function  <£(f,  •)  is  strictly  convex  if  $  (t  ,  (u  +  v)  /2)  <  (<&('.  M)  +  $ 
(f  ,  u))/2  for  all  M,  u  e  7£,  w  ^  u. 

Lemma  4.  Suppose  that  $*  e  Af  .  77z<?ft  f/zere  exfo  a  number  £  >  1  onrf  nonnegative 
function  f*  with  $>  o  2/*  e 


t-a.€.  t  eT  andu>  /*(*). 

Proo/.  Lemma  4  for  £  =  L1  was  proved  in  [1],  the  proof  in  general  case  is  similar. 
The  proof  of  the  next  lemma  is  similar  to  the  proof  of  Theorem  7  in  [9]  . 

Lemma  5.  IfE  e  (UC1)  ,  <t>  €  Af  and  <J>>  0,  ften  £$  €  (UC-1)  . 
Theorem  2.  The  following  assertions  are  true: 

(i)  //£<»  e  (/J),  tfzen  <D  e  A|  . 

(ii)  Ler  £  €  (UM),  <l>  e  Af  ,  <X>  >  0  anJ  O*  e  Af  .  Den<9?e  ^  /*  the  function  from 
Lemma  4.   ///or  fji-a.e.   t  e  T,  0>(r,  -)  iy  rfncrfy  convex  function  on  the  interval 


(iii)  Assume  that  E  €  (UM),  <f>  e  Af ,  <J>  >  0  awd  O*  €  A2.  r/z^n  £<j>  6  (/*). 
(iv)  //£  e  (/J),  $  €  Af  0/zd  <J>  >  0,  rAe/i  £<D  e  (ft). 

Proof  (i)  If  <3>  £  Af ,  then  £$  contains  an  order  isomorphically  isometric  copy  of/00  (see 
[10]).  But  QJ)  =>  (OC)  (see  the  proof  of  necessity  of  Theorem  1).  Because/00  £  (OC), 
then  Z°°  $  (ft)  and  so  £4,  £  (ft). 

(ii)  Let  s  >  0  be  arbitrary.  Take  x,xneB  (£$)  with  sep{jcn}  >  5.  There  exists  an  index  k 
with  ||*  —  *jtlU  >  £/2.  Denote  /i  =  e/2.  Then  the  definition  of  Luxemburg  norm  yields 

>  1.  (12) 


By  the  assumption  that  4>*  €  Af  and  0  is  strictly  convex  function  on  the  interval  [0,  /*  (01 
for  />6-a.e.  t  e  T,  applying  Lemma  4,  it  is  easy  to  show  that  there  exist  numbers  A.  €  (0,  1), 


Calderon-Lozanowskil and  Orlicz-Lorentz  spaces 
6  (0,  1)  and  a  nonnegative  function  f\  with    O  o  -7/1       <  |  such  that 

cf> 


329 


(13) 


for  /x-a.e.  f  €  T  and  M,  t>  >  0  and  satisfying  conditions  max(w,  u)  >  /i(f)  and  |w  —  v\ 
>  &l££\u  +  v\  (see  [1]  and  [6]).  Moreover,  E  e  (UM),  so  £  €  (OC).  Then,  by 
<I>  >  0,  so  we  conclude  that,  there  exist  a  number  k  >  2  and  nonnegative  function  /2  with 

and 


*o/2£< 


(14) 


for/x-a.e.  /  €  T  and  w  >  /2(r)  (see  Lemma  2  in  [10]).  Define  /(O  =  max{/i(f),  /a(OJ- 
Then  J  *  o  ^/  j     <  |-  Denote 

A     = 


Then 


Moreover 


C     =     A 


-f 


<  l- 


/jc  —  jic^A  II        /2    \ 

<Do    -    xr\5      <  po  I  -/  J 

V     M     /  E      II        V^    / 


Consequently 


^;^c  < 


<i-f. 


By  (12)  we  get    <I>  o  (:L-~M  Xc       >  T-  Take  a  natural  numbers  with  ^  <  2m.  Applying 
the  convexity  of  <£  and  using  (14),  we  obtain 


Xc 


< — 

~~   2 


So 


(15) 


Furthermore  2/(0  <  |jc(/)  -^(01  <  2max{|jc(f)l ,  1^(011  for  every  t  €  C.  Then  the 
inequality  ( 1 3)  yields 

<  ~(<J>oJc  +  OojcA)-  "(Oo^-f  <J> 

2,  2* 


/j 

$0v 


330  Pawel  Kolwicz 

Denote  by  p(-)  the  modulus  of  uniform  monotonicity  of  the  Banach  lattice  E.  From  (15) 
and  (16)  we  conclude 


Finally  we  get  ||(;c  +  jc*)  /2\\®  <l—q,  where  q  e  (0,  1)  depends  only  on  p  (see  Lemma 
3  in  [10]). 

(iii)  The  proof  is  analogous  as  in  (ii). 

(iv)  Since  E  €  (ft)  ,  then  by  Theorem  1  we  get  E  e  (UC1)  .  Furthermore  <B  €  Af 
and  4>  >  0.  By  Lemma  5  we  conclude  that  E$>  e  (UC1)  .  Basing  on  Theorem  1, 
it  is  enough  to  show  that  E$  6  (/J+).  Denote  by  E$  the  positive  cone  of  £$.  Let 
s  e  (0,  1)  and  x  6  B(E%).  Take  C*,,)^  C  B(E%)  such  that  septa  }  >  £•  Since  £  6 
(FP),  /$(•)  is  left  continuous.  So,  by  the  definition  of  the  Luxemburg  norm,  we  get 
II*  o  X\\E  <  1  and  ||<I>  o  xn\\E  <  1.  Moreover  (ft)  =»  (OC)  .  Hence  the  assumptions 
that  E  e  (OC),  <£  e  A|  and  $>  >  0  imply  that  there  exists  a  number  a(s)  e  (0,1) 
such  that  ||<I>  o  (xn  -  xm)\\E  >  cr(e)  for  every  natural  n  +  m  (see  Lemma  6  in  [10]). 
The  function  O  is  superadditive  on  7£+,  so  |<I>  o  (xn  —  xni)\  <  |4>  o  xn  —  4>  o  xm\  , 
and  consequently  ||  <£  o  xn  —  4>  o  xm  \\  E  >  cr  (s)  for  every  n  ^  m  .  By  the  property  (/3) 
of  £  we  conclude  that  there  exist  a  number  S  =  8  (0(e))  >  0  and  a  number  k  €  M 
such  that  ||  $  o  jc  +  $  o  jc*  ||  E  <  2  (1  -  S)  .  Furthermore 


'I  2  IE 

Finally,  (see  Lemma  3  in  [10]),  there  exists  a  number  y  =  y(S)  €  (0,  1)  such  that 


Remark  2.  The  condition  E  e  (ft)  is  not  necessary  for  Eq>  €  (ft).  It  is  sufficient  to  take 
E  =  Ll  over  the  finite  measure  space  (T,  E ,  jjt)  and  the  function  3>  which  does  not  depend 
on  the  parameter  /.  Then  L®  e  (ft)  iff  O  e  Aa  and  <!>  is  uniformly  convex  on  the  interval 
[UQ,  oo]  for  every  w0  >  0  (see  [7]).  But  L1  £  (0),  because  it  is  an  Orlicz  space  generated 
by  the  function  <f>(w)  =  \u\  which  is  not  uniformly  convex. 

Now  we  will  assume  that  <l>  does  not  depend  on  the  parameter  t.  The  function  <3>  is 
strictly  convex  if  0>  ((u  +  u)  /2)  <  (O(w)  +  4>(v))/  2  for  all  u,v  £  H,  u  ^  v.  The 
function  0  is  uniformly  convex  (uniformly  convex  for  large  arguments),  if  for  any  a  >  0 
(a  >  0  and  KO  >  0)  there  exists  a(a)  >  0  (<5  =  8(a,  UQ)  >  0)  such  that  $  ((M  -I-  «w)/2)  < 
(1-8)  ($>(u)  -f  <b(au))/  2  holds  true  for  every  u  >  0  (for  every  u  >  «o)- 

The  implication  (UC)  =»  ()8)  can  be  reversed  in  Orlicz  function  spaces  over  the  finite 
measure  space.  It  was  shown  for  both  Luxemburg  and  Orlicz  norm  ([7]).  Here  we  will 
extend  this  result  to  the  case  of  Orlicz-Lorentz  spaces  over  the  finite  and  infinite  measure 
space. 

Theorem  3.  Let  E  =  Aw  with  y  =  oo  an d  assume  that  <b  does  not  depend  on  t.  Then  the 
following  statements  are  equivalent: 

(a)  (AJ*  €  (UC). 
0>)  (A«)*  €  (ft). 


Calderon-Lozanowskiiand  Orlicz-Lorentz  spaces  33 1 

(c)  <f>  is  uniformly  convex,  <f>  satisfies  the  ^-condition  for  all  arguments  and  the  function 
a)  is  regular. 

Proof    The  implication  (UC)  =>  (/?)  holds  in  any  Banach  space.    It  is  also  true  that 
(c)  <»  (a)  (see  [17]).  Note  that  if  y  —  oo  and  w  is  regular,  then  /^°  a>(t)6t  =  oo.  It  is 
enough  to  prove  the  implication  (b)  =»  (c).  Assume  then  that  (A^)®  e  (ft).  First  we  will 
show  that  w  is  regular. 
Take  any  t  >  0.  Since  CD  is  locally  integrable,  there  exists  a  number  af  >  0  such  that 

<*>(*,)  /     a>(s)ds  =  &(at)S(2t)  =  1.  (17) 


Divide  the  interval  [0,  2t]  into  two  intervals  G\  =  [0,  t]  and  G^  =  [t,  2t].  Suppose  that 
the  sequence  of  intervals 


p1  =  [0,  2f/2w~1],  G^"1  =  [2t/2n-\2tf2n~2]> ...,  G^_\  =  [2r  • 

n  >  2,  is  already  defined.  We  divide  each  set  Gp1  =  [(i  -  \)2t/2n~\  I2t/2n~ll  i  = 
1, 2, ...,  2n~l  into  two  subsets  G^j,  G^-  such  that 

and 

G"2i  =  [(i  -  l)2r/2n~1  +  f/2'l~l,  i2t/2n~{]9  (i  =  1,  2, ...,  2"~!). 

In  such  a  way,  we  obtain  a  partition  (G",  GJ, ..-,  C^) ,  n  =  1,  2, ...  of  the  interval  [0,  2t] 
such  that  v(G?)  =  2~n+lr,  (n  =  1,  2, ...,  i  =  1, 2, ...,  2n)  and  v  denotes  the  Lebesgue 
measure.  Define 


where  £u  =    U  ^-p     £2,«  =    U  G^,  (n  =  1,2, ...).  We  get  /«>(*,)  = 

A:=l  A=l 

/02r  o>(5)d5  =  1.  Moreover  jc*r  =  atX[0,2t]  for  every  n  6  AT.  Consequently  /4»(jcn,r)  =  1- 
Furthermore  (jcrtf(  -  jrw,r)*  =  2a;X[0,r]  for  everY  w,  m  €  AT,  n  ^  m.  Hence  Iv(xnj  - 
jcOTt/)  =  <I>(2a;)  /J  co(5)d^.  Taking  into  account  that  0  is  convex  and  <w  is  nonincreasing 
function,  in  view  of  (17),  we  get 

ft  j     f2t 

Q(2at)  I    a>(s)ds  >  2<E>fe);r  /     Q>(s)ds  =  1. 
Jo  2  Jo 

for  every  f  >  0.  Then 

inf  <D(2a,)  f  a)(s)ds  >  1.  (18) 

^>o  Jo 

Thus  I jcrt,r  -  jcm,r  J  0  >  1  for  every  t  >  0  and  n,  w  €  A/\  n  ^  m.  We  have  constructed 
for  every  f  >  0  an  element  xt  e  S(Aa),<i> )  and  a  sequence  (xn,t)^\  €  5(ACl),cD  )  with 
sepfe,/}  >  1.  By  the  property  (ft)  of  (A<y)cj>,  there  exists  a  number  8  =  6(1)  >  0  and  an 
index  fc  for  which 


332  Pawel  Kolwicz 

Notice  that  (  Xt^'*nJ  \    =  atX[Q,t]  for  every  n  e  A/*.  Thus,  in  view  of  (19)  we  get 


This  shows  the  regularity  of  the  weight  co. 

By  the  assumption  (A^)^  e  (/?).  Then  <3>  satisfies  the  suitable  A|  condition  (Theorem 
2(i)  in  the  case  when  O  does  not  depend  on  the  parameter  was  proved  in  [12]),  i.e.  there 
exists  a  number/:  >  0  such  that  for  every  u  e  71  we  have  <f>(2w)  <  £<I>(w)  (see  also  [16]). 
We  get  in  particular  that  3>  >  0.  Moreover,  by  Theorem  1.13  in  [5],  for  every  /  >  1  there 
exists  ki  >  1  such  that  for  every  u  e  7£  we  have 

<D(/M)  <ki®(u).  (20) 

Now  we  will  show  that  $  is  uniformly  convex.  At  first  we  will  prove  that  <l>  must  be 
strictly  convex.  Suppose  conversely  that  <2>  is  affine  on  the  interval  [M,  v].  The  weight 
function  a)  is  locally  integrable,  so  there  exists  a  number  a  >  0  such  that  0  <  M  o  = 
<I>(iO  /Q'  <w(0d?  <  1.  Moreover,  if  we  define  ^(A)  —  fA  a)(t)dt,  then  we  conclude  thai 
/L6W  is  non-atomic.  By  the  Lapunov's  theorem  {^(A)  :  A  is  Lebesgue  measurable}  = 

[0,  oo).  Consequently  for  every  X  >  0  there  exists  a  number  y  >  0  such  that  f^  co(t)dt  = 

i+n 
f^*   o)(r)dr.  Take  numbers  y,  A.  >  0  satisfying 

(i)     MI  =  4>(u)  /ax  o)(0dr  -f  0(w)  /xx  co(t)dt  <  1  -  M0, 
(ii)     £ 


Then  we  find  a  number  c  >  i;  with  3>(c)  f£  a)(t)dt  +  MI  =  1.  Define  a  partition 
(Gp  Gj,  ...,  G^n)  ,n  ~  1,2,  ...  of  the  interval  [X,  y]  in  the  same  way  as  in  the  previ- 
ous part  of  the  proof.  Let 

X  =  CX[Q,a]  +  VX[aM  +  «X[X,y]   and  Xn  = 


2«~i  277"1 

where  ElfB  =  \J  G^_,,    E2,n  =  U  G^,  (n  =  1,  2,  ...).  We  get  7<i>(jc)  =  1.  Moreover 
*=1  Jk=l 

*  M-f  V 

-^n  =  CX[0,a]  +  —  ^—  XMX+y)/2]  +  «X[^+x)/2,y]- 

Then,  by  (ii)  and  the  linearity  of  the  function  <D  on  the  interval  [M,  v],  we  conclude 

=  0. 


r 
-  / 

A 


Furthermore  (xn  -  xm)*  =  (v  -  w)/2x[o,(/~A)/2]  for  every  n,  ;w  e  A/",fl  7^  m.  But 
||(u  -  ^O/^xtoxy-x^Oio  =  ^  for  some  q  >  0.  We  have  defined  an  element*  € 
Jt),^  )  and  a  sequence  Oc/OJjta  €  ^(A^,^  )  with  sep{jc,,}  >  ^.  On  the  other  hand 


V     4 


Calderon-Lozanowskii  and  Orlicz-Lorentz  spaces  333 

for  every  n  €  AT.  Applying  the  fact  that  $  is  affme  on  the  interval  [w,  u],  by  (ii),  we  get 


for  every  n  e  AT.  Hence  ||  ^^  ||  =  1  for  every  n  €  A/".  But  this  is  a  contradiction  with  the 
property  (ft). 

To  finish  the  proof  it  is  enough  to  show  that  <J>  is  uniformly  convex.  Suppose  that  this  is 
not  true  i.e.  there  exists  a  sequence  u^  of  positive  numbers  and  a  constant  b  €  (0,  1)  such 
that 


=  f  ( M*+m*  )+$buk. Consequently, 
applying  the  convexity  of  4>,  it  is  easy  to  prove  that 

<J> 
and 

/^UL  -4-h/y/A          /          4\    /^  1  \ 

(22) 


If  there  is  a  subsequence  of  (M*)  approaching  a  number  w  >  0,  then  <£  is  affine  on  the 
interval  [bu,  u]  and  thus  4>  is  not  strictly  convex.  Consequently,  without  loss  of  generality, 
we  assume  that  Uk  ~>  0  or  uk  ->  oo.  The  proof  will  be  done  only  for  the  case  M*  ->•  0, 
in  another  case  it  is  analogous.  The  weight  function  co  is  locally  integrable,  so  there  exists 
a  number  a  >  0  such  that  0  <  MO  =  $(u\)  f£  <t)(t)dt  <  1/2.  Then,  similarly  as  in 
the  proof  of  the  strict  convexity  of  4>,  for  every  k  €  N  there  exist  numbers  A.*  >  0  and 
>  0  satisfy  ing 


t*  r- 

I     co(t)dt=  I          fi>(0d;,  (23) 

Ja  Jxk 

M0<Mk<l-  Mo,  (24) 

where  M*  =  Qfa)  f£k  co(t)dt  -f  &(buk)  f^  a>(t)dt.  Then  take  a  sequence  c*  >  ut  with 

Affc  =  1.  (25) 


Then  there  exists  a  number  p  >  0  such  that 

-  b)  /2)  S  ((yk  -  A*)/2)  >  p  (26) 


for  all  k  e  AT.  Indeed,  suppose  conversely  that  <I>  (wjt  (1  -  6)  /2)  -S  ((yjk  -  A,^)/2)  ->  0. 
But  O  €  A2-  Putting  /  =  -^  in  inequality  (20)  and  denoting  fa  =  ^,  u  =  jz£«,  we  get 


334  Pawel  Kolwicz 


-  b)v/2)  >  ftb$(v)  for  every  v  e  72,  Thus  O  (uk)  S  ((yk  -  Xk)/2)  ->  0.  Moreover, 
by  (23)  and  (24),  taking  into  account  that  co  is  nonincreasing  and  <3>  is  a  convex  function 
we  obtain 


r^ 
I 

Jrt 


<D  (it*)  5  (0*  -  A,*)/2)  +  6d>  (m)  25  ((yk  -  A.*)  /2)  ->  0, 


but  this  is  a  contradiction,  so  (26)  is  true.  For  every  k  €  AT  let  (G*'*,  G*'/z,  ...,  G 
n  =  1,  2,  ...  be  a  partition  of  the  interval  [kk,  yk]  constructed  in  the  same  way  as  in  the 
previous  part  of  the  proof.  Define  for  every  k  e  M  an  element  jc*  €  S(Aa>t^  )  and  a 
sequence  (xfo^  e  S(Aa>,<s>  )  by 


and 


where  E^fl  =  U^1  G^p  £^  =  U^1  G%\  (n  =  1,  2,  ...).  Then,  by  (26),  we  get 
/o  (x*  -  A:*  )  >  p  for  every  A:  6  A/"  and  n  ^  m.  So  there  is  a  number  q  >  0  such  that 
II  xn  -  xm  I  CD  >  ^  for  all  -A:  6  M  and  n  /  m.  Moreover,  by  (21),  (22),  (23)  and  (25),  we 
get  /<D  (  (xk  +  *„}/  2)  >  1  -  4/fc  for  every  n  e  AT  .  The  Ai-condition  implies  that  there 
exists  a  sequence  (ak)f=l  c  U  with  lim^oo  ak  =  0  such  that  ||  (jc*  +  **)  /2||  ^  >  1  -  ^, 
n  =  1,  2,  ...  (see  [10]).  This  contradiction  shows  that  the  property  (ft)  implies  that  <J>  is 
uniformly  convex  on  the  whole  real  line.  D 

For  y  finite  one  can  prove  in  the  similar  way  as  Theorem  3  the  following: 

Theorem  4.  Let  E  =  Aw  with  y  <  oo  and  assume  that  $  does  not  depend  on  t.  Then  the 
following  statements  are  equivalent: 


(a)  (Aa,)*  €  (UC) 

(b)  (A  Jo  €  08) 

(c)  <J>  is  uniformly  convex  for  large  arguments,  <E>  satisfies  the  ^-condition  for  large 
arguments  and  the  function  a)  is  regular. 

Taking  CD  =  1  in  Theorems  3  and  4  we  get  the  following  characterization  for  Orlicz 
spaces  equipped  with  the  Luxemburg  norm  over  finite  or  infinite  measure  space  (see  [7]  for 
the  finite  measure  space). 

COROLLARY  2 

Let  $  be  an  Orliczfunction  and  let  L<t>  be  the  Orlicz  function  space  over  the  finite  or  infinite 
measure  space.  Then  the  following  statements  are  equivalent: 


Calderon-Lozanowskiiand  Orlicz-Lorentz  spaces  335 

(a)  Lo>  e(UC). 

(b)  U  e(ft). 

Using  Theorems  3  and  4  for  <D(w)  =  \u\  we  get  immediately 

COROLLARY  3 

Let  y  <  oo  or  y  =  oo.  The  Lorentz  space  A&  does  not  have  the  property  (ft). 

Acknowledgements 

We  wish  to  thank  the  anonymous  referee  for  his  valuable  remarks  which  led  to  substantial 
improvements  of  the  paper. 

References 

[1  ]  Alherk  G  and  Hudzik  H,  Uniformly  non-/^  Musielak-Orlicz  spaces  of  Bochner  type,  Forum 

Math.  I  (1989)403-410 
[2]  Bennett  C  and  Sharpley  R,  Interpolation  of  Operators  (New  York:  Academic  Press  Inc.) 

(1998) 
[3]  Calderon  A  P,  Intermediate  spaces  and  interpolation,  the  complex  method,  Studia  Math. 

(1964)  113-190 

[4]  Cerda  J,  Hudzik  H  and  Mastyto  M,  On  the  geometry  of  some  Calderon-Lozanowskii  inter- 
polation spaces,  Indag.  Mathem.  N.S.  6(1)  (1995)  35-49 
[5]  Chen  S,  Geometry  of  Orlicz  spaces,  Dissertations  Math.  356  (1996)  1-204 
[6]  Chen  S  and  Hudzik  H,  On  some  convexities  of  Orlicz  and  Orlicz-Bochner  spaces,  Comm. 

Math.  Univ.  Carolinae  29(1)  (1988)  13-29 
[7]  Cui  Y,  Phiciennik  R  and  Wang  T,  On  property  (ft)  in  Orlicz  spaces,  Arch.  Math.  69  (1997) 

57-69 
[8]  Domingues  T,  Hudzik  H,  Mastyto  M,  Lopez  G  and  Sims  B,  Complete  characterizations  of 

Kadec-Klee  properties  in  Orlicz  spaces  (to  appear) 
[9]  Foralewski  P,  On  some  geometric  properties  of  generalized  Calderon— Lozanowskii  spaces, 

Acta  Math.  Hungar.  80(1-2)  (1988)  55-66 
[10]  Foralewski  P  and  Hudzik  H,  Some  basic  properties  of  generalized  Calderon-Lozanowskii 

spaces,  Collectanea  Math.  48(4-6)  (1997)  523-538 
[11]  Hudzik  H  and  Kaminska  A,  On  uniformly  convexifiable  and  B  -convex  Musielak-Oriicz 

spaces,  Comment.  Math.  25  (1985)  59-75 
[12]  Hudzik  H,  Kaminska  A  and  Mastyto  M,  Geometric  properties  of  some  Calderon-Lozanowskii 

spaces  and  Orlicz-Lorentz  spaces,  Houston  J.  Math.  22  (1996)  639-663 
[13]  Hudzik  H,  Kaminska  A  and  Mastyto  M,  Monotonicity  and  rotundity  properties  in  Banach 

lattices,  Rocky  Mountain  J.  Math.  30,  3  (2000)  933-950 
[14]  Hudzik  H  and  Landes  T,  Characteristic  of  convexity  of  Kothe  function  spaces,  Math.  Ann. 

294(1992)117-124 
[15]  Huff  R,  Banach  spaces  which  are  nearly  uniformly  convex,  Rocky  Mountain  J.  Math.  10 

(1980)473-749 

[16]  Kaminska  A,  Some  remarks  on  Orlicz-Lorentz  spaces,  Math.  Nachr.  147  (1990)  29-38 
[17]  Kaminska  A,  Uniform  convexity  of  generalized  Lorentz  spaces,  Arch.  Math.  56  (1991)  181- 

188 

[18]  Kantorovich  L  V  and  Akilov  G  P,  Fund.  Anal.  Nauka  (Moscow)  (1977)  in  Russian 
[19]  Kutzarowa  D  N,  A  nearly  uniformly  convex  space  which  is  not  a  (ft)  space,  Acta  Univ. 

Carolinae,  Math,  et  Phys.  30  (1989)  95-98 
[20]  Kutzarowa  D  N,  An  isomorphic  characterization  of  property  (ft)  of  Rolewicz,  Note  Mat.  10(2) 

(1990)347-354 
[21]  Kutzarowa  D  N,  On  condition  (ft)  and  A-uniform  convexity,  C.  R.  Acad.  Bulgar.  Sci.  42(1) 

(1989)  15-18 


336  Pawel  Kolwicz 

[22]  Lindenstrauss  J  andTzafriri  L,  Classical  Banach  spaces  II  (Springer- Verlag)  (1979) 
[23]  Lozanowskli  G  Ya,  On  some  Banach  lattices,  Sibirsk.  Math.  J.  12  (1971)  562-567 
[24]  Maligranda  L,  Orlicz  spaces  and  interpolation,  Seminars  in  Math.  5  (Campinas)  (1989) 
[25]  Rolewicz  S,  On  drop  property,  Studio  Math.  85  (1987)  27-35 
[26]  Rolewicz  S,  On  A-uniform  convexity  and  drop  property,  Studia  Math.  87  (1987)  181-191 


Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  11 1,  No.  3,  August  2001,  pp.  337-350. 
©  Printed  in  India 


On  oscillation  and  asymptotic  behaviour  of  solutions  of  forced 
first  order  neutral  differential  equations 

N  PARHI  and  R  N  RATH* 

Department  of  Mathematics,  Berhampur  University,  Berhampur  760  007,  India 
'"Department  of  Mathematics,  Govt.  Science  College,  Chatrapur  761  020,  India 

MS  received  26  June  2000 

Abstract.    Jn  this  paper,  sufficient  conditions  have  been  obtained  under  which  every 
solution  of 


[v(0  ±  y(t  -  r)]'  ±  Q(t)  G(y(t  -  a))  =  /(r),     t  >  0, 


oscillates  or  tends  to  zero  or  to  ±00  as  t  -+  oo.  Usually  these  conditions  are  stronger 
than 

oo 
j  Q(t)  dt  =  co.  (*) 

0 

An  example  is  given  to  show  that  the  condition  (*)  is  not  enough  to  arrive  at  the  above 
conclusion.  Existence  of  a  positive  (or  negative)  solution  of 


-  y(t  -  *)     +  2(0  G(y(t  -  or))  =  /(r) 
is  considered. 
Keywords.     Oscillation;  nonoscillation;  neutral  equations;  asymptotic  behaviour. 

1.  Introduction 

In  a  recent  paper  [8],  the  authors  have  obtained  necessary  and  sufficient  conditions  so  that 
every  solution  of 

[y(t)  -py(t-  r)]'  +  Q(t)  G(y(t  -  or))  =  /(O 

oscillates  or  tends  to  zero  as  t  -»  co  on  various  ranges  of  p,  where  G  6  C(R,  R),  Q  € 
C([0,  oo),  [0,  oo)),  /  6  C([0,  oo),  R)t  r  >  0  and  a  >  0.  They  have  studied  the  similar 
problem  in  [9]  for  equations  of  the  form 

-  P(t)  y(t  ~  r)]  ±  Q(t)  G(y  (t  -  a))  =  /(r) 

for  different  ranges  of  p  €  C([0,  oo),  R),  where  /,  G,  Q,  r  and  a  are  same  as  above.  In 
these  results,  the  primary  assumption  is 


oc 


(1) 


338  N  Parhi  and  R  N  Rath 


However,  these  results  don't  hold  good  for  the  critical  case  p(t)  =  1  or  p(t)  =  -  L  In 
this  paper,  an  attempt  is  made  to  study  oscillatory  and  asymptotic  behaviour  of  solutions 
of  equations  of  the  form 

[y(!)  ±  y(t  -  r)]'±  2(0  G(y(t  -  a))  =  /(r),  (2) 

where xG(x)  >  0  for  x  ^  0  and  G  is  nondecreasing.  We  assume  that 


CO 

/|/(0|dr 


<  00. 


In  most  of  our  results,  the  assumptions  are  stronger  than  (1).  It  seems  that  it  is  possible  to 
obtain  an  example  of  a  neutral  differential  equation  in  the  critical  case  such  that  (1)  holds 
but  the  equation  admits  a  nonoscillatory  solution  which  does  not  tend  to  zero  as  t  ->  oo. 
A  similar  example  is  obtained  in  the  discrete  case  by  Yu  and  Wang  [13]. 

Several  open  problems  are  stated  in  [2]  (see  6.12.9  and  6.12.10,  pp.  161)  for  equations 
of  the  type 

[y(t)  ±  y(t  -  T)]'  +  2(0  y(t  -  a)  =  0- 

In  a  recent  paper  [10],  Piao  has  solved  one  open  problem  with  an  extra  condition.  Indeed, 
he  showed  that  every  nonoscillatory  solution  of 

0(0  +  y(*  -  r)]'  +  2(0  y(t  -  a)  =  0 

tends  to  zero  as  t  ->  oo  if  (1)  holds  and  Q(t  +  r/n)  <  Q(t )  for  t  e  [0,  oo)  where  n  is  any 
fixed  positive  integer.  However,  Ladas  and  Sficas  [6]  have  shown  that  every  solution  of 

0(0  -  y(*  - *)]'  +  2(0  y(*  -  <O  =  o  (3) 

oscillates  if  (1)  holds.  Chuanxi  and  Ladas  [1]  posed  the  open  problem  that  whether  (1)  is 
a  necessary  condition  for  the  oscillation  of  all  solutions  of  (3).  In  other  words,  whether 


oc 

/ 


2(0  dt  <oo 


implies  that  (3)  admits  a  nonoscillatory  solution.  Liu  et  al  [7]  (see  also  [11])  gave  an 
example  to  show  that  the  open  problem  is  not  true.  They  have  shown  that  a  stronger 
condition,  viz, 


oo 

/< 


(2(0  dr  <oo 


implies  that  (3)  admits  a  bounded  nonoscillatory  solution. 

By  a  solution  of  eq.  (2)  on  [T9  oo),  T  >  0,  we  mean  a  function  y  €  C([T  -  r,  oo),  R) 
such  that  y(t)  ±  y(t  -  T)  is  continuously  differentiable  and  (2)  is  satisfied  identically  for 
t  >  T,  where  r  =  max{r ,  a }  and  T  is  depending  on  y .  Such  a  solution  of  (2)  is  said  to  be 
scillatory  if  it  has  arbitrarily  large  zeros;  otherwise,  it  is  called  nonoscillatory. 


Asymptotic  behaviour  of  solutions  339 

2.  Sufficient  conditions 

In  this  section  we  obtain  sufficient  conditions  so  that  every  solution  of  (2)  oscillates  or  tends 
to  zero  or  to  ±  oo  as  t  -*  oo . 

Theorem  2.1.  Suppose  that 

G(x)  +  GOO  >  «G(JC  +  y),  x  >  0,  y  >  0 
and 

GOO  +  GOO  <  0GOc  +  y),  *  <  0,  y  <  0,  (Hi) 

vv/zere a  >  0 a«d @  >  Qare  constants.  If 

••oo,  (H2) 

w/iere  Q*(f )  =  min{Q(r),  fi(r  -  r)},  r/ie;z  every  solution  of 

[y(t)  +  y('  -  r)];  +  2(0  G(y(t  -  a))  =  /(r)  (4) 

oscillates  or  tends  to  zero  as  t  -»  oo. 

Proof.  Let  y(/)  be  a  nonoscillatory  solution  of  (4)  on  [Ty,  oo),  Ty  >  0.  Hence  there  exists 
a  r0  >  Ty  such  that  y  (0  >  0  or  <  0  for  r  >  *0.  Let  y  (/)  >  0  for  r  >  *0.  Setting 


and 

/ 

(*)  dJ,  (5) 


o 
for  f  >  fj  >  tQ  +  r,  we  obtain  z(r)  >  0  and 

^(0  =  -Q(t)G(y  (t  -  a))  <  0  (6) 


forf  >  fi.  Hence  u;(r)  >  0  or  <  0  for  t  >  ^  >  fj.  If  ^(0  >  0  for  /  >  ^2,  then  lim 

f—  >oo 

exists.  If  w(t)  <  0  for  t  >  r2,  then  0  <  y(0  <  z(r)  <  F(t)  implies  that  y(f)  is  bounded 

and  hence  u;(0  is  bounded.  Thus  lim  w(t)  exists.  In  either  case  lim  z(t)  exists.  We 

f-Kx>  r-^-oo 

claim  that  lim  z(t)  =  0.  If  not,  then  z(0  >  A.  >  0  for  r  >  ^  >  *2.  From  (4)  we  obtain, 

/~»00 

forr  >  r4  >  *3  +  cr  +  r, 


-f  z'a  -  r)  +  G(*  )0(y  (r  -  ^))  +  C(^  -  r)G(y  (t  -  r  -  or)) 
+  z'(t  -  T)  -f  Q*(r)(G(y  (r  -  cr))  +  G(y  (r  -  T  -  a))) 
-h  zx(r  -  T)  -f  afi*(OG(y  (t  -  a)  +  y(r  -  T  -  a)) 
-f  z;(r  -  r)  +  a 
+  z'(r  -  r)  +  a 


340  N  Parhi  and  R  N  Rath 

Hence, 

z(r)  +  z(t  -  r)  <  z(*4)  +  z(r4  -  r) 


f(s)dS  +  J    /(J-T)dS 

'4  '4 


implies  that  z(r)  <  0  for  large  r,  a  contradiction.  Hence  the  claim  holds.  Consequently, 
lim  y(t)  =  0.  Similarly,  when  y(t)  <  0  for  r  >  r0l  we  obtain  lim  j>(0  =  0.  Thus  the 

f->co  t—  >oo 

theorem  is  proved. 

Remark.  Clearly,  (H2)  implies  (1). 

Remark.  If  G(«)  =  wy  ,  where  y  >  0  is  a  ratio  of  odd  integers,  then  (Hi)  is  satisfied  due 
to  well-known  inequalities 

(\a\+\b\)"<\a\"+\b\p,0<p<l, 
(\a\+\b\)i><2^(\a\»+\b\»),p>l, 

where  a  and  b  are  any  two  real  numbers.  If  G(M)  =  \U\Y  sgn  u,  where  y  >  0,  then  (Hi)  is 
also  satisfied. 

Remark.  Clearly,  (1)  and  Q  (t  +  -  j  <  g(r)  for  t  6  [0,  oo),  where  n  is  any  fixed  positive 

integer,  imply  (H2)  because  2(0  >  Q(t  +  r),t  e  [0,  oo).  Hence  Theorem  2.1  may  be 
regarded  as  an  improvement  and  generalization  of  the  work  in  [10]. 


Theorem  2.2.  Ifd&i)  holds,  then  every  solution  of  (4)  oscillates  or  tends  to  zero  as  t  ->  oo  , 
where  (Hs)  is  stated  as  follows: 

(Hs)     For  every  sequence  (a/>  C  (0,  oo),  a/  ->  oo  ^5-  z  ->  oo;  and  for  every  rj  >  0,  . 
//iflf  r/i^  intervals  (a/  -  y,  or,-  +  v),  i  =  1,  2,  .  .  .  ,  and  nonoverlapping, 


'•=0 


Proof.  If  y(0  is  a  nonoscillatory  solution  of  (4)  on  [Tyt  oo),  Ty  >  0,  then  y(t)  >  0  or  <  0 
forr  >  r0  >  Tv.  Lety(r)  >  0,  r  >  r0.  Settingz(r)  andu;(r)  as  in  (5)  for  r  >  Tj  >  r0  +  r3 
we  obtain  (6).  Hence  w(t)  >  0  or  <  0  for  t  >  T2  >  T{.  Proceeding  as  in  Theorem  2.1, 

we  show  that  lim  w(t)  and  lim  z(r)  exist.  Since  y(t)  <  z(0,  then  limsupy(0  exists. 

r-^oo  r-xx)  r-»oo 

We  claim  that  lim  sup  y  (t)  =  0.  If  not,  then  lim  sup  y(t)  =  a,  0  <  a  <  oo.  Hence  there 

r-s^oo  r-*oo 

exists  a  sequence  (/n)  C  [T,  oo),  T  >  J2,  such  that  rn  ->  oo  as  n  ->  oo  and  y(tn)  ->  a  as 

»  ->  oo.  Thus,  for  large  N\  >  0,  y(tn)  >  /?  >  Oifn  >  ^^  Since  .v(0  i  s  continuous  at  tn, 
then  there  exists  8n  >  0  such  that  y  (0  >  ft  for  r  6  (/„  -  8n,  tn  +  8n)  and  lim  inf  8n  >  0. 

n->oo 

Hence  ^  >  ^  >  0  forn  >  AT2.  Choosing  N  =  max  {A^  ,  AT2},  we  obtain 


Asymptotic  behaviour  of  solutions  34  1 


fQ(t)G(y(t-a))dt 
T 

oo 


j 


"~yv  tn-8n+<r 
~        tn+S-Hf 


2(0  df 
which  implies  that 


T 

by  (Hy.  However,  integrating  (6)  we  obtain 


Q(s)  G(y  (s  -  or))  dr  =  ^(T)  -  w(t). 

T 

Thus 

oo 

f  Q(!)G(y(t-cr))dt  <oo, 
r 

a  contradiction.  Hence  our  claim  holds.  Consequently,  lim  y(t)  =  0.  The  proof  is  similar 

/—  >oo 

for.y(f)  <  0,  t  >  TO.  This  completes  the  proof  of  the  theorem. 


Remark.  Clearly,  (Hs)  implies  (1).  From  the  following  example  it  is  clear  that  (1)  does  not 
imply 


Remark.  Theorem  2.2  holds  if  we  assume  that 

-oo  <  liminf  F(t)  <  iimsup  F(t)  <  oo 

instead  of 


/ 


:  oo, 
o 

where  F(t )  is  given  by  (5).  In  the  following  we  give  an  example  to  show  that  the  condition 
(1)  is  not  enough  to  arrive  at  the  conclusion  of  Theorem  2. 

Example.  Consider 


-  I)]'  +  Q(t)y(t  -  1)  =  h'(f)  +  h'(t  -!),/>  1, 


342  N  Parhi  and  R  N  Rath 

where 

Q(r)  =  (e2  +  <?)  [e'+lh(t  -  1)  +  e2]"    >  0,  t  >  1, 

and  /z  €  Cl  ([0,  oo),  [0,  oo))  defined  by 
f  0,  f  6  [0,  1] 


and  extended  to  oo  by  the  periodicity  A(r)  =  h(t  +  2),  t  >  0.  Clearly,  y  (0  =  A(0  + 

is  a  positive  solution  of  the  equation  with  lim  sup  v(0  =  limsupA(0  =  — .  Further, 

f-»oo  *-»oo  1° 

/°°  oo        /.' 

Q(Odf>]T   J    2(0  dr  = 


Thus  (1)  holds  but  the  equation  admits  a  nonoscillatory  solution  which  does  not  tend  to 
zero  as  t  -*  oo.  This  suggests  that  stronger  conditions  are  needed  to  show  that  every 
nonoscillatory  solution  of  (4)  tends  to  zero  as  t  ->•  oo  . 

Example.  Consider 

[)>(0  +  y(t  -  TT)]'  +  (r  -  7r)"1/2  >'0  -  *)  =  /(O,  r  >  2n 

where 

cosf        2sin^        2sinr         cost  sinf 


ir(ri^""ir"-(r^^"a 

Since 


lS  (0  =  min 
then 

oo 

f  Q*(t)At  =  oo. 

27T 

From  Theorem  2.1  it  follows  that  every  solution  of  the  equation  oscillates  or  tends  to  zero 
as  t  .+  oo.  in  particular,  y(t)  =  sin  t/t2  is  such  a  solution  of  the  equation.  We  may  note 
that  Theorem  2.2  fails  to  hold  for  this  equation  because 


oo  oo 


/    Q(t)  dt  =  2^[(ai  +  r)  -*)'*-  (cr;  -r,-x) 

'  +  ^  -  ^'^  +  (cr/  -  1  -  Jr)'/2] 


i=0 


i=0 

<  oo 


Asymptotic  behaviour  of  solutions  343 

for  a  sequence  (cr/)  ==  (z4)  C  [27T,  oo). 

Example.  Consider 

[y(t)  +  y(t  -  l)];  +  y(t  -  l)  exp(y(r  -  1))  =  /(r),  r  >  1, 
where 


Since  Q(t)  ==  1,  then  (H3)  holds  trivially.  Thus  every  nonoscillatory  solution  of  the 
equation  tends  to  zero  as  t  -+  oo  by  Theorem  2.2.  In  particular,  y(t)  =  e~l  is  such  a 
solution.  However,  Theorem  2.1  cannot  be  applied  to  this  equation  because 

G(u  +  v)  =  (u  +  v)eu+v  >  ueli  +  vev  =  G(u)  +  G(v) 
for  w  >  0  and  v  >  0  and  hence  (Hi)  fails  to  hold. 

Theorem  2.3.  Every  unbounded  solution  of  (4)  oscillates.  In  other  words,  every  nonoscil- 
latory solution  of  (4)  is  bounded. 

Proof.  Let  y(t)  be  an  unbounded  nonoscillatory  solution  of  (4).  Let  y(t)  >  0  for  t  > 
t$  >  0.  The  case  y(t)  <  0  for  t  >  t$  >  0  may  be  dealt  with  similarly.  Setting  z(t)  and 
w(t)  as  in  (5)  we  obtain  (6).  If  w(t)  >  0  for  large  f  ,  then  z(t)  is  bounded  and  hence  y(t) 
is  bounded,  a  contradiction.  If  w(t)  <  0  for  large  t  and  is  bounded,  then  z(f)  is  bounded 
and  hence  y(t)  is  bounded,  a  contradiction.  Thus  w(t)  <  0  for  large  t  is  unbounded. 

Consequently,  lim  w(t)  =  —  oo  which  implies  that  z(t)  <  0  for  large  t,  a  contradiction. 

t—  »oo 

Hence  the  theorem  is  proved. 

Theorem  2.4.  If  (I)  holds,  then  every  solution  of 

[XO  -  y(t  -  t)]'  -f  Q(t)G(y(t  -  a))  =  0  (7) 

oscillates. 

Proof.  If  possible,  let  y(t)  be  a  nonoscillatory  solution  of  (7)  on  [Ty,  oo).  Without  any  loss 
of  generality,  we  may  assume  thaty  (0  >  Oforf  >t$  >  Ty.  Setting  z(t)  =  y(t)  —  y(t  —  r) 
for  t  >  t\  >  to  +  r,  we  obtain 


Hence  z(t)  >  0  or  <  0  for  t  >  ti  >  t\  .  lfz(t)  >  0,  t  >  *2»  then 
Q(t)G(y(t-a))dt  <  z(t2)  <  oo. 


>2 

On  the  other  hand,  z(f)  >  0  for  r  >  ft  implies  that  y(0  >  y(t  -  T)  and  hence  lim  inf 

/— >oo 

y(t)  >  0.  Thus  y(t)  >  a  >  0  for  t  >  {3  >  fc.  Then 


344  N  Parhi  and  R  N  Rath 

implies  that 

CO 

fi(r)GCy(r-or))d/  =  oo, 

f3+cr 

a  contradiction.  Therefore,  z(t)  <  0  for  t  >  ^2,  that  is,  y(t)  <  y(t  -  T),  r  >  t2.  Then  >'(f) 
is  bounded  and  hence  lim  inf  y(f)  and  lim  z(t)  exist.  From  Lemma  1.5.1  of  [2]  it  follows 

t-*00  f-»00 

that  lim  z(/)  =  0,  a  contradiction  because  z(0  <  0  and  monotonic  decreasing.  Hence  the 

f->00 

theorem  is  proved. 

Remark.  Theorem  2.4  generalizes  Theorem  6.4.1  due  to  Gyori  and  Ladas  [2]. 

Remark.  In  [7],  an  example  is  given  to  show  that  the  condition  (1)  is  not  necessary  for 
oscillation  of  all  solutions  of  (7).  They  have  proved  that  every  bounded  solution  of  (7) 
oscillates  if  and  only  if 


00 

/ 


(H4) 
o 
We  may  note  that  (1)  is  stronger  than  (H4). 

Theorem  2.5.  //(Hs)  holds,  then  every  solution  of 

[XO  -  y(t  -  r)]'  +  Q(t)G(y(t  -  cr))  =  /(r)  (8) 

oscillates  or  tends  to  zero  ast  -»  oo. 

Proof.  Let  v(r)  be  a  solution  of  (8)  on  [Ty,  oo),  Ty  >  0.  If  y(t)  oscillates,  then  there  is 
nothing  to  prove.  Let  y(t)  be  nonoscillatory.  Hence  y(t)  >  0  or  <  0  for  t  >  TQ  >  Ty.  Let 
y(t)  >  Oforf  >  TO-  Setting 


and 

r 
u;(0  =  z(/)  -  F(0,  F(r)  =  f  /(j)  dj, 

o 
for  t  >  TI  >  TO  -f  r,  we  obtain 


If  ty(r)  >  0  for  t  >  TI  >  T\  ,  then  lim  w(t)  exists.  If  w(t)  <  0  for  r  >  72  is  unbounded, 

f-»OO 

then  lim  u;(0  =  -oo  and  hence  z(r)  <  0  for  large  r,  that  is,  y(t)  <  y(r  -  r)  for  large  r. 
Thus  >•(*)  is  bounded,  which  implies  that  u;(f)  is  bounded,  a  contradiction.  Hence  w(t)  <  0 
fort  >  T2  ^  bounded.  Then  lim  w(t)  exists.  We  claim  that  lim  sup  y(t)  =  0.  If  not, 

/->oo  fm>00 

then  limsupy(0  =  or,  0  <  or  <  oo.  There  exists  a  sequence  (tn)  C  [Ii,  oo)  such  that 


Asymptotic  behaviour  of  solutions  345 


tn  ->  oo  and  }>(f/j)  ->  ct  as  n  -*  ex:.  Hence  y(/«)  >  /3  >  Qforn  >  N\  >  0.  Since 

is  continuous  at  fn,  there  exists  <$„  >  0  such  that  y(t)  >  p  for  t  e  (?„  —  £„,  fn  -f  <$„)  and 

lim  inf  5,2  >  0.  Then  8n  >  8  >  0  for  n  >  Ni  >  0.  Choosing  Af  =  max  {N{  ,  AT-?}  and  then 

n—  KX) 

proceeding  as  in  the  proof  of  Theorem  2.2,  we  arrive  at  a  contradiction  due  to  (Hs).  Hence 

our  claim  holds.  Thus  lim  y(t)  =  0.  Similarly,  we  may  show  that  lim  y(t)  =  0  when 

r—  >co  r—  »-oo 

<  0  for  r  >  TQ.  This  completes  the  proof  of  the  theorem. 


Theorem  2.6.  Suppose  that  (Hi)  and  (H2)  A0/J.  //XO  «•  a  solution  of 

b(0  +  y(*  -  t)]'  -  fi(OG(?(f  -  or))  =  /(o,  (9) 

f/zen  >?(0  oscillates  or  tends  to  zero  as  t  ->  oo  or  lim  sup  |y  (t)  \  =  -{-oo. 

f~>00 

Proof.  If  possible,  let  y(t)  be  nonoscillatory.  Hence  there  exists  ?o  >  0  such  that  j(r)  >  0 
or  <  0  for  t  >  fo-  Let  y(t)  >  0  for  r  >  ro.  Setting  z(0  and  w(t)  as  in  (5),  we  obtain 
z(t)  >  0  and 


for  ^  >  fi  >  fo  +  ^-  lfw(t)  <  0  for  r  >  ti  >  t\  ,  then  lim  w(t)  exists  and  hence  lim  z(t) 

r->oo  r-»cxD 

exists.  If  w(t)  >  0  for  t  >  ^2  is  bounded,  then  lim  w(t)  and  lim  z(t)  exist.  We  claim 

r-*oo  r->oo 

that  lim  z(0  =  0.  If  not,  then  z(r)  >  A  >  0  for  t  >  t<$  >  t2.  Using  (9)  and  (HO  we  may 

r-»oo 

write,  for  t  >  U  >  t$  +  r, 

/(O  +  /a  -  T)  <  zf(t)  +  zf(t  -  r)  -  G*(0(G(X*  -  ^))  +  G(y(r  -  r  -  a))) 
<  z'(t)  +  zr(t  -  r)  -<*Q*(t)G(z(t  -  a)) 


This  implies,  due  to  (H2),  that  lim  z(t)  =  oo,  a  contradiction.  Hence  our  claim  holds. 

?-»00 

Sincez(0  >  y(0,then  lim  v(r)  =  0.  Ifuj(0  >  0,  r  >  ^2*  is  unbounded,  then  lim  w(t)  — 


+00.  Hence  v(f)  is  unbounded.  Similarly,  if  v(f  )  <  0  for  r  >  fo>  then  lim  v(f)  =  0  or 

r-»oo 

>'(0  is  unbounded.  Thus  lim  y(t)  =  0  or  lim  sup  !>'(/)  I  =  -hoc.  This  completes  the  proof 

f-»oo  ^oo 

of  the  theorem. 
COROLLARY  2.7 


7f(Hi)  anc(  (H2)  /io/J,  ?/z<?n  every  bounded  solution  of  (9)  oscillates  or  tends  to  zero  as 
t  ->  oo. 
This  follow  from  Theorem  2.6. 

Theorem  2.8.  Let  (Hs)  /zo/d.  7jf>'(0  is  a  solution  of  (9),  then  it  oscillates  or  tends  to  zero 

ast  -*  oo  or  lim  sup  \y(t)  \  =  +00. 
/->oo 

The  proof  is  similar  to  that  of  Theorem  2.2. 

COROLLARY  2.9 

holds,  then  every  bounded  solution  of  (9)  oscillates  or  tends  to  zero  as  t  -*  oo. 


346  N  Parhi  and  R  N  Rath 

Theorem  2.10.  Suppose  that  (H?,)  holds.  Ify(t)  is  a  solution  of 

[>'(0  -  y(t  -  T)]'  -  G(0  G(y(t  -  a))  =  /(/),  (10) 

then  y(t)  oscillates  or  tends  to  zero  or  \y(t)  -»  +00  as  t  -»  oo. 
The  proof  is  similar  to  that  of  Theorem  2.5. 

COROLLARY2.il 

holds,  then  every  bounded  solution  of  (10)  oscillates  or  tends  to  zero  as  t  ->  oo. 


Remark.  Some  of  our  results  partially  answer  the  open  problems  stated  in  6.12.9  and 
6.12.10  [2]. 

3.  Existence  of  nonoscillatory  solutions 

In  this  section  we  obtain  necessary  and  sufficient  conditions  for  the  existence  of  a  bounded  j 

positive/negative  solution  of  the  eq.  (8).  [ 

Theorem  3.1.  Let  f(t]  >  0  with  \ 


/(/)dr<oo.  (11) 

Then  eq.  (8)  admits  a  bounded  negative  solution  if  and  only  if 

C(/)dr<oo.  (H5) 


Proof.  Suppose  that  eq.  (8)  admits  a  bounded  negative  solution  y(t)  on  [7V,  oo),  Ty  >  0. 
Setting  z(t)  =  y(t)-y(t-r)  and  w(t)  =  z(t)  +  F(r),  where 


for  r  >  ?o  >  Ty  +  r,  we  obtain  w(t)  >  z(0,  iu(0  is  bounded,  F(t)  ->  0  as  f  ->  oo  and 
w;(0  =  -fi(OG(X^-cr))>0.  (12) 

Hence  iy(/)  >  0  or  <  0  for  t  >  t\  >  tq.  If  w(t)  >  0  for  /  >  t\9  then  lim  u;(0  exists  and 

/-*oo 

hence  lim  z(t)  exists.  Since liininf y(0  (or  limsupyO)  I  exists,  then  lim  z(0  =  Oby 

f->oo  r-»oo  y        t-+<x>  )  f->0° 

Lemma  1.5.1  in  [2].  Thus  lim  w(t)  =  0,  a  contradiction  to  the  fact  that  iy(f)  >  0  and 

t— >00 

nondecreasing.  Hence  iy(/)  <  Oforf  >  fi.  Consequently,  lim  iy(0  exists  and  z(t)  <  0 

>  ,  ^-^oo 

for  t  >t\.  There  exists  a  >  0  such  that  >?(0  <  -a  for  r  >  t\.  Integrating  (12)  from  5-  to 
t  (s  >  t  >  ti  >  t\  +  a)  and  then  taking  limit  as  s  ->  co  we  obtain 


oo 

<G(-d)  I  fi(«)du, 


Asymptotic  behaviour  of  solutions 


347 


that  is, 


y(t-r)>  y(t) 


CO 

-G(-a)j 


Q(s)ds. 


Putting  the  values  of  t  successively  one  may  obtain 


t  "7 

y(t-r)>  y(t  +  nr)  +  ^2    I    /($)  dy  -  G(-a)  £    /     Q(s)ds. 

A=°,«r     .  *=0, 

Since  y(t)  is  bounded,  then  using  (1  1)  we  get 


7 

/ 


Q(s)  As  <  oo. 


From  this  (H5)  follows. 

Next  we  assume  that  (Hs)  holds.  It  is  possible  to  choose  m  >  0  sufficiently  large  such 
that 


k=m 


Q(t)dt<- 


1 


and 


k=m 


kr 


°? 


/(Odr<-, 


that  is, 


-1 


and 


where  T  =  mr.  Define 


L(0  = 


0,     0  <  t  <  T 

00  00 

-  /  f&ds, 


Hence  L(0  <  0  for  r  >  7.  Further,  define 
0,     0  <  r  <  T 


Thus  w(0  <  0  and  w(r)  -  u(t  -  r)  =  L(t),  t  >  T.  For  /  >  T,  there  exists  an  integer 
k  >  0  such  that  T  +  kr  <  t  <  7  +  (*  4-  l)r.  Hence  r  <  t  -  kr  <  T  -j-  r  and 
r  -  T  <  /  -  (Jk  +  l)r  <  T.  Then 


348  N  Parhi  and  R  N  Rath 

M(0  =  1(0  +  L(t  -  T)  +  •  •  -  +  L(t  -  kr) 

00  OO  OO  OO 


Q(s)As-j  /(j)(b  +  ...  +  G(-I)     (    Q(s)ds-    f    f(s)ds 

i  t-kr  t-kr 

00  00  00 

-    /*    f(s)ds  +  ---  +  G(-l)  f  Q(s)ds.-  f 

> 

/ 


T+kr  T+kr  T  T 

oo        <*>  *> 


Let  X  =  BC([T,  oo),  R),  the  space  of  all  real- valued,  bounded  continuous  functions 
on  [T,  oo).  It  is  a  Banach  space  with  respect  to  supremum  norm.  Let  K  —  {x  €  X  : 
x(t)  >  0,  t  >  T}.  For  M,  v  e  X,  we  define  u  <  v  if  and  only  if  v  —  u  e  K.  Thus  X  is  a 
partially  ordered  Banach  space  (see  pp,  30,  [2]).  Define 

M  =  {x  €  X  :  u(t)  <  x(t)  <  0}. 

Clearly,  u  e  M  and  u  =  inf  M.  If  <p  c  A  C  M,  then  A  =  {*  6  M  :  u(t)  <  v(f)  < 
^(0  <  w(0  <  0}.  Setting  wo(t)  =  sup{u;(0  :  jc(0  <  w(/)  <  0,  ,Y  e  A},  we  notice  that 
w0  =  sup  A  and  tu0  e  M.  Define  S  :  M  ~>  X  by 

00  00 

c(f  -  r)  -f  /    2(^)^(^(5  -  cr))d,s  -  /   /(5-)d5-,  r  >  T{ 
J  J 

'  *  (13) 


where  TI  =  T+r  andr  =  max{r,  a}.  Clearly,  Sx  is  continuous  on  [T,  co)  and  Sjc(f)  <  0 
forf  >  T.  For?  >T\, 


CO  CO 


5  x(t)  >  jc(r  -  r)  +  G(-l)  f  Q(s)ds  -  f  f(s)ds 


>  u(t  -  T) 
ForT  <f  <7i, 


"  T! 

Thus  5  :  M  -»  Af .  Moreover,  ;q  >  jca  implies  that  S  ;ci  >  S  X2-  From  the  Knaster-Tarski 
fixed-point  theorem  (see  pp.  30,  [2])  it  follows  that  S  has  a  fixed  point  y  €  M  which  is  a 
solution  of  (8)  on  [T\ ,  oo).  Since  y(T\  -  T)  <  0,  then  from  (13)  it  follows  that 


-r)-f 


<  y(Ti  -r)-       f(s)ds  <  0. 


Asymptotic  behaviour  of  solutions  349 


Thus  y(t)  <  0  for  t  e  [T,  Tj].  For  f  €  [7i,  T\  +  T],  XO  <  0.  Consequently,  y(0  <  0 
for  t  >  T[  .  This  completes  the  proof  of  the  theorem. 


Theorem  3.2.  Let  f(t)  <  0 

f(t)dt  >  ~oo. 


Then  eq.  (8)  admits  a  bounded  positive  solution  if  and  only  ij 
The  proof  is  similar  to  that  of  Theorem  3.1. 

Remark.  Theorems  3.1  and  3.2  hold  if  f(t)  ==  0.  Hence,  we  have  the  following  corollary. 

COROLLARY  3.3 

Every  bounded  solution  of(l)  oscillates  if  and  only  if 

'  oo    •  °? 

T \        / 

=  00. 


This  follows  from  Theorems  3.1  and  3.2. 
Remark.  We  may  note  that 

00  00 

I     «  .    .   I 

<  OO 


and 


/   |/(r)|  df  <  oo  implies  that   /   |/(r)|  dt 


00         «  °? 

/    )2(0  df  <  oo  implies  that   /   g(r)  df  <  oo. 

=    ikr  0 

References 

[1  ]  Chuanxi  Q  and  Ladas  G,  Oscillation  of  neutral  differential  equations  with  variable  coefficients, 

Appl.  Anal.  32  (1989)  215-228 
[2]  Gyori  I  and  Ladas  G,  Oscillation  Theory  of  Delay  Differential  Equations  with  Applications 

(Oxford:  Clarendon  Press)  (1  99  1  ) 
[3]  Ivanov  A  F  and  Kusano  T,  Oscillations  of  solutions  of  a  class  of  first  order  functional  differ- 

ential equations  of  neutral  type,  Ukrain.  Mat.  Z.  51  (1989)  1370-1375 

\  [4]  Jaros  J  and  Kusano  T,  Oscillation  properties  of  first  order  nonlinear  differential  equations  of 

f  neutral  type,  Diff.  Integral  Eq.  5  (1991)  425^36 

[5]  Kitamura  Y  and  Kusano  T,  Oscillation  and  asymptotic  behaviour  of  solutions  of  first  order 

functional  differential  equations  of  neutral  type,  Funkcial  Ekvac  33  (1  990)  325-343 
[6]  Ladas  G  and  Sficas  Y  G,  Oscillation  of  neutral  delay  differential  equations,  Can.  Math.  Bull. 

29(1986)438-445 
[7]  Liu  X  Z,  Yu  J  S  and  Zhang  B  G,  Oscillation  and  nonoscillation  for  a  class  of  neutral  differential 

equations,  Diff.  Eq.  Dynamical  Systems  1  (1993)  197-204 


350  N  Parhi  and  R  N  Rath 

[8]  Parhi  N  and  Rath  R  N,  On  oscillation  criteria  for  a  forced  neutral  differential  equation,  Bull. 

Inst.  Math.  Acad.  Sinica  28  (2000),  59-70 
[9]  Parhi  N  and  Rath  R  N,  Oscillation  criteria  for  forced  first  order  neutral  differential  equations 

with  variable  coefficients,  J.  Math.  Anal.  Appl.  256  (2001),  525-541 
[10]  Piao  D,  On  an  open  problem  by  Ladas,  Ann.  Diff.  Eq.  13  (1997)  16-1 8 
[11]  Yu  J  S,  The  existence  of  positive  solutions  of  neutral  delay  differential  equations.  The  Pro- 

ceeding  of  Conference  of  Ordinary  Differential  Equations  (Beijing:  Science  Press)  (1991) 

263-269 
[12]  Yu  J  S,  Wang  Z  C  and  Chuanxi  Q,  Oscillation  of  neutral  delay  differential  equations,  Bull 

Austral  Math.  Soc.  45  (1992)  195-200 
[13]  Yu  J  S  and  Wang  Z  C,  Asymptotic  behaviour  and  oscillation  in  neutral  delay  difference 

equations,  Funkcial.  Ekvac.  37  (1994),  241-248 


Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  Ill,  No.  3,  August  2001,  pp.  351-363. 
©  Printed  in  India 


Monotone  iterative  technique  for  impulsive  delay  differential 
equations 

BAOQIANG  YAN  and  XILIN  FU 

Department  of  Mathematics,  Shandong  Normal  University,  Ji-Nan,  Shandong 
250  014,  People's  Republic  of  China 

MS  received  29  May  2000 

Abstract.  In  this  paper,  by  proving  a  new  comparison  result,  we  present  a  result  on 
the  existence  of  extremal  solutions  for  nonlinear  impulsive  delay  differential  equa- 
tions. 

Keywords.  Contraction  mapping  theorem;  extremal  solutions;  impulsive  delay 
differential  equations. 

1.  Introduction 

In  this  paper,  we  discuss  the  impulsive  retarded  functional  differential  equation  (IRFDE) 

/(',*),  te[0,T],tt*k', 

/=^  =  /*(*(**)),     *=l,2,...,m;  (1.1) 

O, 

where  $  e  PC([~r,  0],  R)  =  {x,  x  is  a  mapping  from  [T,  0]  into  R,  x(t~)  =  x(t)  for  all 
t  e  (-T,  0],  x(t+)  exists  for  all  t  e  [— T,  0),  and  x(t+)  =  x(t)  for  all  but  at  most  a  finite 
number  of  points  t  e  [— T,  0)}  and  M([— T,  0],  R)  =  {x,  x  is  a  bounded  and  measurable 
function  from  [— r,0]  into  R]  with  norm  ||;c||  =  sup,€(_r0]  |;c(r)|,  T  >  0,  *,(#)  =  x(t  4- 
0),0  €  [-t,0],0  =  r0  <  t\  <  t2  <  •••  <  tm  <  T,  J  =  [0,  T],  ]'  =  J  -  {ti}^.  It  is 
easy  to  see  that  FC0([-T,  0],  R)  c  M([-r,  0],  /?)  and  M([-r,  0],  J?)  is  a  Banach  space. 
Now  we  suppose  that  /  e  C(J  x  M([-T,  0],  R),  R),  lk  €  C(/?,  tf)(fc  =  1,  2,  . . . ,  m) 
throughout  this  paper. 

In  [1]  and  [2],  some  existence  and  uniqueness  results  were  obtained  for  eq.  (1.1)  by  the 
Tonelli's  method  or  fixed  point  theorems.  And  it  is  well-known  that  the  method  of  upper  and 
lower  solutions  and  its  associated  monotone  iteration  is  powerful  technique  for  establishing 
existence-comparison  for  differential  equations  (see  [4, 5,  6]).  But  to  impulsive  differential 
equations  with  delay  as  eq.  (1.1),  this  method  has  not  been  used  yet  as  far  as  we  know.  In 
this  paper,  we  discuss  eq.  (1.1)  by  the  method  and  we  can  find  that  the  delay  and  impulses 
make  the  discussions  more  difficult. 

2.  Main  results 

Assume  M ([-r,  T],  R)  =  {x,  x  is  a  bounded  and  measurable  function  from  [— r,  T]  into 
R}  with  norm  \\x  \\  =  supfe[_r?:r]  |jc(OI,  PCo([-t,  T],  R)  =  {x,x  is  a  mapping  from  [r,  0] 
into  R,  *(*-)  =  jc(r)  for  all  \  e  (-r,  0],  *(/+)  exists  for  all  t  e  [-r,  0),  x(t+)  =  x(t) 


352  Baoqiang  Van  and  Xilin  Fu 

for  all  but  at  most  a  finite  number  of  points  t  e  [—  r,  0),  and  x(t)  is  continuous  at  t  € 
[0,  T]  -  {r/lJLj  left  continuous  at  t  =  tk,  and  x(t£)  exists  (k  =  1,  2,  .  .  .  ,  m)}. 

DEFINITION  2.1 

A  function  x  e  PCo([—  T,  I],  /?)  is  said  to  be  a  solution  of  (1.1)  if  x  satisfies  the  first 
expression  of  eq.  (1.1)  for  all  t  e  J  except  on  a  set  of  Lebesgue  measure  zero  (the 
exceptional  points  will  generally  include  but  may  not  be  limited  to  impulse  times  /*)  and 
satisfies  the  second  one  of  eq.  (1.1)  for  all  t  e  {**}/[Lp  and  x  is  piecewise  absolutely 
continuous  on  [0,  T]  with  *o  =  <£. 

DEFINITION  2.2 

A  function  G  :  M([—  r,  0],  R)  ->  R  is  said  to  be  weakly  continuous  at  0o  G  M([—r,  0],  R) 
if  for  any  {<£„}  c  M([-r,  0],  #)  with  lim  0n(y)  =  0o(*),  a.e.  5  6  [-r,  0],  then 

tt->+OO 

lim 

n-»+o 

And  G  is  said  to  be  weakly  continuous  on  M([—  r,  0],  /?)  if  G  is  weakly  continuous  at 
0forany0eM([-r,0],jR). 

Remark  2.1.  This  condition  is  more  direct  than  that  in  [1]  and  is  different  from  that  in  [2], 
which  need  that  f(tt  VO  is  continuous  at  each  (t,  ^o)  €  (0,  T]  x  L1  ([-r,  0],  Rn). 


Lemma  2.1.  A55wme  ?/iaf  a  function  g  :  J  x  M([—  r,  0],  R)  -+  R  is  continuous  at  evety 
t  €  J  for  each  fixed  (j)  e  M([—  T,  0],  #)  and  is  weakly  continuous  at  every  $  €  M([—  r,  0], 
#)/0r  each  fixed  t  e  J.  Then  for  every  x  e  PC([-r,  T],  #),  ^(/,  ;cf)  w  measurable  on 
[0,  T]. 

Proo/  Choose  a  continuous  function  sequence  {xn}  such  that 
lim   xn(t)=x(t),   for  all  t  e  [-r,  J]. 

n->-+oo 

By  Lemma  4  in  [3],  xnt  is  continuous  at  /  6  [0,  T].  So  g(r,  xnt  )  is  measurable  on  [0,  T]. 
Since  lim  ^^(s1)  =  xt(s),  for  all  5  e  [-r,  0],  then 

71^+00 

rtJj^oo  ^(f'  ^^  =  ^(f'  ^  )j    f°r  a11  r  €  f°'  r]' 
So  g(r,  ^:r)  is  measurable  on  [0,  T].  D 

Set  J?  e  M([-r,  0],  ^)*.  Moreover,  suppose  that  there  exists  aye  LI([—T,  0],  /?) 
with  x(r)  >  0  for  all  most  t  e  [-r,  0]  such  that 


/•° 

for  all  VT  e  A/([-T,  0],  R)  and  ||fi||  =  /     y(r)dr. 

J— T 

Now  we  list  a  main  lemma. 


Impulsive  delay  differential  equations  353 

Lemma  2.2  (Comparison  result).  Assume  that  p  e  PC([—r,  T],  R)  Pi  Cl(J',  R)  satisfies 

p{  <  -Mp(t)  -  Bpt,     t  e  J,  t  ^  tk 

(2.1) 


vv/zere  constants  Af  >  0,  0  <  L*  <  1  (fc  =  1,  2,  .  .  .  ,  m)  tf/zd  M0  =   /     e~M'  y  (f)d/. 
suppose  fu  rt  her  that 

(a)  either  p(fy  <  p0fa)  <  0,  ^  €  [-T,  0]  and 


(2'2) 


=  max{/i,  ^2  —  fi,  .  .  .  ,  T  -  /OT};  or 


(b)  ;;(0)  >  -X,  po  6  PC0([-~r,  0],  JR)nC1(//,  R)  where  I'  =  [-r,  0]-^}^,  {r/JfJ., 
w  the  set  of  the  discontinuous  points  of  PQ,  p'(t)  < 


(2.3) 


inf    p(s)  =  —  A,  <  0 

,ve[-T,0] 


n?1   r(i-W 
MoA2  -  i  +  r"     n"  (i-^)'  (2'4) 

1  ^  2^j=-r  nk=j^1       ^k) 

where  AI  =  max{f_r  +  t,  f_r+]  —  fr,  .  .  .  ,  -f_i,  fi,  ti  -  t\9  .  .  .  ,  T  -  r,n}.  Then  p(t)  <  0 
fora.e.  t  e  J. 

Proof.  Now  let  u(r)  =  eMrw(r),  r  €  [-r,  0].  By  the  definition  of  5,  the  eq.  (2.1)  can  be 
listed  as 


<  -  / 

Jt- 


teJ.t^  tkl 

t-r  \*"J) 

==tk  <  -Lkv(tk),  (k=  l,2,...,m). 

Now  we  will  prove  v(t)  <0,t  e  [—  r,  T]. 

In  fact,  if  there  exists  a  0  <  f  *  with  u(r*)  >  0,  we  might  well  suppose  t*  ^  t[  ,  h  ,  .  .  .  ,tm 
(otherwise,  we  can  choose  a  t  nearing  t*  enough  with  v(t)  >  0),  let 

inf     v(t)  =  -&.  (2.6) 

-•c<t<t* 

First  we  consider  the  case  (a). 

(A)  In  case  ofb  =  0:  v(t)  >  0,  t  e  [0,  r*].  Then  t/(0  <  0,  t  e  [0,  f*].  So  u;(r*)  <  0. 
This  is  a  contradiction. 

(B)  In  case  ofb>0:  Assume  t*  €  (f/,  f/+i].  It  is  clear  that  there  exists  a  0  <  f*  <  t* 
with  v(t*)  =  -&,  where  r*  in  some  Jj(j  <  i)  or  n(rt)  =  —b.  We  may  assume  that 
ufe)  =  —  &  (in  case  of  i>(rt)  =  —  b,  the  proof  is  similar).  By  mean  value  theorem,  we 
have 


354 


Baoqiang  Yan  and  Xilin  Fu 


On  the  other  hand,  for  t  €  [0,  f  *] 


Now  from  (2.1),  we  get 


and 


,  (*  =  1,  2  .  .  .  ,  m), 


which  implies 


Moreover, 


0+25 


{ 

V 


(2.7) 


(2.8) 


which  contradicts  (2.2). 
By  virtue  of  (A)  and  (B),  v(f  )  <  0,  f  6  J. 
Next  we  consider  the  case  (b). 

(A')  If  —  b  =    inf    u(r),  we  can  obtain  a  contraction  similarly  as  (a). 


(BO  If  -fc  <     inf    u(r),  then  b  =  X  and  there  exists  a  f*  €  (f-/-i,  t-/]  with  u(r*)  =  ~ 
re[0,r*] 

(or  v(t*j_  j)  =  —  £,  the  proof  is  similar).  So 


Impulsive  delay  differential  equations 
u(f*)-u(^)  =  v/(?/)a*-ft), 


355 


r'  <  ?,  < 


(2-9) 


By  (2.9)  and  (2.3),  one  has 


which  implies 

0  <  u( 


~bn'k=_j(l  -  Lk) 


Similarly  we  get 


which  contradicts  (2.4). 
By  virtue  of  (A')  and  (Bx),  i/(f)  <  0,  a.e.  t  €  7.  And  the  proof  is  complete. 


D 


Lemma  2.3.  Let  a,rj  e  M([-r,  T],  R).  Then  x  e  PCo([-r,  T],  #)  is  a  solution  of  the 

equation 

x1  -f  MX  +  #;tr  =  cr(/),  t  €  /,  ?  7^  tk, 

&x\t=:tk  =  4(^)  —  ^JtUfe)  —  »7(^)]f     (fc  =  1,  2,  . . . ,  m),  (2.10) 

if  and  only  ifx  e  PCo([-r,  T],  jR)  is  a  solution  of  the  following  integral  equation 

x(t)  =  O(0)e""Mr  +  /    e~M(r~~'y)[aCy)  -  Bxs]ds 
Jo 


-  Lk[x(tk)  - 


,  t  €  7, 


(2.11) 


whew  xt(s)  =  jc(f  4-  s)  =  4>(r  +  51)  ift  +  s  <  0. 


356  Baoqiang  Yan  andXilin  Fu 

Proof  Assume  that  x  €  PC0([-T,  T]9  R)  is  a  solution  of  IRFDE  (2.10).  Let  z(t)  = 
x(t)e-Mt.  Then  z  €  PC([-T,  T],  #)  and 

z;(0  =  [*(0  -  toOle-"1  ,  r  6  [0,  7],  t  ^  tk  (k  =  1,  2,  .  .  .  ,  m). 


Since  (a(r)  —  Bxt)e~Mt  is  measurable  on  [0,  T],  it  is  easy  to  establish  the  following 
formula: 


I  z' 

-70 


-  z(0)  +       z'(s)ds  +  [z(£)  -  z(fc)L  t  €  [0,  T]. 


And  from  the  second  expression  of  (2.  1  0),  we  have 

z(£)  -  2(4)  =  Wtfoftk))  -  Lk[x(tk)  - 
Consequently, 


=  O(0)  +  /   [a(0  -  Bxs]ds 

i  -  Lk[x(tk)  -  r](tk)]}zMtk ,  r  €  [0,  T]9 


i.e.,  jc(0  satisfies  (2.  11). 

Conversely,  if  jc  e  PC([-r,  T])  is  a  solution  of  eq.  (2.11),  by  direct  differentiation,  it 
is  easy  to  see  the  first  expression  of  (2.10)  is  true  for  all  t  e  [0,  T]  -  {^IJL-i  except  on  a 
set  of  Lebesgue  measure  zero  and  the  second  one  and  the  third  one  of  (2.  1  0)  are  true.  The 
proof  is  complete.  D 

Lemma  2.4.    Equation  (2.  1  1)  has  a  unique  solution  in  PCo([—  T,  T],  R)  with  XQ  =  <J>. 
Proof  For*  e  C([0,  *i],  R),  let  \\x\\  =  max{e-A/''|jc(0|,  t  e  [0,  t{]}  and 

s,  t  e  J, 


wherex(f+,y)  =  Q(t+s)ift+s  <  OandAfi  =  \\B\\+l.  Obviously  AI  :  C([0,^i],  R) 
C"([0,  fj],  y?)  is  a  continuous  operator.  For  jc,  y  e  C([0,  t\],  R\ 


>(0-(Ai>0(OI 

=      /  [(Bxs)  -  (By5)]ds 

JQ 

['  [*Mr)_  ,g 

JO     J-T 

r°  rf 

=      /      /    l*Xr)-Mr)|y(r)<fcdr 

J-T  JO 

rO    /»r 


-rJO 


Impulsive  delay  differential  equations 

fO     ft+r 

=      I      I        \x(s)-y(.s)\dsy(r)dr 

J—T  Jr 

/O     /»f+r 
/        |*(5)-y(j)|cfcy(r)dr 
-T  Jo 

<      I     I  \x(s)-y(s)\Asy(r)dr 

J-T  JO 

=      I   |*(5)-y(j)|<b  /    y(r)dr 

JO  J-T 


357 


\\B\\ 


So 


i.e., 


MI 


MI 


(2.12) 


By  contraction  mapping  theorem,  A\  has  a  unique  fixed  point  X[  e  C([0,  t\],  R).   For 
Jc  €  C([/i,  f2],  /?),  let  ||jc||  =  maxfe-^'ijc^)!,  /  €  [rj,  r2]}  and 


(2.13) 


r,  t  e 


where  jc(r  +  .9)  =  O(f  +  j)  if  f  +  5  <  0,  jc(r  +  j)  =  jci(r  +  s)  if  f  +  s  €  (0,  fj]  and 
M2  =  H2?  ||  +  1.  Similarly,  A2  has  a  unique  fixed  point  JC2  in  C([fi,  ?z],  R)-  So  forth  and 
so  on,  for*  6  C([tn,  T],  R),  let  ||jc||  =  maxfe-^"^'  |*(/)|,  t  e  [tn,  T]}  and 


+ 


f 

Jo 


j,  r  €  [ft,,  T], 


(2.14) 


t+s  €  (fn-^fJan 
Similarly  An+\  has  a  unique  fixed  point  xn+\  €  C([tn,  T],  R).  Let 


x*(t)  = 


*(0, 


e  (0, 


xn+i(t),    te(tn,T]. 


Then  A:*  e  PC([-r,  T],  7?)  is  a  solution.  If  y*  e  PC([-r,  T],  R)  is  another  solution  of 
equation,  by  jc*(f)  =  y*(t)fort  e  [-r,  0],  it  is  easy  to  verify  jc*(f)  =  y*(f)forr  €  [0,fi]. 


358  Baoqiang  Yan  and  Xilin  Fu 

And  so  on,  jc*(0  =  y*(t)  for  t  e  (t\,  fi].  Continuing  as  before,  we  get  jc*(0  =  y*(t)  for 
t  6  (tn,  T].  Therefore  jc*  =  y*.  The  proof  is  complete.  D 

Now  we  list  some  independent  conditions  for  convenience. 
(Ai)  There  exist  w,  v  e  PC0([-r,  T],  R)  satisfying  u(t)  <  v(t)  (t  €  J)  and 


Aw|r=r,  <  Ik(u(tk)),     (k  =  1,  2,  .  .  .  ,  77t), 
MO  <  $. 


Moreover,  $  —  wo  and  DO  —  $  satisfy  either  the  assumption  (a)  or  (b)  of  Lemma  2.1. 
(A2)  There  exist  constants  M  >  0  such  that 

/(r,  0)  -  /(r,  VO  >  -Af  (0(0)  -  V(0))  -  B((t>  -  VO, 

whenever  r  €  /,  0,  ^  E  {xt,  u(t)  <  x(t)  <  v(f),  ^  €  J}  with  0  >  ^r. 
There  exist  constants  0  <  L^  <  1  (k  =  1,  2,  .  .  .  ,  m)  such  that 


whenever  wfe)  <  v  <  jc  <  ufo),  (^  =  1,2,...,  m). 

(A4)  f  :  J  x  M([-r,  0],  #)  ->  ^  is  continuous  at  every  f  €  /  for  each  fixed  0  € 
M([-r,  0],  /?)  and  is  weakly  continuous  at  every  0  €  M([—  T,  0],  7?)  for  each  fixed 
te  J. 

Theorem  2.1.  Let  the  conditions  (Ai)-(A4)  be  satisfied  and  f  e  C([0,  T]  x  M([-r,  0], 
/?),  /?)  a^  [M,  v]  c  JPCo([-r,  0],  J?).  77z^  //zere  ^/j/  monotone  sequence  {w,,},  {v/z}  ^ 
PC'oCt—  ^»  ^]>  R)  which  converge  on  J  to  the  minimal  and  maximal  solutions  x*tx*  € 
PCQ([-T,  T],  R)  in  [M,  u]  respectively.  That  is,  ifx  €  PCo([-T,  T],  /?)  w  any  solution 
satisfying  x  e  [M,  v], 


u(t)<u\(t)  <...  <jc*(0  <jc(0  <Jc*(0  <... 

For  any  77  6  [w,  u],  consider  the  linear  eq.  (2.10),  where 
or(0  =  /(r,  ty)  4-  Af  iy(0  +  5??,,  t  €  J. 

By  the  condition  (A4)  and  Lemma  2.1,  one  has  cr  e  M([—  r,  T],  R).   By  Lemma  2.3, 
IRFDE  (2.10)  has  a  unique  solution  x  e  jPCo([-r,  7*],  J?)  with  XQ  =  3>.  Let 

^€7.  (2.15) 


Then  A  is  a  continuous  operator  from  [w,  v]  into  PCo([-r,  T],  JR).  Now  we  show 

(a)  u  <  Au,  Av  <  v; 

(b)  A  is  nondecreasing  in  [M,  v]. 


Impulsive  delay  differential  equations  359 

To  prove  (a),  we  set  w  i  =  Au  and  p  =  u  —  u  [  .  By  Lemma  2.3,  we  have 


But,     t  e  J,t  ^tk, 
AM  il*=*  =  /*(«('*))  -  I*  [*i  ft*)  -  «ft*)L  A  =  1,  2,  .  .  .  ,  m,         (2.16) 


So 

t=tk  =  A«|r=,,  -  AIM  |,=r,  <  -Lkp(tk),     (k  =  1,  2, . . . ,  m)  (2.17) 

.  =  WQ  -  WIG  <  0, 

which  implies  by  virtue  of  Lemma  2.2  that  p(t)  <  0  for  t  e  J,  i.e.   u  <  u\  =  Au. 
Similarly,  we  can  show  v\  =  Av  <v. 

To  prove  (b),  for  171,772  e  [u,  v]  with  rj[  <  772,  let/?  =  x\  -  *2,  where  ^i  =  A?7i,  ^2  = 

2.  From  Lemma  2.2,  we  get 


M(?72ft)  - 


)  -  7?i  ft))  + 
Mp  —  Bpt 
Mp(t)-Bpt,te  Jt 


-  Lk[x\(tk)  - 

»7lftjfc)]}  - 


and 


Hence,  by  Lemma  2.2,  pft)  <  0  for  all  t  e  J,  i.e.,  Arn  <  Ar?2,  and  (b)  is  proved. 
Let  un  =  Awn-i  ,  and  urt  =  Aun_i  (n  =  1,  2,  .  .  .  ,  m).  By  (a)  and  (b),  we  get 

Mft)  <  tt!ft)  <  .  .  .  <  Wnft)  <  -  -  •  <  ^nft)  <...<-•.<  Vlft)  <  Vft),   /  €  J,      (2.18) 


and  wn,  vn  e  PCo([-r,  T],  /?)  with  wno  =  vno  =  <&,  w  =  1,  2, . . ..  So  there  exist  x*  and 
jc*  such  that 

x  u,,ft)  ->  j:*ft),  r  €  [-r,  7],  n  -*  +00,  (2.19) 

vnft)  ->  ;c*(0,  t  e  [-T,  J],  n  ~>  +00.  (2.20) 


Therefore 


«ni  CO  ->  JC*r(j),  teJ.se  [-r,  0],  rc  ->  +00, 
>  *?(•*),  teJ.se  [-T,  0],  n  ->  -foe. 


360  Baoqiang  Yan  and  Xilin  Fu 

So 

->  f(t,  **r )  4-  M**(0,  n  -»  +00. 

By  the  Lebesgue  dominated  convergence  theorem,  we  get 
rt 
JQ  e 

"*  Jo  C 
So 

/•/ 

;t*(0  =  0(0)e-"'  4-  /    z~M(t~x}[f(s,  x*s)  4-  Afjt*(5)]dy,  t  e  [0,  *i],     (2.22) 
Jo 

where  .x*o  =  <!*.  And  by  virtue  of  the  continuity  of  1\ ,  we  get 

/i(«n(*i))  ->  /i(Jc*ai)),  «  ~>  +00.  (2.23) 

Similarly,  one  has 


-  (Buns  -  Bun 

(2.21) 


Tt-s[f(s,  Jc«)  4-  M;c*(5)]d5,  r  €  (rl?  /2],  (2.24) 

where  ^*o  =  <^>.  So  forth  and  so  on, 

*„(/)  =  [x*(rn)  +  /rtfe(fn))]e-M^-r«) 

+  f  eTM(t-*[f(s,  x*,)  +  Af  jc*(5)]<fc,  f  6  (rn,  T],  (2.25) 

Jtn 


where  X*Q  =  <l>.  Then 

fc'~ft) /*(**(**)),  ^  e  /.  (2.26) 

By  the  similar  proof,  we  get 


_  (2.27) 

where  JCQ  =  <I>. 

Finally,  if  x  e  PC([-r,  T],  7?)  is  a  solution  of  eq.  (1.1)  in  [u,  v],  Now  let  p  =  M«  -  jc 
and  use  mathematics  induction.  Obviously  u  <  x.  Suppose  wn_i  <  jc.  Then 


=  f(t,  Kn_lf)  -  Af  (wn(f)  -  un-i(t))  -  (Bttnf  -  B,!^)  -  f(t,xt) 
=  -Up  -  £p,  -  [/(r,  x,)  -  /(/,  un-it)} 

+M(-jc(r)  +  Mn-i(r))  4-  (~£jcr  4-  BII^I,) 
<  -Up  -  Bpf,  t^J,  t^tkt 


Impulsive  delay  differential  equations 


361 


-  Lk[un(tk)  -  K/,- 


and 


Hence,  by  Lemma  2.2,  p(r)  <  0  for  all  r  6  /,  i.e.  wn(0  <  x(f),  r  €  /.  So  «„(*)  <  jc(f  ), 
f  €  7,  n  =  1,  2,  ....  By  the  same  proof,  we  can  show  x(t)  <  v^(t)t  t  6  J,  n  =  1,  2,  .  .  .. 
Consequently,  jc*(f)  <  ;c(0  <  x*(0,  t  6  /.  The  proof  is  complete.  D 


3.  An  example 

We  consider. 


sin2?- 


where 


1,      f  €[-!,-!), 


,r  6(0,1]; 


(3.1) 


Conclusion.  IRFDE  (3.1)  admits  minimal  and  maximal  solutions. 
Proo/  Let 


and 


1, 


It  is  easy  to  see  that  u,  v  are  not  solutions  of  eq.   (3.1)  and  u(t)  <  v(t),  1  6  [—1 ,  1]. 
Moreover, 


362  Baoqiang  Yan  and  Xilin  Fu 


1          1 


M'(0=0,  r  €[0,1]; 


Then 


144 


. 

sin"*  - 


,  re  [0,1]. 


t  6(0, 


1     /I 


24 


:=ft     >    ~~V    \    "Z    \    i 

L  j 


i.e.  the  condition  (As)  is  true. 
By  mean  value  theorem,  we  get 

72  y 

40  X  >       ~     8 

and 

((sin2 1  —  x)3  -  (sin2 1  —  y)3) 

For  any  $  e  Af  ([-1,  0],  R\  let 


Then 


48J_] 
1 


*,  0)  -  f(t,  if)  >  - 


for  all  0,  T/T  €  {;cf,  w(0  <  x(f)  <  u(0,  ^  e  [0,  1]}  with  (j>  < 
So  the  condition  (A?)  is  true. 


V2/' 
1 


-  y). 


Impulsive  delay  differential  equations  363 

So  the  condition  (As)  is  true.  So  M  =  — ,  LI  =  -,  Aj  =  -,  A2  =  1, 

24  6  2 


Forpi(f)  =  u(t)  -  0>(t),t  €  [-1,0],  we  get 

L-i  =  -,A  = 
and 

/^(j)  =  0  <  Mo,  r  €    -1,-r  )  n  (  — ,0   • 
L          2/      V    2     J 

Moreover, 

5  (1-L-OO-^O 

Af0Ai  <  —  =  — 

For  p2  =  4>(0  —  u(0>  we  g^ 


=  ~~  <  p2(0, 


and 


And  thus  it  is  easy  to  see  that  (A4)  is  true.  By  Theorem  2.1,  eq.   (3.1)  has  a  maximal 
solution  and  a  minimal  solution.  The  proof  is  complete.  D 

Remark.  Our  result  can  be  extended  to  impulsive  delay  differential  equations  in  Banach 
spaces. 

Acknowledgement 

This  project  was  supported  by  the  National  Natural  Science  Foundation  of  China  ( 1 977 1 054) 
and  YNF  of  Shandong  Province  (Q99A14). 

References 

[1]  Ballinger  George  and  Liu  Xinzhi,  Existence,  uniqueness  results  for  impulsive  delay  differential 

equations,  Dynamics  of  continuous,  discrete  and  impulsive  systems  5  (1999)  579-591 
[2]  Fu  Xilin  and  Yan  Baoqiang,  The  global  solutions  of  impulsive  retarded  functional  differential 

equations,  Int.  Appl  Math.  2(3)  (2000)  389-398 

[3]  Hale  J  K,  Theory  of  functional  differential  equations  (New  York:  Springer- Verlag)  (1977) 
[4]  Ladde  G  S,  Lakshmikantham  V  and  Vatsala  A  S,  Monotone  iterative  technique  for  nonlinear 

differential  equations  (Pitman  Advanced  Publishing  Program)  (1985) 
[5]  Lakshmikantham  V,  Bainov  D  D  and  Simeonov  P  S,  Theory  of  impulsive  differential  equations 

(Singapore:  World  Scientific)  (1989) 
[6]  Lakshmikantham  V  and  Zhang  B  G,  Monotone  iterative  technique  for  impulsive  differential 

equations,  Appl  Anal.  22  (1986)  227-233 


r 


On  Initial  conditions  for  a  boundary  stabilized  hybrid 
Euler-Bernoiilli  beam 


SUJITKBOSE 

BE-  188,  Salt  Lake  City,  Kolkata  700  064,  India 
MS  received  1  1  July  2000 

Abstract  We  consider  here  small  flexural  vibrations  of  an  Euler-Bernoulli  beam 
with  a  lumped  mass  at  one  end  subject  to  viscous  damping  force  while  the  other  end  is 
free  and  the  system  is  set  to  motion  with  initial  displacement  _y°(;c)  and  initial  velocity 
y  [  (x)  .  By  investigating  the  evolution  of  the  motion  by  Laplace  transform,  it  is  proved 
(in  dimensionless  units  of  length  and  time)  that 


f[  fl 

I  ylt  d*  <  / 

Jo  Jo 


t  >  t0, 


where  fy  rnay  be  sufficiently  large,  provided  that  {y°,  y1}  satisfy  very  general  restric- 
tions stated  in  the  concluding  theorem.  This  supplies  the  restrictions  for  uniform 
exponential  energy  decay  for  stabilization  of  the  beam  considered  in  a  recent  paper. 

Keywords.  Euler-Bernoulli  beam  equation;  hybrid  system;  initial  conditions;  small 
deflection;  exponential  energy  decay. 

1.  Introduction 

In  a  recent  paper,  Gorain  and  Bose  [2]  investigated  the  possibility  of  stabilization  of  trans- 
verse vibrations  of  a  hybrid  system  consisting  of  an  Euler-Bernoulli  beam  held  by  a  lumped 
mass  movable  hub  attached  to  one  of  its  ends.  The  beam  is  assumed  to  be  initially  set  in 
vibration  by  a  displacement  y0  and  velocity  y1  in  the  transverse  direction  and  stabilization 
is  sought  by  applying  viscous  damping  force  to  the  moving  lumped  mass.  The  system 
equations  for  simplicity  can  be  written  in  dimensionless  form  by  suitably  choosing  the 
units  of  length  and  time.  If  y  (x  ,  t)  be  the  transverse  displacement  of  a  point  of  the  beam 
distant  x  from  the  lumped  mass  at  time  /,  the  equations  are  [2] 

ytt(x,  t)  +  yxxxx(x,t)  =  0,     0  <  x  <  1,  t  >  0,  (1) 
along  the  length  of  the  beam,  while  at  the  lumped  mass  and  free  ends, 

yxxx  (0,  0  +  <xytt  (0,  t)  +  Xy,  (0,  0  =  0,      yx  (0,  r  )  =  0,      t  >  0,  (2) 

3^(1,0=0,     3^(1,0  =  0,     r>0,  (3) 

where  a  is  the  dimensionless  mass  of  the  lump  and  similarly  A.  the  damping  coefficient. 
The  system  is  set  to  vibration  with  initial  conditions 

y(;c,0)  =  >'0(;c),      yt(x,0)=yl(x)t     0<x<l.  (4) 


366  Sujit  K  Bose 

We  note  in  (l)-(4)  that  without  loss  of  generality  we  can  assume 

y°(0)=0.  (5) 

Such  hybrid  systems  for  general  y°(x)  and  y  l  (x  )  have  been  investigated  in  detail  in  search 
of  uniform  exponential  decay  of  total  energy  (kinetic  and  potential)  for  proving  stability 
of  the  process.  However  Littman  and  Marcus  [5]  and  Chen  and  Zhou  [1]  have  found  by 
calculating  the  eigenvalues  of  their  hybrid  systems  that  uniform  stabilization  is  not  possible 
because  infinitely  large  wave  number  k,  during  the  passage  of  a  wave  along  the  beam  are 
present  in  the  general  case.  Rao  [6]  arrives  at  the  same  conclusion  by  applying  semigroup 
theory  to  the  evolving  system. 

In  [2]  it  was  noted  that  eq.  (1)  is  arrived  at  by  assuming  that  the  beam  remains  ap- 
proximately straight  during  vibration,  precluding  infinitely  large  wave  numbers.  From  this 
observation,  heuristically  an  additional  condition  was  suggested,  which  in  nondimensional 
form  is 

&d*.     *>r0,  (6) 

where  fy  may  be  as  large  as  we  please.  Subject  to  this  condition,  it  was  proved  in  [2],  that 
uniform  exponential  decay  of  total  energy  indeed  takes  place. 

The  condition  (6)  places  restrictions  on  the  initial  conditions  y°(x),  y[(x)  from  which 
the  system  evolves.  It  is  the  purpose  of  this  paper  to  determine  them  by  investigating  the 
actual  evolution  of  the  system  (l)-(5)  by  Laplace  transformation  in  the  complex  frequency 
domain  s  and  invoking  the  final  value  theorem  for  the  system  behaviour  for  t  tending  to 
infinity. 

2.  System  evolution 

Let  the  Laplace  transform  of  y(x  ,  t)  be 

jc,r)e-Jfdr,  (7) 

then  according  to  the  final  value  theorem,  if  s  be  complex  (with  jc  fixed)  and  Y(xt  s)  be 
analytic  in  Re{s}  >  c,  c  <  0, 

lim  y(x,t)  =  UmsY(x,s)  (8) 

r-»oo  ,y-+o 

and  so  we  would  be  interested  in  the  transformed  quantities  as  s  ~>  0.  The  transformation 
of  equations  (l)-(4)  in  the  usual  way  yield 

2Y(xt  s)  =  syQ(x)  +  y[(x),  (9) 


with  boundary  conditions,  using  (5): 

Yxxx(Q,s)  +  as2Y(Qts)+teY(Q,s)=ayl(Q),  YX(Q,  s)  =  0,  (10) 

r«(l.*)  =  0,  Yxxx(l.s)=Q.  (ID 

In  order  to  solve  (9)-(l  1),  we  introduce  'wave  number'  k  by  the  relation 

s  =  ~ik2  :  s2  =  -k4.  (12) 


Rule r- Bernoulli  beam  equation 


367 


The  general  solution  of  (9)  is  then 

F(JC,  —ik2)  =  Qsin&.r  +  C[  coskx  -f  C2sinhkx  -f-  C3  coshkx 


.     (13) 


For  the  differentiability  of  the  particular  solution  of  (9)  represented  by  the  integral  in  (13) 
we  require  that  };000  andy1^)  areC1  smooth.  The  boundary  conditions  (10),  (11)  yield 
for  the  coefficients  C0,  C\,  C2,  C3  the  four  equations 


CQ  =  —C?, 

-k2(ak2  +  iX)(Ci  -r-  C3)  +  2k3  C2  =  ayl(Q 
—C\  cos  k  -f  C2(sin  /:  4-  sinh  £)  H-  C3  cosh  k  = 
1 


(14a) 
(14b) 

(I4c) 
.  (14d) 


The  exact  solution  of  (14)  can  be  explicitly  written  down  by  Cramer's  rule.  But  here  we 
are  interested  in  the  solution  for  large  t,  that  is  to  say,  for  small  s  or  k  and  so  we  expand  the 
determinants  formally  in  powers  of  k  and  do  the  same  for  the  trigonometric  and  hyperbolic 
functions  appearing  in  (13).  Thus,  restoring  s  in  place  of  A:  defined  in  eq.  (12)  we  obtain, 

1 


sin  A:  4-  C2(cos  k  +  cosh  fc)  +  C3  sinh  /:  = 

1     f  l 

~~^H       t-^2>;0(?)  + 
2/c-  ^ 


O(s2)] 


-T 


ISX~ 


I 

2, 


(is^ 
1  + 
•^^ 


where 


r\ 

=  I    [sy°( 
J n 


(15) 


(16) 


In  §4  we  shall  prove  that  poles  of  Yx  (x  ,  s)  for  each  x  lie  in  Refa}  <  c,  c  <  0  when  A  >  0. 
Hence,  by  the  final  value  theorem  of  Laplace  transform,  we  find  that  since  A.  ^  0, 


lim 


,  r)  =  lim  sYx(x,s)  —  0. 


(17) 


368  Sujit  K  Bose 

The  limiting  operation  in  (1 5)  is  essentially  justified  by  expansion  in  powers  of  s  therein  and 
the  assumed  C l  continuity  of  yQ(x )  and  yl(x).  The  limit  (17)  means  that  in  the  presence  of 
the  viscous  damping,  as  t  becomes  large,  the  beam  approaches  its  original  straight  shape. 

3.  Validity  of  condition  (6) 

In  order  to  prove  that  condition  (6)  holds  for  the  motion,  consider  the  functions  tyxx  (x ,  t) 
and  t2yxt  (x,t).  The  Laplace  transforms  of  the  two  functions  are  respectively 


a  a2 

——\Yxx(xts)]     and     —-=• 
951  d  s2 

Hence  by  the  final  value  theorem, 

t  •  '      J  0    '  Xt  i  •  ~      ii;  .>  i  /  t  r>\ 

lim        ° /.      =  bm  — S; ^ — .  (18) 


The  limit  of  the  numerator  in  (18),  from  equations  (15),  (16)  turns  out  to  be 


/!       1  Of       }  /  .t3N 

JQ  A,  \  3 

while  that  of  the  denominator  turns  out  to  be 


1  a  rx  no 

yl(^)d§  -  -V(0)(l  +  x2)+  /     y°(?)(§  ~^c)d§  I  dx.  (20) 

A.  J0  J 

If  the  latter  limit  vanishes,  it  follows  by  differentiating  twice  that 

y°(x)  =  -J/    v1(jc)djc-f  av!(0)l  =  0,      0  <  :c  <  1,  (21) 

since  .v°(0)  =  0.  If  this  is  the  case,  (19)  and  (20)  respectively  become 

and     [T^rfr^  +  iy+ll  (22) 


Hence  the  limit  in  (18)  exists  finitely  even  in  the  case  when  the  initial  values  jy°(*)  and 
yl(x)  satisfy  (21)  together  with  the  provision  that  vl(0)  ^  0.  This  last  condition  means 
that  the  velocity  at  the  end  where  viscous  damping  is  applied  should  not  vanish  when  the 
initial  displacement  is  zero.  Let  the  limit  in  (18)  be  /  >  0.  It  then  follows  that  given  €  >  0 
however  small,  there  exists  /o  such  that 

f^y^dx      i  +  t     i  +  € 

-~- <  —      <  —-,      for  t  >  tQ. 


Elder-Bernoulli  beam  equation  369 

Hence  for  t  >  tQ  >  */l  +  €,  the  condition  (6)  must  hold.  Thus  we  have  proved  the  following 
theorem. 

Theorem.  Let  y(x  ,  t)  be  the  solution  of  the  system  (l)-(5)  corresponding  to  the  initial 
conditions  {y®(x),  y*(x)}  which  are  Cl[Q,  1]  continuous.  Then  condition  (6)  holds,  pro- 
vided that  if  yQ(x)  =  0  on  [0,  1]  then,  either  /Ql  yl(x)dx  ^  -ay!(0)  or  f^  y[(x)dx  = 
-ay1  (0)5*0. 

4.  Poles  of  Fjcfjr,  s) 

When  51  is  considered  complex,  Y(x  ,  s)  given  by  (13)  together  with  (12)  has  poles  at  those 
of  the  coefficients  C0  ,  Clf  C2,  C3.  These  are  at  zeroes  of  the  determinant  of  the  coefficients 
on  the  right  hand  side  of  the  equations  (14b)-(14d),  satisfying  the  equation  (in  terms  of  A:), 

=0.     (23) 


When  a  differentiation  of  (13)  is  performed,  k  =  0  no  longer  remains  a  pole  of  Yx(xt  s)  as  is 
reflected  in  (15).  The  poles  of  Yx  (x  ,  s)  are  thus  the  nonzero  zeroes  of  (23).  We  investigate 
their  domain  by  a  method  similar  to  that  of  Krall  [4]  as  given  in  Gorain  [3]. 

The  zeroes  of  (23)  result  from  (I4b)-(14d)  when  the  right  hand  sides  are  taken  zero.  In 
other  words,  they  crop  up  from  the  boundary  value  problem  (9)-(l  1)  with  the  right  hand 
sides  set  to  zero: 

YxxxxW*  s)  +  s2Y(x,s)  =0,     S  =  U  +  IVT£Q,  (24) 


,  s),      7X(0,  s)  =  0,  (25a) 

7^(1,^=0,      Yxxx(l,s)=Q.  (25b) 

If  we  multiply  (24)  by  the  complex  conjugate  F*  and  then  take  its  conjugate,  we  obtain 

Y*YXXXX+S*\Y\2  =  0    and     YY*XXX  +^2|F|2  =  0- 
Subtracting  one  from  the  other  and  integrating  from  0  to  1,  we  have 


Integrating  by  parts  and  applying  boundary  conditions  (25),  we  obtain  from  the  above  after 
simplification, 

(s2  -  s*2)  I    \Y\2dx  =  -(s  -  j*)  [a  (a-  +  5*)  +  A.]  |F(0,  s)\2. 


If  now  s  -  s*  =  2f  v  ^  0,  it  follows  that 


<0.  (26) 

)|2 

In  (26)  «  ^  0,  since  otherwise  7(0,  s)  =  0  and  then  (24),  (25)  yield  F(x,  j)  identical  to 
zero. 


370  Sujit  K  Bose 

lfs—s*=2iv  =  0,  we  have  s  =  u  and  the  boundary  value  problem  (24),  (25)  becomes 
one  of  real  value.  Equation  (24)  then  yields 


Integrating  by  parts  from  0  to  1  and  applying  the  boundary  conditions  (25)  with  u  in  place 
of  s,  we  obtain  since  7(0,  s)  ^  0  as  before, 


In  (27)  M  7^  0,  since  otherwise  /J  T^d*  =  0,  which  implies  that  Yxx  =  0,  that  is  to  say, 
}'A-JC  =  0  on  0  <  x  <  1,  t  >  0,  meaning  that  the  beam  is  not  bent. 

References 

[1]  Chen  G  and  Zhou  J,  The  wave  propagation  method  for  the  analysis  of  boundary  stabilization 

in  vibrating  structures,  SIAM  J.  Appl  Math.  50  (1990)  1245-1283 
[2]  Gorain  G  C  and  Bose  S  K,  Boundary  stabilization  of  a  hybrid  Euler-Bernoulli  beam,  Proc. 

Indian  Acad.  Sci.  (Math.  ScL)  109  (1999)  41  1-416 
[3]  Gorain  G  C,  Exact  vibrau'on  control  and  boundary  stabilization  of  a  hybrid  internally  damped 

elastic  structure,  Ph.D.  thesis  (Jadavpur  University)  (1999) 
[4]  Krall  A  M,  Asymptotic  stability  of  the  Euler-Bernoulli  beam  with  boundary  control,  /  Math. 

Anal.  Appl.  137  (1989)  288-295 
[5]  Littman  W  and  Markus  L,  Stabilization  of  a  hybrid  system  of  elasticity  by  feedback  boundary 

damping,  Ann.  Mat.  Pura  Appl  152  (1988)  28  1-330 
[6]  Rao  B,  Uniform  stabilization  of  a  hybrid  system  of  elasticity,  IEEE  Trans.  Autom.  Contr.  33 

(1995)440^54 


Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  1 11,  No.  4,  November  2001,  pp.  371-379. 
©  Printed  in  India 


Cyclic  codes  of  length  2" 


MANJU  PRUTHI 

Department  of  Mathematics,  M.D.  University  P.O.  Regional  Centre,  Rewari  123  401, 

India 

E-mail:  m.pruti22@yahoo.com 

MS  received  15  March  2000;  revised  26  March  2001 

Abstract.  In  this  paper  explicit  expressions  of  m  +  1  idempotents  in  the  ring  R  = 
Fq[X]/(X2>n  -  1}  are  given.  Cyclic  codes  of  length  2m  over  the  finite  field  Fq,  of 
odd  characteristic,  are  defined  in  terms  of  their  generator  polynomials.  The  exact 
minimum  distance  and  the  dimension  of  the  codes  are  obtained. 

Keywords.  Cyclotomic  cosets;  generator  polynomial;  idempotent  generator;  [n,  k, 
d]  cyclic  codes. 

1.  Introduction 

Throughout  in  this  paper  we  consider  Fq  to  be  a  field  of  odd  characteristic  and  the  ring 
R  =  F(j[X]/(X^  —  1).  The  ring/?  can  be  viewed  as  semi-simple  group  ring  F^Cy  where 
Com  is  a  cyclic  group  of  order  2"1  generated  by  x.  It  is  assumed  that  reader  is  familiar 
with  the  properties  of  cyclic  codes  based  on  the  theory  of  idempotents  [3].  In  §2  of  this 
paper  complete  set  of  equivalence  classes  (modulo  2'")  is  given  and  also  the  construction 
of  explicit  expressions  of  idempotents  is  given.  In  §3,  we  completely  describe  the  cyclic 
codes  of  length  2'"  in  terms  of  their  generator  polynomials.  In  §4  we  obtain  g-cyclotomic 
cosets  (modulo  2"2)  when  order  of  q  modulo  2m  =  2m~2.  An  example  has  been  given  to 
illustrate  the  results. 

2.  Construction  of  idempotents 

For  any  positive  integer  m,  consider  the  set  S  =  {1,  2,  3,  .  .  .  ,  2m  -  1).  Divide  the  set  S 
into  disjoint  classes  S/  (modulo  2"7)  as  follows: 
For  1  <  i  <  m,  consider  the  set 


St  =  {21'-1,  2/~13,  .  .  .  ,  2/~1(2n/  -  1)},  1  <  n(-  <  2"1"'' 

Clearly  the  elements  of  S/  are  incongruent  to  each  other  modulo  2m  .  Note  that  the  elements 
of  Si  are  the  product  of  2l~[  with  odd  numbers.  So  these  are  divisible  by  2'~  l  but  no  higher 
power  of  2.  In  the  set  5,  the  number  of  elements  divisible  by  2'~  [  but  no  higher  power  of 
2  are 

(2m-'"+1  -  1)  -  (2m-/  -  1)  =  2m~i+[  -  2m~l  =  2m~/(2  -  1)  =  2'"~/. 
Hence  the  number  of  elements  in  the  set  S/  is 
#St  =  2"'-''. 

371 


372  Manju  Pruthi 

Clearly  for  z  ^  j,  S/  n  Sj  =  O  and  so 

y  s  «1  /7i 

#  (  U  S;  )  =  Y>S)  =  Y(2"1-1)  =  2f"  -  1. 
V=      /      -t—  '  t—f 


/=i 

/  =  ! 


Hence  the  sets  S-t  (  1  <  z  <  m)  form  the  partitioning  of  the  set  S  (modulo  2"'). 
For  1  <  z  <  m,  define  the  element  $/(*)  as 


Let  a  be  a  primitive  2"zth  root  of  unity  in  an  extension  of  the  field  Ff/.  To  prove  the  main 
theorem  we  require  the  following  facts: 


Fact  2.1    For  I  <i  <  m, 


0         if     2"'-''  /; 
-2m~l     if    ;  =  2'w-/ 
2"'-''      if     2WI-/+1| 


Proof.  By  definition,  for  1  <  z  <  m, 


E2'~~'(2/?/  —  1)  j_  Y"^    ,2'~1(2/j/— 2)        \~^     2'~I(2/?/— 2) 
jc  -j-  /    x  ./     x 

/  <•  /  ^ 

n/=l  «/  =  l  //,-  =  ! 

9»7— /+! I  *)m—i 

Therefore, 

Si(&J)  =     2^     (ex2'    J)k  ~  V^  or  •/(-/z/~~1\  (1) 

Jfc=0  ///=! 

Caw  1.  If2m-//y,then2m-l/2/-1;  S02'-1;  =£  OCmod^Jhencea2'''1^  ^  1.  Similarly 
or'.'  ^  1.  Therefore  (1)  gives  that 


a2'"1-/  -  1  a2'-/  -  1 

(denominator  being  non-zero).  This  proves  the  Case  1 . 

Case  2.    If;  ^  2m~i ,  then  2''-1;  =  27""1  and  2l  j  =  2m.  Since  a  is  a  primitive  2mth  root 
of  unity  in  an  extension  of  Ffjt  so  a2'-7"  =  a2'"  =  1  and  a2'"1''  =  a2"7"1  =  -1.  Again  (1) 


Codes  of  length  2m  373 

gives  that 


*  =  0  7Z/=0 

This  proves  the  Case  2. 

Case  3.     If  2m~i+{ /j  then  2m/2i~[j  implies  that  or  "^  =  1  and  also  or  •>  =  1.  Again 
from  (1)  we  have 

2'"-'  +  i_i 

A'=0  n/=0 

—     2m~i+l  —  2m~l  =  2m~*  (2  —  1)  =  2m~l . 
This  proves  the  Fact  2.1. 

Fact  2.2.    For  0  <  i  <  m  -  1 , 

,   ,    V    --./x_        °        if     WXl 
1  +   LJ  ' 

Proof.  By  definition 


If  2m~'1  /  j  then  2m  /  2/:  j  implies  that  a2'-1  ^  1  .  Hence  the  required  sum  takes  the  value 
zero.  Secondly  if  2m~l  /j,  then  2m  /2l  j  implies  that  or'7  =  1  in  the  extension  field  and 
hence  the  required  sum  takes  the  value 


This  proves  the  Fact  2.2. 

Our  construction  of  idempotents  is  based  on  the  following  two  facts  developed  in  §2  and 
3  of  chapter  8  of  [3]. 

Fact  2.3.    An  expression  e(x)  in  R  is  an  idempotent  iff  <?(a-7)  =  0  or  1. 

Fact  2.4.    An  idempotent  ei  (jt)  is  primitive  iff 

,    ,\          1     if  /  6  Yr  for  some  r,  0  <  r  <  m 
e;(ctj)  =  j        r  —     — 

•  0     otherwise, 

where  Yr  is  some  #-cyclotomic  coset  (modulo  2m)  with  7o  =  {0}. 


374 


Manju  Prut  hi 


Theorem  2.5.  The  following  polynomial  expressions  are  (m  +  1)  idempotents  in  the 
ring  R, 


7=0 


and  for  1  <  i  <  m 


1 


2/n  — /'+! 


1  + 


V 


/V00/  By  Fact  2.2 

0  if     2'"/7 

1  if     2m  1 7 

0  if     7  €  ft 

1  if     2m  1 7     ' 

By  Fact  2.4,  <?oU)  is  a  primitive  idempotent  with  single  non-zero  a°  =  1 .  For  1  <  i  <  m, 
Facts  2.1  and  2.2  show  that 

0  if      2;"™77 

1  if    2'"-''  =  7 
0     if    2/w-/+1|7 


Thus  for  1  <  ?  <  m,  £/(aJ")  =  0  or  1  and  e {(&•')  =  1  only  if  7  =  2m  *  or  equivalently  by 
definition  only  if  7  €  Sm_;+i.  Hence  by  the  Fact  2.3  the  expressions  e/(;c)  are  idempotents. 

3.  Cyclic  codes  of  length  2m 

Let  for  0  <  i  <  m,  £/  denotes  the  cyclic  code  of  length  2m  with  idempotent  generator 
ei(x).  By  (Theorem  56,  [4]),  (Remark  6. 3,  [6])  the  generator  polynomial  ^/(jc)  of  the  cyclic 
code  EI  is  given  by 


gi(x)  = 


),x-    -  1). 


(2) 


Define 


and  for  1  <  i  <  m, 


2m—\ 


;=0 


Then  to  show  #/(*)  (0  <  i  <  m)  is  the  generating  polynomial  of  the  cyclic  code  £/.  In 
view  of  (2)  it  is  sufficient  to  prove  the  following  two  facts: 


Fact  3.1.  gi  (a-0  =  0  iff  <?/  (a>)  =  0. 


Codes  of  length  2m 


Fact  3.2.  g/W/Jc-  1. 

To  prove  the  Fact  3.1,  consider  for  1  <  i  <  m, 


375 


'/to     = 


Thus  for  1  <  i  <  m,  ei(x)  is  a  constant  multiple  of  gi(x):  Also  by  definition  €Q(X)  is  a 
constant  multiple  of  go  (;c).  Hence  g/fa-7')  =  Oiffe/Ca-7')  =  0. 
To^prove  the  Fact  3.2,  consider  for  0  <  /  <  m, 


Thus  gi(x)  is  a  factor  of  (1  —  x2'").  Hence  the  assertion  follows. 
Theorem  33.  £/  w  a  [2m,  2''"1,  2m~/-f  l]  cyclic  code  over  GF(q}. 

Proof.  By  Corollary  3  ([3],  p.  218)  (generalized  to  non  binary  case)  for  0  <  i  <  m, 
dim£/  =  #aj  such  that  a  (ctj)  -  1. 

By  Theorem  2.5,  we  have  e£-(a-/')  =  1  only  if  j  e  Sm-i+{.  So  dim£/  =  #5m_/+i  = 
2''-1. 

As  shown  in  [5,  6,  1]  it  is  easy  to  prove  that  the  repetition  code  £"/  generated  by  gt(x) 
has  the  minimum  distance  2m~i+{  and  d(E$)  =  2m  =  #  non-zero  terms  in  gQ(x). 

4.  £-Cyclotomic  cosets  (modulo  2m)  when  order  (q)  =  2m~~2 

First  note  that  such  a  ^  exists  due  to  the  following  facts  [2].  Obviously  in  this  case  m  >  3. 
So  throughout  this  section  assume  that  m  >  3. 


376  Manju  Pmthi 

Fact  4. 1 .  The  integer  2m  has  no  primitive  root. 

-)in  —  2 

Fact  4.2.  Let  a  be  any  odd  integer,  then  it  is  always  true  that  cr       =  1  (mod  2m). 

Fact  4.3.  If  ord(fl)  =  2  (mod  23)  and  a2  •£  1  (mod  24),  then  ordfc)  =  2/;i~2  (mod  2m)  for 
every  m  >  3. 

Computation  of  g-cyclotomic  cosets  (modulo  2m)  depend  upon  the  following  facts: 

Fact  4.4.  If  ord(<?)  =  2m~2  (modulo  2"' )  for  every  m  >  3,  (Fact  4.3),  then  #'  ^  - 1  (mod 
2m)  for  1  <  r  <2';l~2. 

Proo/  For/  >  2//J~2,  we  have  ^r  ==  1  (mod  2/n). 

If  possible  let  q1  =  —1  (mod  2'")  for  some  non-negative  integer  t  <  2m~2,t\iQnq2t  ==  1 
(mod  2'").  But  ord(ry)  =  2;//-2  implies  that  2"'~2|2f  or  2m~3|r  =>  r  =  2m~3a,  but 
/  <  2'"~2.  So  we  must  have  a  =  1.  So  we  have 

=>q~m~*  =  ~l(inod2'") 

=>q2'"~*  =  -I(mod2'""1).  (3) 

But  we  are  assuming  that  ord(^)  =  2m~2  for  all  m  >  3.  So  we  have 

q2  '  '  ~  l(mod  2'""1).  (4) 

From  (3)  and  (4) 

-1  ~  I(mod2'""1)     for  all  m  >  3 
which  is  not  possible.  Hence  the  result  follows. 

Fact  4.5.  Thus  in  this  case  q  cyclotomic  cosets  modulo  2m  are  given  by: 
For  1  <  /  <  m, 


Remark  4.6.  By  definition  of  £/  it  is  clear  that  for  1  <  i  <  m, 


Note  that  integers  of  the  type  q  =  8X  -f  3  (A.  >  0)  satisfy  the  above  facts.  In  particular 
we  may  consider  q  =  3,  then  order  (3)  =  2m~~2  (modulo  2m)  for  all  m  >  3.  In  this  case 
observe  the  following. 

Fact  4.7.  For  1  <  i  <  m  -  2, 


or 


Codes  of  length  2m  377 

Fact  4.8.  Since  3  is  primitive  root  of  unity  modulo  4 

32  ==  1  (mod  22)  =»  2"'-232  =  2'"-2(modulo  2/n). 
Fact  4.9.  Since  3  ==  -I  (mod  22), 


and 

2"'~2.32  =  -2"'-2.3(modulo2'"). 
Fact  4.  1  0. 

1  =  -I  (mod  2), 

=»2"1-1  s  -2'"-  '(mod  2'"). 

Using  the  facts  of  §4,  the  3-cyclotomic  cosets  modulo  2m  are  given  as  follows: 
For  1  <  i  <  m  —  2, 


and 

^m-i     -     X*  _!  -  {2m-2,  2/7'-2.3}  =  {-2m~2,  -2/;'-2. 


Example.  Consider  ^  =  5  and  C2s  be  a  cyclic  group  of  order  25  generated  by  jc.  Then  the 
4-cyclotomic  cosets  (modulo  25)  are  given  by 

X{  =  {1,5,25,29,  17,21,9,  J3}, 

Xt  =  {-J,  -5,  -25,  -29,  -17,  -21,  -9,  -13} 

=  {31,27,7,3,  15,  11,23,  19}, 

X2  =  {2,10,18,26}, 

X^  =  {30,22,14,6} 

*3  =  {4,20}, 

X;  =  {28,  12}, 

^4  -  {8}, 


X5  =     {6}  =  X;. 
By  Remark  4.6, 

S{  =  {1,3,5,7,9,  11,  13,  15,  17,19,21,23,25,27,29,31}, 

52  =  {2,6,10,14,18,22,26,30}, 

53  =  {4,12,20,28}, 

54  =  {8,  24}, 
Ss  =  {16}. 


378  Manju  Pruthi 

The  six  distinct  idempotents  in  this  case  can  be  read  as  follows: 


e\(x)    =     •( 


^-{ 

^( 

~{ 


The  important  parameters  of  the  codes  EQ,  £1,  £2,  £3,  £4,  £5  of  length  25  over  the  field 
GF(5)  are  listed  in  the  table  below. 

Code  Non-zero  Dimension      Minimum       Generator 

K  distance,  d      polynomial,  gi(x) 

_. a()=  i  j  —  1+JC  +  JC2+  ...  +  j.31 

£,  a16  1  25  (1 -x){l  +  52  +  53  +  54  +  55} 

E2  a8, a24  2  24  (1  -JC2){1  +  53  +  54  +  55} 

£3  ft4,  of'2,  ft2^,  Of28  4  2^  (1  —  ^c4){  1  +  Jf^  +  x     +  Jf    } 

£4  a2,  of6,  a10,  a14,  Oflx,a22,  a2fi,o!?0  8  22  (1  -  JC8){1  +  x16} 

E5  ujj€S\  16  2  (1  -AJfi) 

Example.  Consider  q  =  3  and  C?  be  a  cyclic  group  of  order  23  generated  by  *.  Then  the 
^-cyciotomic  cosets  (modulo  23)  are  given  by 

Xj  =  {1,3}, 

^T  =  (5,7), 

*2  =  {2,6}, 

X3  =  {4}, 

^o  -  {0}. 

The  five  primitive  idempotents  in  the  group  algebra  GF(3)  C3  are  given  with  their  non- 
zeroes: 


Primitive  idempotents  Non-zeroes 


«3(Jt)  =  ^{(1  -  X3)  -  (X,  -  Xt)}W  a-'',  ;  6 


Codes  of  length  2m  379 

References 

[I]  Arora  S  K  and  Pruthi  Manju,  Minimal  cyclic  codes  of  length  2pn,  Finite  fields  and  their 

applications,  5(1999)  177-187 

[2]  Burton  David  M,  Elementary  number  theory,  2nd  ed.  (University  of  New  Harsheri) 
[3]  Mac  Williams  F  J  and  Sloane  N  J  A,  Theory  of  error-correcting  codes  (Amsterdam:   North 

Holland)  (1977) 
[4]  Piess  V,  Introduction  to  the  theory  of  error  correcting  codes  (New  York:  Wiley-Interscience) 

(1981) 
[5]  Pruthi  Manju  and  Arora  S  K,  Minimal  codes  of  prime  power  length,  Finite  fields  and  their 

applications,  3(1997)99-113 
[6]  Vermani  Lekh  R,  Elements  of  algebraic  coding  theory  (UK:  Chapman  and  Hall)  (1992) 


Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  Ill,  No.  4,  November  2001,  pp.  381-397. 
©  Printed  in  India 


Unitary  tridiagonaiization  in  M(49  C) 

VISHWAMBHAR  PATI 

Stat.-Math.  Unit,  Indian  Statistical  Institute,  RVCE  P.O.,  Bangalore  560  059,  India 
MS  received  7  April  2001 ;  revised  4  September  2001 

Abstract.  A  question  of  interest  in  linear  algebra  is  whether  all  n  x  n  complex 
matrices  can  be  unitarily  tridiagonalized.  The  answer  for  all  n  ^  4  (affirmative  or 
negative)  has  been  known  for  a  while,  whereas  the  casen  =  4  seems  to  have  remained 
open.  In  this  paper  we  settle  the  n  =  4  case  in  the  affirmative.  Some  machinery  from 
complex  algebraic  geometry  needs  to  be  used. 

Keywords.  Unitary  tridiagonaiization;  4x4  matrices;  line  bundle;  degree;  algebraic 
curve. 

1.  Main  Theorem 

Let  V  =  C\  and  (  ,  }  be  the  usual  euclidean  hermitian  inner  product  on  V.  U(V)  =  U(ri) 
denotes  the  group  of  unitary  automorphisms  of  V  with  respect  to  {  ,  ).  {<?/  }"=1  will  denote 
the  standard  orthonormal  basis  of  V.  A  e  M(nt  C)  will  always  denote  an  n  x  n  complex 
matrix. 

A  matrix  A  =  [#/;-]  is  said  to  be  tridiagonal  if  fl//  =  0  for  all  1  <  z,  j  <  n  such  that 
\i  -  j\  >  2.  Then  we  have: 

Theorem  1.1.  For  n  <'4,  and  A  G  M(n,  C),  there  exists  a  unitary  U  e  U(ri)  such  that 
UAU*  is  tridiagonal. 

Remark  1.2.  The  case  n  =  3,  and  counterexamples  for  n  >  6,  are  due  to  Longstaff,  [3]. 
In  the  paper  [1],  Fong  and  Wu  construct  counterexamples  for  n  —  5,  and  provide  a  proof 
in  certain  special  cases  for  n  =  4.  The  article  §4  of  [1]  poses  the  n'—  4  case  in  general 
as  an  open  question.  Our  main  theorem  above  answers  this  question  in  the  affirmative.  In 
passing,  we  also  provide  another  elementary  proof  for  the  n  —  3  case. 

2.  Some  Lemmas 

We  need  some  preliminary  lemmas,  which  we  collect  in  this  section.  In  the  sequel,  we  will 
also  use  the  letter  A  to  denote  the  unique  linear  transformation  determined  by  the  matrix 
A  =  [au]  (satisfying  Aej  =  ^l=1  a^e^. 

Lemma  2.1.  Let  A  e  M(n,  C).  For  all  nt  the  following  are  equivalent: 
(i)  There  exists  a  unitary  U  €  U(n)  such  that  UAU*  is  tridiagonal. 

381 


382  Vishwambhar  Pali 

(ii)  There  exists  a  flag  (=  ascending  sequence  of  C-sub  spaces)  ofV  =  Cn: 
0  =  W0  C  W,  C  W2  C  .  .  .  C  Wn  =  V 

such  that  dim  W/  =  z,  AW/  C  Wm  and  A*  W/  C  Wi+ifor  all  0  <  z  <  n  -  1. 
(iii)  TTiere  ejdsts  ajflag  m  V: 

0  =  W0  C  W{  C  W2  C  .  .  .  C  Wn  =  V 
such  that  dim  Wt  =  z,  AW/  c  WiJr[  and  A(Wj^})  c  W^  for  all  0  <i  <n-l. 

Proof,    (i)  =»  (ii).    Set  W/  =  C-span(/i,  /2,  .  .  .  ,  //),  where  //  =  U*et  and  */  is  the 
standard  basis  of  V  =  C72.  Since  the  matrix  [6/y]  :=  UAU*  is  tridiagonal,  we  have 

Aft  =  bi-ufi-\  +  bnfi  +  Z?/+i,  ///+!,      for    1  <.i  <n 

(where  Z?//  is  understood  to  be  =  0  for  z,  ;  <  0  or  >  n  +  1).  Thus  AW/  C  W/+i.  Since 
{//  }"=1  is  an  orthonormal  basis  for  V  —  C/z,  we  also  have 

-A*//  =bij-ifi-i  +bnfi  +bu+\fi+i      I  <i  <n 

which  shows  A*(W,-)  c  W/+i  for  all  f  as  well,  and  (ii)  follows. 

(ii)  =»  (iii).  A*W/  C  W/+i  implies  (A*1^/)1  D  W^j  for  1  <  f  <  n  -  1.  But  since 
(A*^-)-1  =  A-^W;1),  we  have  A(W^{)  C  Wr1  for  1  <  /  <  n  -  1  and  (iii)  follows. 

(iii)  =>  (i).  Inductively  choose  an  orthonormal  basis  //  of  V  =  Cn  so  that  W/  is  the 
span  of  {/i,...,  //}.  Since  A(W/)  C  W/+i,  we  have 


Since  /•  6  (Wf-i)1,  and  by  hypothesis  A(Ty^1)  c  W^,2,  and  W/i2  =  C-span(//_i,  //  , 
•  •  •  ,  /n)?  we  also  have 

Aft  =  ai-\tifi-{+aufi  H-  ----  hflm7n  (2) 

and  by  comparing  the  two  equations  (1),  (2)  above,  it  follows  that 


for  all  z,  and  defining  the  unitary  U  by  U*ei  =  ./}  makes  [/At/*  tridiagonal,  so  that  (i) 
follows.  0 


Lemma  2.2.  Letn  <  4.  /jf  f/zere  exto  a  2-dimensional  C-sub  space  W  ofV  =  Cn  such  that 
AW  cW  andA*W  c  W,  then  A  is  unitarily  tridiagonalizable. 

Proof.  If  n  <  2,  there  is  nothing  to  prove.  For  n  —  3  or  4,  the  hypothesis  implies 
that  A  maps  W1  onto  itself.  Then,  in  an  orthonormal  basis  {//}"=1  of  V  which  satisfies 
W  =  C-span(/!,  /2)  and  WL  =  C-span(/3,  .  .  .  ,  /n)  the  matrix  of  A  is  in  (1,  2)  (resp. 
(2,  2))  block-diagonal  form  for  n  =  3  (resp.  n  =  4),  which  is  clearly  tridiagonal.  D 


Lemma  2.3.  £very  marra  A  €  M(3,  C)  w  unitarily  tridiagonalizable. 


Unitary  tridiagonalization  in  M(4,  C)  383 

Proof.  For  A  e  Af  (3,  C),  consider  the  homogeneous  cubic  polynomial  in  v  =  (v\,  vi,  1*3) 
given  by 

F(v\,  i»2,  us)  :=  det(u,  Au,  A*i>). 

Note  11  A  Ai>  A  A*v  =  F(i>i,  u?,  ^3)^1  A<?9  A  £3.  By  a  standard  result  in  dimension  theory 
(see  [4],  p.  74,  Theorem  5)  each  irreducible  component  of  V(F)  C  PQ  is  of  dimension 
>  1,  and  V(F)  is  non-empty.  Choose  some  [v\  :  v^  :  1^3]  €  V(.F)»  and  let  v  =  (uj,  i>2,  1^3) 
which  is  non-zero.  Then  we  have  the  two  cases: 

Case  1.  u  is  a  common  eigenvector  for  A  and  A*.  Then  the  2-dimensional  subspace 
W  =  (Cu)-1"  is  an  invariant  subspace  for  both  A  and  A*,  and  applying  the  Lemma  2.2  to 
W  yields  the  result. 

Case  2.  v  is  not  a  common  eigenvector  for  A  and  A*.  Say  it  is  not  an  eigenvector  for  A 
(otherwise  interchange  the  roles  of  A  and  A*).  Set  Wj  =  Cv,  W?  =  C-span(u,  Au),  W^  = 
V  ~  C3.  Then  dim  W/  =  z  ,  for  /  =  1,  2,  3,  and  the  fact  that  v  A  Au  A  A*u  =  0  shows  that 
A*Wi  c  W?.  Thus,  by  (ii)  of  Lemma  2.1,  we  are  done.  D 

Note.  From  now  on,  V  =  C4  and  A  e  M  (4,  C). 

Lemma  2.4.  TfA  anJ  A*  /zav^  <3  common  eigenvector,  then  A  is  unitarily  tridiagonaliiable. 


Proof.  If  v  7^  0  is  a  common  eigenvector  for  A  and  A*,  the  3-dimensional  subspace 
W  =  (Cu)1  is  invariant  under  both  A  and  A*,  and  unitary  tridiagonalization  of  A\w  exists 
from  the  n  =  3  case  of  Lemma  2.3  by  a  U\  €  U(W)  =  17(3).  The  unitary  £7  =  1  0  U{  is 
the  desired  unitary  in  £/(4).tridiagonalizing  A.  D 

Lemma  2.5.  Tjf  f/ze  mam.  theorem  holds  for  all  A  €  S,  where  S  is  any  dense  (in  the  classical 
topology)  subset  ofM(4,  C),  then  it  holds  for  all  A  e  M(4,  C). 

Proof.  This  is  a  consequence  of  the  compactness  of  the  unitary  group  17(4).  Indeed,  let  T 
denote  the  closed  subset  of  tridiagonal  (with  respect  to  the  standard  basis)  matrices. 

Let  A  e  M(4,  C)  be  any  general  element.  By  the  density  of  5,  there  exist  An  €  S  such 
that  An  ->  A.  By  hypothesis,  there  are  unitaries  Un  €  17(4)  such  that  UnAnU*  =  Tn, 
where  Tn  are  tridiagonal.  By  the  compactness  of  17(4),  and  by  passing  to  a  subsequence 
if  necessary,  we  may  assume  that  Un  -+  U  e  17(4).  Then  UnAnU*  ~>  UAU*.  That 
is  Tn  ->  UAU*.  Since  T  is  closed,  and  Tn  €  T,  we  have  [7  AC/*  is  in  T,  viz.,  is 
tridiagonal.  D 

We  shall  now  construct  a  suitable  dense  open  subset  S  C  M(4,  C),  and  prove  tridiago- 
nalizability  for  a  general  A  6  S  in  the  remainder  of  this  paper.  More  precisely: 

Lemma  2.6.  There  is  a  dense  open  subset  S  C  M(4,  C)  such  that: 
(i)  A  is  nonsingularfor  all  A  €  S. 
(ii)  A  has  distinct  eigenvalues  for  all  A  6  5. 


384  Vishwambhar  Pali 

(iii)  For  each  A  e  S,  the  element  (r0/  +  ri  A  -f  ft  A*)  €  M(4,  C)  has  rank  >  3  for  all 
(to,t\,t2)  ^(0,  0,0)  wC3. 

Proo/.  The  subset  of  singular  matrices  in  M(4,  C)  is  the  complex  algebraic  subvariety 
of  complex  codimension  one  defined  by  Z\  =  {A  :  det  A  =  0}.  Let  Si,  (which  is  just 
GL(4,  C))  be  its  complement.  Clearly  S\  is  open  and  dense  in  the  classical  topology  (in 
fact,  also  in  the  Zariski  topology). 

A  matrix  A  has  distinct  eigenvalues  iff  its  characteristic  polynomial  0/\  has  distinct  roots. 
This  happens  iff  the  discriminant  polynomial  of  0/\,  which  is  a  4th  degree  homogeneous 
polynomial  A  (A)  in  the  entries  of  A,  is  not  zero.  The  zero  set  Z2  =  V(A)  is  again  a 
codimension-1  subvariety  in  M(4,  C),  so  its  complement  £2  =  (V(A))C>  is  open  and  dense 
in  both  the  classical  and  Zariski  topologies. 

To  enforce  (iii),  we  claim  that  the  set  defined  by 

Z3  :=  {A  €  M(4,  C)  :  rank(f0/  +  t\  A  +  M*)  <  2  for  some  (?0,  *i,  ^2) 
^(0,0,0)  in  C3} 

is  a  proper  real  algebraic  subset  of  M(4,  C).  The  proof  hinges1  on  the  fact  that  three  general 
cubic  curves  in  PQ  having  a  point  in  common  imposes  an  algebraic  condition  on  their 
coefficients. 

Indeed,  saying  that  rank(fo/  -f  t\A  +  ft  A*)  <  2  for  some  (^o,/i,ft)  ^  (0,0,0)  is 
equivalent  to  saying  that  the  third  exterior  power  /\*(tQl  +  t\  A  +  ^A*)  is  the  zero  map, 
for  some  (fy,  t\,  ft)  i=-  0.  This  is  equivalent  to  demanding  that  there  exist  a  (fo,  t\  ,  ft)  ^  0 
such  that  the  determinants  of  all  the  3  x  3-minors  of  (to!  +  t\A  +  ft  A*)  are  zero. 

Note  that  the  (determinants  of)  the  (3  x  3)-minors  of  (/Q/  -f  t\  A  +  ft^*)>  denoted  as 
M//(A,  0  (where  the  z'th  row  and  y'th  column  are  deleted)  are  complex  valued,  complex 
algebraic  and  C-homogeneous  of  degree  3  in  t  =  (?o,  t\  ,  ^2),  with  coefficients  real  algebraic 
of  degree  3  in  the  variables  (A//,  A//)(or,equivalenlly,  inRe  A//,  Im  A//),  whereA  =  [A//]. 

We  know  that  the  space  of  all  homogeneous  polynomials  of  degree  3  with  complex 
coefficients  in  (?o,  t\,  ro)  (up  to  scaling)  is  parametrized  by  the  projective  space  PQ  (the 
Veronese  variety,  see  [4],  p.  52).  We  first  consider  the  complex  algebraic  variety: 

X  =  {(P,  2,  R,  [t])  e  P£  x  P9C  x  P^  x  P£  :  P(0  =  2(0  =  R(t)  =  0}, 


where  [r]  :=  [>o  :  ?i  :  fr]»  and  (P,  Q,  /?)  denotes  a  triple  of  homogeneous  polynomials. 
This  is  just  the  subset  of  those  (P,  2,  R,  [r])  in  the  product  P^xP^  xP^xPg  such  that  the 
point  [/]  lies  on  all  three  of  the  plane  cubic  curves  V(P),  V(Q).  V(R)-  Since  Z  is  defined 
by  multihomogenous  degree  (1,1,1,3)  equations,  it  is  a  complex  algebraic  subvariety  of 
the  quadruple  product.  Its  image  under  the  first  projection  Y  :=  n\  (X)  C  PQ  x  P£  x  P^  is 
therefore  an  algebraic  subvariety  inside  this  triple  product  (see  [4],  p.  58,  Theorem  3).  Y  is 
a  proper  subvariety  because,  for  example,  the  cubic  polynomials  P  =  t$,Q=tf,R  =  t\ 
have  no  common  non-zero  root. 

Denote  pairs  (z,  jf)  with  1  <  i,  j  <  4  by  capital  letters  like  7,  J,  K  etc.  From  the  minorial 
determinants  Af/(A,  r),  we  can  define  various  real  algebraic  maps: 

:  M(4,  C)     ->     P£  x  P^  x  P^ 
A    H> 


Unitary  tridiagonalization  in  M(4,  C) 


385 


for/,  J,  K  distinct.  Clearly,  /\3(fo7-HiA-K2A*)  =  Oforsomer  =  (Jo, '1,^2)  ^  (0,0,0) 
iff  ©///c(A)  lies  in  the  complex  algebraic  subvariety  Y  of  P^  x  P^  x  P^,  for  all  /,  J,  A: 
distinct.  Hence  the  subset  Z^  c  M(4,  C)  defined  above  is  the  intersection: 


where  7,  /,  7£  runs  over  all  distinct  triples  of  pairs  (/,  ;),   1  <  z,  j  <  4. 

We  claim  that  Z^  is  a  proper  real  algebraic  subset  of  M(4,  C).  Clearly,  since  each 
MI  (A,  t)  is  real  algebraic  in  the  variables  Re  A//,  Im  A//  the  map  0//#  is  real  algebraic. 
Since  Y  is  complex  and  hence  real  algebraic,  its  inverse  image  ®JjK(Y),  defined  by  the 
real  algebraic  equations  obtained  upon  substitution  of  the  components  M/ (A,  f),  My  (A,  0, 
MK(A,  t)  in  the  equations  that  define  F,  is  also  real  algebraic.  Hence  the  set  Z^  is  a  real 
algebraic  subset  of  M(4,  C). 

To  see  that  Z^  is  a  proper  subset  of  M(4,  C),  we  simply  consider  the  matrix  (defined 
with  respect  to  the  standard  orthonormal  basis  {£/}£_  j  of  C4): 


A  = 


0100 
0010 
0001 
0000 


For  t  =  (tQ,  t\ ,  r2)  ^  0,  we  see  that 

tQ     t\     0     0 

/o      ^0      ^1       0 

0    r2    /o    ^i 
0     0    r2    'o 

For  the  above  matrix  the  minorial  determinant  M4i(A,  0  —  fj*,  whereas  Af]4(A,  0  =  f|. 
The  only  common  zeros  to  these  two  minorial  determinants  are  points  [tQ  :  0  :  0].  Setting 
t[  =  r2  =  0  in  the  matrix  above  gives  M//(A,  r)  =  ?Q  for  1  <  z  <  4.  Thus  ?o  must  also 
be  0  for  all  the  minorial  determinants  to  vanish.  Hence  the  matrix  A  above  lies  outside  the 
real  algebraic  set  Z^. 

It  is  well-known  that  a  proper  real  algebraic  subset  in  euclidean  space  cannot  have  a  non- 
empty interior.  Thus  the  complement  Z%  is  dense  and  open  in  the  classical  and  real-Zariski 
topologies.  Take  63  =  Z%. 

Finally,  set 


:=  5i  H  52  0  S3  = 


which  is  also  open  and  dense  in  the  classical  topology  in  M  (4,  C).  Hence  the  lemma.     D 


Remark2.1.  One  should  note  here  that  for  each  matrix  A  6  M (4,  C),  there  will  be  at  least  a 
curve  of  points  [t]  =  Oo  :  t\  :  t2]  e  P^  (defined  by  the  vanishing  of  det(^/H-^i  A +  ^  A*)), 
on  which  (IQ!  4- 1\ A  +  tiA*)  is  singular.  Similarly  for  each  A  there  is  at  least  a  curve  of 
points  on  which  the  trace  tr  (  /\"  (t$I  +  t\  A  -f-  £2^*))  vanishes,  and  so  a  non-empty  (and 
generally  a  finite)  set  on  which  both  these  polynomials  vanish,  by  dimension  theory  ([4], 


386  Vishwambhar  Pali 

Theorem  5,  p.  74).  Thus  for  each  A  e  M(4,  C),  there  is  at  least  a  non-empty  finite  set 
of  points  [t]  such  that  (fo/  -h  t\  A  +  ^2^*)  has  0  as  a  repeated  eigenvalue.  For  example, 
for  the  matrix  A  constructed  at  the  end  of  the  previous  lemma,  we  see  that  the  matrix 
(fo/  -f  t\  A-hfrA*)  is  strictly  upper-triangular  and  thus  has  0  as  an  eigenvalue  of  multiplicity 
4  for  all  (0,  t\>  0)  ^  0,  but  nevertheless  has  rank  3  for  all  (r0,  *i ,  ^2)  7^  (0,  0,  0). 

Indeed,  as  (iii)  of  the  lemma  above  shows,  for  A  in  the  open  dense  subset  5,  the  kernel 
ker(Yo/  -f  riA  +  foA*)  is  at  most  1  -dimensional  for  all  [t]  =  [to  :  t[  :  ti]  e  PQ. 

3.  The  varieties  C,  T,  andD 

Notation  3.1.  In  the  light  of  Lemmas  2.5  and  2.6  above,  we  shall  henceforth  assume  A  e  S. 
As  is  easily  verified,  this  implies  A*  e  S  as  well.  We  will  also  henceforth  assume,  in  view 
of  Lemma  2.4. above,  that  A  and  A*  have  no  common  eigenvectors.  (For  example,  this 
rules  out  A  being  normal,  in  which  case  we  know  that  the  main  result  for  A  is  true  by  the 
spectral  theorem.)  Also,  in  view  of  Lemma  2.2,  we  shall  assume  that  A  and  A*  do  not  have 
a  common  2-dimensional  invariant  subspace. 

In  PQ,  the  complex  projective  space  of  V  =  C4,  we  denote  the  equivalence  class  of 
v  €  V  \  0  by  [v].  For  a  [v]  6  PQ,  we  define  W([u])  (or  simply  W(v)  when  no  confusion 
is  likely)  by 

W([u])  :=  C-span(u,  AD,  A*u). 

Since  we  are  assuming  that  A  and  A*  have  no  common  eigenvectors,  we  have  dim  W([u])  > 
2  for  all  [u]  e  P;L 

Denote  the  four  distinct  points  in  PQ  representing  the  four  linearly  independent  eigen- 
vectors of  A  (resp.  A*)  by  E  (resp.  E*).  By  our  assumption  above,  E  fi  E*  =  <p. 

Lemma  3.2.  Let  A  e  Af(4,  C)  be  as  in  3.1  above.  Then  the  closed  subset: 
C  =  {[v]  e  P3C  :  v  A  Av  A  A*i>  =  0} 

is  a  closed  projective  variety.  This  variety  C  is  precisely  the  subset  of[v]  e  PQ  for  which 
the  dimension  dim  W([u])  =  dim  (C-span  {i>,  Av,  A*v})  is  exactly  2. 

Proof.  That  C  is  a  closed  projective  variety  is  clear  from  the  fact  that  it  is  defined  as  the  set 
of  common  zeros  of  all  the  four  (3  x  3)-minorial  determinants  of  the  (3  x  4)-matrix 


A  := 


v 
Au 

A*u 


(which  are  all  degree-3  homogeneous  polynomials  in  the  components  of  v  with  respect  to 
some  basis).  Also  C  is  nonempty  since  it  contains  E  U  E*. 

Also,  since  A  and  A*  are  nonsingular  by  the  assumptions  in  3.1,  the  wedge  product 
v  A  Au  A  A*v  of  the  three  non-zero  vectors  i>,  Au,  A*i>  vanishes  precisely  when  the  space 
W([v])  =  C-span(v,  AD,  A*u)  is  of  dimension  <  2.  Since  by  3.1,  A,  A*  have  no  common 
eigenvectors,  the  dimension  dim  W([v])  >  2  for  all  [v]  e  PQ,  so  C  is  precisely  the  locus 
of  [v]  €  P£  for  which  the  space  W([v])  is  2-dimensional.  D 


Unitary  tridiagonalization  in  Af  (4,  C)  387 

Now  we  shall  show  that  for  A  as  in  3.  1  ,  the  variety  C  defined  above  is  of  pure  dimension 
one,  For  this,  we  need  to  define  some  more  associated  algebraic  varieties  and  regular  maps. 

DEFINITION  3.3 

Let  us  define  the  bilinear  map: 

B  :  C4  x  C3     -»     C4 


We  then  have  the  linear  maps  B(vt  -)  :  C3  -»  C4  for  v  €  C4  and  B(-,  t)  :  C4  ~*  C3  for 

reC3. 

Note  that  the  image  Im  B(v,  —)  is  the  span  of  (i>,  Av,  A*v},  which  was  defined  to  be 
W(v).  For  a  fixed  t,  denote  the  kernel 

K(t)  :=  ker(£(-,  0  :  C4  ->  C4). 
Denoting  [/o  :  t\  :  ft]  by  [f  ]  and  [v\  :  V2  :  i>3  :  ^4]  by  [u]  for  brevity,  we  define 


Finally,  define  the  variety  D  by 

D  C  P£  :=  {[*]  €  P£  :  det  £(-,  r)  =  det  (f0/  +  1\  A  H-  r2A*)  =  0}. 
Let 

JTI  :  P3C  x  P°6  -^  P3  ,     7T2  :  P3C  x  P95  -*  P°c 
denote  the  two  projections. 

Lemma  3.4.  We  /zav£  the  following  facts: 
(i)  jri(r)  =  C,am/7r2(r)  =  D. 
(ii)  TTi  :  F  ->•  C  w  1-1,  and  the  map  g  defined  by 


w  a  regular  map  so  that  F  is  the  graph  of  g  and  isomorphic  as  a  variety  to  C. 
(iii)  D  C  PQ  is  a  plane  curve,  of  pure  dimension  one.  The  map  7t2  :  F  ~>  D  w  1-1, 
and  the  map  n\  on^~{  :  D  -+  C  is  the  regular  inverse  of  the  regular  map  g  defined 
above  in  (ii).  Again  F  is  also  the  graph  of  this  regular  inverse  g"1,  and  D  and  F  are 
isomorphic  as  varieties.  In  particular,  C  and  D  are  isomorphic  as  varieties,  and  thus 
C  is  a  curve  in  PQ  of  pure  dimension  one. 

(iv)  Inside  PQ  x  PQ,  each  irreducible  component  of  the  intersection  of  the  four  divisors 
DI  :=  (5/(v,  0  =  0)  for  i  =  1,  2,  3,4  (where  B/(v,  t)  is  the  i-th  component  of 
B(v,  t)  with  respect  to  a  fixed  basis  ofC4)  occurs  with  multiplicity  1.  (Note  that  F  is 
set-theoretically  the  intersection  of  these  four  divisors,  by  definition). 


388  Vishwambhar  Pali 

Proof.  It  is  clear  that  7r\(T)  =  C,  because  B(v,  t)  =  fyv  +  fi  An  +  tiA*v  =  0  for  some 
fro  :  ?i  :  ft]  €  PQ  iff  dim  W(v)  <  2,  and  since  A  and  A*  have  no  common  eigenvectors, 
this  means  dim  W(v)  =  2.  That  is,  [v]  e  C. 

Clearly  [f]  e  JT2(F)  iff  there  exists  a  [v]  €  PQ  such  that  £(u,  r)  =  0.   That  is,  iff 
dim  ker  £(-,  t)  >  1,  that  is,  iff 

G('o,fi,f2)  :=det  B(-,f)  =  0. 

Thus  D  =  7T2(F)  and  is  defined  by  a  single  degree  4  homogeneous  polynomial  G  inside 
PQ.  It  is  a  curve  of  pure  dimension  1  in  PQ  by  standard  dimension  theory  (see  [4],  p.  74, 
Theorem  5)  because,  for  example  [1  :  0  :  0]  &  D  so  D  ^  P|.  So  ^(F)  =  D,  and  this 
proves  (i). 

To  see  (ii),  for  a  given  [v]  6  C.  we  claim  there  is  exactly  one  [t]  such  that  ([i>],  [t])  e  F. 
Note  that  ([i>],  [f])  e  F  iff  the  linear  map: 

B(vt  -)  :  C3     -*     C4 


has  a  non-trivial  kernel  containing  the  line  G.  That  is,  dim  Im  B(v,  —  )  <  2.  But  the 
image  Im  #(u,  -)  =  W(u),  which  is  of  dimension  2  for  all  v  e  C  by  our  assumptions. 
Thus  its  kernel  must  be  exactly  one  dimensional,  defined  by  ker  B(v,  —  )  =  Ct.  Thus 
([u],  [/•])  is  the  unique  point  in  F  lying  in  n^l[v],  viz.  for  each  [u]  6  C,  the  vertical  line 
[f]  x  Pj,  intersects  F  in  a  single  point,  call  it  ([v],  g[v]).  Son\  :  F  ->  C  is  1-1,  and  F 
is  the  graph  of  a  map  g  :  C  ->  D.  Since  #([>])  =  7T27rj~l([u])  for  [v]  e  C,  and  F  is 
algebraic,  g  is  a  regular  map.  This  proves  (ii). 

To  see  (iii),  note  that  for  [t]  6  D,  by  definition,  the  dimension  dim  ker  B(—t  t)  >  1. 
By  the  fact  that  A  e  5,  and  (iii)  of  Lemma  2.6,  we  know  that  dim  ker  B  (—,/)<  1  for  all 
fr]  €  P£.  Thus,  denoting  tf(r)  :=  ker  £(-,  r)  for  [r]  e  D,  we  have 


dim  £(0  =  1     for  all     t  e  D.  (3) 

Hence  we  see  that  the  unique  projective  line  [v]  corresponding  to  Cv  =  K(t)  yields  the 
unique  element  of  C,  such  that  ([v],  [/])  6  F.  Thus  #2  :  F  -»  J9  is  1-1,  and  the  regular 
map  TT  i  o  jr^"1  :  D  -»  C  is  the  regular  inverse  to  the  map  g  of  (ii)  above.  F  is  thus  also  the 
graph  of  g~]  and,  in  particular,  is  isomorphic  to  D.  Since  g  is  an  isomorphism  of  curves, 
and  D  is  of  pure  dimension  1,  it  follows  that  C  is  of  pure  dimension  one.  This  proves  (iii). 

To  see  (iv),  we  need  some  more  notation. 

Note  that  D  c  Pg  \  {[1;  0;  0]},  (because  there  exists  no  [u]  €  P;L  such  that  Lv  =  0!). 
Thus  there  is  a  regular  map: 

e\D    ->     P<L 
fro  :  t{  :  *2]     H>     [fj  :  /2].  (4) 

Let  A(f  i  ,  ^2)  be  the  discriminant  polynomial  of  the  characteristic  polynomial  <ptl/\+t2A*  ^ 
t\A  +  tiA*.  Clearly  A(fi,  r2)  is  a  homogeneous  polynomial  of  degree  4  in  (t\  ,  ^2),  and  it  is 
not  the  zero  polynomial  because,  for  example,  A  (  1  ,  0)  ^  0,  for  A  (  1  ,  0)  is  the  discriminant 
°f  0/1,  which  has  distinct  roots  (=ihe  distinct  eigenvalues  of  A)  by  the  assumptions  3.1  on 
A.  Let  S  c  PQ  be  the  zero  locus  of  A,  which  is  a  finite  set  of  points.  Note  that  the  fibre 
^(U  :  At])  consists  of  all  [t  :  1  :  /x]  €  D  such  that  -t  is  an  eigenvalue  of  A  +  /M*, 


Unitary  tridiagonalization  in  Af  (4,  C)  389 

which  are  at  most  four  in  number.  Similarly  the  fibres  0~]  ([X  :  1])  are  also  finite.  Thus 
the  subset  of  D  defined  by 

F  :=e~}CZ) 

is  a  finite  subset  of  D.    F  is  precisely  the  set  of  points  [/]  =  [t$  :  t\   :  b]  such  that 
#(—  ,  t)  =  (t$I  •+•  t\  A  -h  /2^*)  has  0  as  a  repeated  eigenvalue. 
Since  7T2  :  F  ->  D  is  1-1,  the  inverse  image: 


is  a  finite  subset  of  F. 

We  will  now  prove  that  for  each  irreducible  component  FQ,  of  F,  and  each  point  x  = 
([«].  [&])  in  Fa  \  FI,  the  four  equations  {Z?/(u,  f)  =  0}4=1  are  the  generators  of  the  ideal  of 
the  variety  Ta  in  an  affine  neighbourhood  of  x,  where  #/  (v,  0  are  the  components  of  B(v,  t) 
with  respect  to  a  fixed  basis  of  C4.  Since  FI  is  a  finite  set,  this  will  prove  (iv),  because 
the  multiplicity  of  Fa  in  the  intersection  cycle  of  the  four  divisors  Z>/  =  (Z?/  (IF,  t)  =  0)  in 
PQ  x  P£  is  determined  by  generic  points  on  FQ,  ,  for  example  all  points  of  ra\F\.  We  will 
prove  this  by  showing  that  for  x  =  ([a],  [b])  €  Fff  \  Fj,  the  four  divisors  (£/(u,  0  =  0) 
intersect  transversely  at  x. 

So  let  Ta  be  some  irreducible  component  of  F,  with  x  =  ([#],  [£>])  eTa\F\. 

Fix  an  «  e  C4  representing  [^]  e  Ctt  :=  jr^Fa),  and  also  fix  Z?  €  C3  representing 
[b]  =  ^([fl])  6  £(Cff).  Also  fix  a  3-dimensional  linear  complement  V\  :—  7[rtj(P^)  C  C4 
to  <7  and  similarly,  fix  a  2-dimensional  linear  complement  VS  =  T[^](PQ)  C  C3  to  b.  (The 
notation  comes  from  the  fact  that  T[V](P'^)  ^  C"+l/Cv,  which  we  are  identifying  non- 
canonically  with  these  respective  complements  V/.)  These  complements  also  provide  local 
coordinates  in  the  respective  projective  spaces  as  follows.  Set  coordinate  charts  0  around 
[a]  e  P^by  [v]  —  <f>(u)  :=  [a  -f  M]»  and  ^around  [Z?]  €  Pgby  [t]  =  VCO  :=  [b+s],  where 
u  €  V\  ~  C3,  and  ^6  VS  —  C2.  The  images  0(Vj)  and  ^(^2)  are  affine  neighbourhoods 
of  [a]  and  [Z?]  respectively.  These  charts  are  like  'stereographic  projection'  onto  the  tangent 
space  and  depend  on  the  initial  choice  of  a  (resp.  b)  representing  [a]  (resp.  [b]),  and  are  not 
the  standard  coordinate  systems  on  projective  space,  but  more  convenient  for  our  purposes. 

Then  the  local  affine  representation  of  B(v,  t)  on  the  affine  open  V\  x  V^  =  C3  x  C2, 
which  we  denote  by  /3,  is  given  by 


Note  that  ker  B(a,  ~)  =  C/?5  where  [Z?]  =  g([a]),  so  that  5(a,  —  )  passes  to  the  quotient 
as  an  isomorphism: 


(5) 

where  W(d)  is  2-dimensional. 

Similarly,  since  B(—,  b)  has  one  dimensional  kernel  Ca  =  K(b)  C  C4,  by  (3)  above, 
we  also  have  the  other  isomorphism: 

£(-,  b):V\-Z=+  Im  B(-,  b),  (6) 

where  Im  £(-,  ^)  is  3-dimensional,  therefore. 


390  Vishwambhar  Pati 

Now  one  can  easily  calculate  the  derivative  D/J(0,  0)  of  ft  at  (u,s)  =  (0,0).    Let 
(X,  F)  e  V\  x  VS.  Then,  by  bilinearity  of  5,  we  have 


)  +  B(a,  Y)  H-  £(X,  F). 
Now  since  5(X,  Y)  is  quadratic,  it  follows  that 

Dp(Q,  0)  :  Vj  x  V2     -*     C4 

(X,  7)     H*     B(X,  fc)  +  5(a,  F).  (7) 


By  eqs  (5)  and  (6)  above,  we  see  that  the  image  of  D/3(0,  0)  is  precisely  Im  B(—  ,  Z?) 
W(a). 

Claim.  For  ([a],  [b])  eroe\Fi,  the  space  Im  £(-,  b)  -f  W(<3)  is  all  of  C4. 


Proof  of  Claim.  Denote!  :=  £(-,/?)  for  brevity.  Clearly  a  6  W(tf)  by  definition  of 
W(a).  Also,<2  €  kerT  =  K(b).  We  claim  that  a  is  not  in  the  image  of  T.  For,  if  a  €  ImT, 
we  would  have  a  =  Tw  for  some  w  &  K(b)  =  ker  T  and  if  ^  0.  In  fact  w  is  not  a 
multiple  of  a  since  Tw  =  a  ^  0  whereas  (7  e  ker  7.  Thus  we  would  have  T2w  =  0,  and 
completing  f{=a  =  Tw,  fa  =  wj  to  a  basis  {//}f=1  of  C4,  the  matrix  of  T  with  respect 
to  this  basis  would  be  of  the  form: 

0  1  *  * 

0  0  *  * 

0  0  *  * 

0  0  *  * 

Thus  T  =  5(—  ,  b)  would  have  0  as  a  repeated  eigenvalue.  But  we  have  stipulated  that 
(M>  [b])  &  FI,  so  that  [6]  £  F,  and  hence  £(-,  6)  does  not  haveO  as  a  repeated  eigenvalue. 
Hence  the  non-zero  vectors  6  W(a)  is  not  in  Im  T.  Since  Im  T  is  3-dimensionaI,  we  have 
C4  =  Im  T  -f  W(fl),  and  this  proves  the  claim.  d 


In  conclusion,  all  the  points  of  Fa  \  FI  are  in  fact  smooth  points  of  Pa,  and  the  local 
equations  for  Fa  in  a  small  neighbourhood  of  such  a  point  are  precisely  the  four  equations 
ft  (M,  j)  =  0,  1  <  /  <  4.  This  proves  (iv),  and  the  lemma.  a 

4.  Some  algebraic  bundles 

We  construct  an  algebraic  line  bundle  with  a  (regular)  global  section  over  C.  By  showing 
that  this  line  bundle  has  positive  degree,  we  will  conclude  that  the  section  has  zeroes  in  C. 
Any  zero  of  this  section  will  yield  a  flag  of  the  kind  required  by  Lemma  2.  1  .  One  of  the 
technical  complications  is  that  none  of  the  bundles  we  define  below  are  allowed  to  use  the 
hermitian  metric  on  V,  orthogonal  complements,  orthonormal  bases  etc.,  because  we  wish 
to  remain  in  the  C-algebraic  category.  As  a  general  reference  for  this  section  and  the  next, 
the  reader  may  consult  [2]. 


Unitary  tridiagonalization  in  M(4,  C)  39  1 

DEFINITION  4.1 

For  0  3=  v  e  V  —  C4,  we  will  denote  the  point  [v]  e  PQ  by  v,  whenever  no  confusion  is 
likely,  to  simplify  notation.  We  have  already  denoted  the  vector  subspace 
C-span(u,  An,  A*u)  C  C4  as  W(v).  Further  define  W3(v)  :=  W(v)  +  AW(v),  and 
:=  W(u)  -f  A*  W(v).  Clearly  both  W3(u)  and  W3(u)  contain  W(v). 


Since  A  and  A*  have  no  common  eigenvectors,  we  have  dim  W(v)  >  2  for  all  v  €  PQ, 
and  dim  W(v)  =  2  for  all  u  6  C,  because  of  the  defining  equations  A  Av  A  A*  v  —  Oof  C. 
Also,  sincedim  W(u)  =  2  =  dim  AW(v)forv  e  C,  and  since  0  ^  An  €  W(u)HAW(v), 
we  have  dim  W3(u)  <  3  for  all  v  e  C.  Similarly  dim  W^(v)  <  3  for  all  u  e  C. 

If  there  exists  a  u  e  C  such  that  dim  W$(v)  =  2,  then  we  are  done.  For,  in  this  case 
Wi(v)  must  equal  W(v)  since  it  contains  W(v).  Then  the  dimension  dim  W$(v)  =  2  or 
=  3.  If  it  is  2,  W(u)  will  be  a  2-dimensional  invariant  space  for  both  A  and  A*,  and  the 
main  theorem  will  follow  by  Lemma  2.2.  If  dim  ^3(1;)  =  3,  then  the  flag: 

0  =  WQ  C  W[  =  Cv  C  W2  =  W(v)  CW3  =  W3(v)  C  W4  =  V 

satisfies  the  requirements  of  (ii)  in  Lemma  2.1,  and  we  are  done.  Similarly,  if  there  exists 
a  v  e  C  with  dim  W^(v)  =  2,  we  are  again  done.  Hence  we  may  assume  that: 


dim  W3(v)  =  dim  W$(v)  =  3    for  all    veC.  (8) 

In  the  light  of  the  above,  we  have  the  following: 

Remark  4.2.  We  are  reduced  to  the  situation  where  the  following  condition  holds:  For  each 
v  €  C,  dim  W(v)  =  2,  dim  W3(u)  =  dim  W3(v)  =  3. 

Now  our  main  task  is  to  prove  that  there  exists  a  v  e  C  such  that  the  two  3-dimensional 
subspaces  W$(v)  and  W^(v)  are  the  same.  In  that  event,  the  flag 

0  =  W0  C  Wi  =  Cu  C  W2  =  W(v)  CW3  =  W(v)  +  AW(v)  =  W(v) 
+A*W(u)  C  W4  =  V 

will  meet  the  requirements  of  (ii)  of  the  Lemma  2.1.  The  remainder  of  this  discussion  is 
aimed  at  proving  this. 

DEFINITION  4.3 


Denote  the  trivial  rank  4  algebraic  bundle  on  PjL  by  O43  ,  with  fibre  V  =  C4  at  each  point 

c 
(following  standard  algebraic  geometry  notation).  Similarly,  O^  is  the  trivial  bundle  on 

C.  In  O43  ,  there  is  the  tautological  line-subbundle  C?p3  (—  1),  whose  fibre  at  v  is  Cu.  Its 

pc  c 

restriction  to  the  curve  C  is  denoted  as  W\  := 


There  are  also  the  line  subbundlesAOp3  (-1)  (respectively  A*OP3  (—  l))of  O43  ,  whose 

c  c  "^c 

fibre  at  v  is  Av  (respectively  A*u).  Both  are  isomorphic  to  OP3  (—1)  (via  the  global  linear 

automorphisms  A  (resp.  A*)  of  V).  Similarly,  their  restrictions  AOc(—  1).  A*Oc(~l)» 
both  isomorphic  to  Oc(—  1).  Note  that  throughout  what  follows,  bundle  isomorphism  over 
any  variety  X  will  mean  algebraic  isomorphism,  i.e.  isomorphism  of  the  corresponding 
sheaves  of  algebraic  sections  as  Ox  -modules. 


392  Vishwambhar  Pali 

Denote  the  rank  2  algebraic  bundle  with  fibre  W(v)  c  V  at  v  e  C  as  W2.  It  is  an 
algebraic  sub-bundle  of  O^,  for  its  sheaf  of  sections  is  the  restriction  of  the  subsheaf 

Ops  (-1)  +  AOpa  (-1)  +  A*0P3  (-1)  C  O* 

C  C  C  "Q 

to  the  curve  C,  which  is  precisely  the  subvariety  of  PQ  on  which  the  sheaf  above  is  locally 
free  of  rank  2  (=rank  2  algebraic  bundle). 

Denote  the  rank  3  algebraic  sub-bundle  of  O%  with  fibre  W^(v)  =  W(v)  +  AW(v) 
(respectively  W$(v)  =  W(v)  +  A*  W(v))  by  Ws  (respectively  Ws).  Both  Ws  and  VVs  are 
of  rank  3  on  C  because  of  Remark  4.2  above,  and  both  contain  W2  as  a  sub-bundle.  We 
denote  the  line  bundles  f\2  Wo  by  £2>  and  /\3  VVa  (resp.  /\3  $3)  by  £3  (resp.  £3).  Then 
£2  is  a  line  sub-bundle  of  A2  #£,  and  £3,  £3  are  line  sub-bundles  of  /\3  O^. 

Finally,  for  X  any  variety,  with  a  bundle  €  on  J£  which  is  a  sub-bundle  of  a  trivial  bundle 
O'x  ,  the  annihilate  r  of  £  is  defined  as 


Ann£  =  {<£  E  hom^(^,  Ox)  :  0(f  )  =  0}. 
Clearly,  by  taking  homx(-  ,  Ox)  of  the  exact  sequence 

0-»£->0'«  _»  O%/£->  0, 
the  bundle 

Ann£  -  homx(^/^,  O/)  =  (O'$/£)*, 
where  *  always  denotes  the  (complex)  dual  bundle. 

Lemma  4.4.  Denote  the  bundle  W3/W2  (resp.  W^/Wi)  /?y  A  (r^p.  A).  Then  we  have 
the  following  identities  of  bundles  on  C: 

(i) 

0    -^W2-»W3-»A-»0 
0    ->  Wo  -+  VV3  ~>  A  -»  0 

0    -»  £3  -V  AnnW2  -^  A*  ->  0 
0    ->  £3  4  AnnW2  A  A*  -»  0, 

(ii) 

£3  ±1  £2  ®  A    and    £3  ~  £2  ®  A, 
(iii) 


(iv) 

A  ~  A, 


Unitary  tridiagonalization  in  M(4,  C)  393 

(v) 


£2  ~  A  <g>  Oc(-\)  ^  A 
(vi) 

homc(£3,  A*)  c±  £*  (2)  A*2  -  £?3  ®  Oc(-2). 

Proof.  From  the  definition  of  A,  we  have  the  exact  sequence: 

0-*W2-*W3-»A-*0 
from  which  it  follows  that: 

0  -»  A  ~>  O£/W2  -»  e>c/W3  ->  0 
is  exact.  Taking  homc(—  ,  OQ)  of  this  exact  sequence  yields  the  exact  sequence: 

0  ->  AnnW3  -*  AnnW2  -»  A*  ->  0. 

Now,  via  the  canonical  isomorphism  /\  V  ~»  V*  which  arises  from  the  non-degenerate 
pairing 

3  4 

/\V®V->/\V-C, 

it  is  clear  that  AnnWa  —  A3  Ws  =  £3. 

Thus  the  first  and  third  exact  sequences  of  (i)  follow.  The  proofs  of  the  second  and  fourth 
are  similar.  From  the  first  exact  sequence  in  (i),  it  follows  that  /\3  Ws  ~  /\2  W2  ®  A. 
This  implies  the  first  identity  of  (ii).  Similarly  the  second  exact  sequence  of  (i)  implies  the 
other  identity  of  (ii). 

Since  for  every  line  bundle  y  ,  y  <8>  y*  is  trivial,  we  get  from  the  first  identity  of  (ii)  that 
£2  ~  £3  <8>  A*.  From  third  exact  sequence  in  (i)  it  follows  that  /\~  AnnWo  ~  £3  (8)  A*, 
and  this  implies  (iii). 

To  see  (iv),  note  that 

W2  +  AW2  _ 


™      W2      ~  AW2n>v2' 

The  automorphism  A~  ]  of  V  makes  the  last  bundle  on  the  right  isomorphic  to  the  line  bundle 
W2/(W2  Pi  A~  !  VV2)  (note  all  these  operations  are  happening  inside  the  rank  4  trivial  bundle 
O*c).  Similarly,  A  is  isomorphic  (via  the  global  isomorphism  A*  ~  J  of  V)  to  the  line  bundle 
W2/(W2nA*-1W2).  Butforeachi;  €  C,  W(u)nA~I  W(v)  =  Cv  =  W^nA*-1  W(v), 
from  which  it  follows  that  the  line  sub-bundles  W2  O  A"1  W2  and  W2  H  A*"1  W2  of  W2 
are  the  same  (=  Wi  ~  Oc(—  1))-  Thus  A  ~  A,  proving  (iv). 

To  see  (v),  we  need  another  exact  sequence.  For  each  v  €  C,  we  noted  in  the  proof  of 
(iv)  above  that  Cv  =  W(v)  O  A~~l  W(u).  Thus  the  sequence  of  bundles: 

Wo 
0  _>  Oc(-l)  ->  W9  ->  -  ~  --  >  0 

^    J        ~ 


is  exact.  But,  as  we  noted  in  the  proof  of  (iv)  above,  the  bundle  on  the  right  is  isomorphic 
to  A,  so  that 

i)->W2->  A-»0 


394  Vishwambhar  Pati 

is  exact.  Hence  £2  =  A2  ^2  —  A  ®  Oc(—  1).  The  other  identity  follows  from  (iv),  thus 
proving  (v). 

To  see  (vi)  note  that  we  have  by  (ii)  C\  ~  C\  ®  A*.  Thus 

home  (£3,  A*)  ^  £3  ®  A*  c±  ££  (8)  A*  ®  A*. 

However,  since  by  (iv),  A  c±  A,  we  have  home  (£3,  A*)  2±  £*  <g>  A*2.  Now,  substituting 
A*  =  £*  <8>  0c(—  1)  fr°m  (v)»  we  have  the  rest  of  (vi).  Hence  the  lemma.  D 

We  need  one  more  bundle  identity: 

Lemma  4.5.  TTzere  is  a  bundle  isomorphism: 


Proof.  When  [t]  =  [/o  :  ^i  '  *2]  =  ^(M),  we  saw  in  (5)  that  the  linear  map 
B(v,  —  )  :  C3  ->•  C4  acquires  a  1-dimensional  kernel,  which  is  precisely  the  line  O,  which 
is  the  fibre  of  GD(—  1)  at  [r].  The  image  of  B(v,  —  )  was  the  2-dimensional  span  Wf(i;)  of 
v,  Av,  A*v,  as  noted  there.  Thus  for  v  £  C,  B(—  ,  —)  induces  a  canonical  isomorphism  of 
vector  spaces: 


which,  being  defined  by  the  global  map  B(—,  —  ),  gives  an  isomorphism  of  bundles: 

0c(-08**(02>/0D(-i))  ^  vw. 

From  the  short  exact  sequence: 

0  ->  0D(-1)  -^  O^  O|>/OD(-I)  -^  0, 
it  follows  that  /\2(O3D/OD(-l))  -  OD(1).  Thus: 


This  proves  the  lemma.  Q 

5.  Degree  computations 

In  this  section,  we  compute  the  degrees  of  the  various  line  bundles  introduced  in  the  previous 
section. 

DEFNITION5.1 

Note  that  an  irreducible  complex  projective  curve  C,  as  a  topologicai  space,  is  a  canon- 
ically  oriented  pseudomanifold  of  real  dimension  2,  and  has  a  canonical  generator  /zc  € 
#2(C,  Z)  =  Z.  Indeed,  it  is  the  image  TT*/^,  where  n  :  C  ->  C  is  the  normalization 
map,  and  /xg-e  ^(C,  Z)  =  Z  is  the  canonical  orientation  class  for  the  smooth  connected 


Unitary  tridiagonalization  in  M(4,  C)  395 

compact  complex  manifold  C,  where  jr*  :  H2(C,  Z)  —  >  //2(C,  Z)  is  an  isomorphism  for 
elementary  topological  reasons. 

If  C  =  (Jra=lCa  is  a  projective  curve  of  pure  dimension  1,  with  the  curves  Ca  as 
irreducible  components,  then  since  the  intersections  Ca  fl  C/j  are  finite  sets  of  points  (or 
empty),  Hi(C,  Z)  =  ®aH2(Ca,  Z).  Letting  /ZQT  denote  the  canonical  orientation  classes 
of  Ca  as  above,  there  is  a  unique  class  JJLC  =  X^  Ma  6  //2(C,  Z).  Thinking  of  C  as  an 
oriented  2-pseudomanifold,  //e  is  just  the  sum  of  all  the  oriented  2-simplices  of  C. 

If  f  is  a  complex  line  bundle  on  C,  it  has  a  first  Chern  class  c  \  (J-)  e  H~(X,  Z),  and  the 
degree  of  JF  is  defined  by 


It  is  known  that  a  complex  line  bundle  on  a  pseudomanifold  is  topologically  trivial  iff  its 
first  Chern  class  is  zero.  In  particular,  if  an  algebraic  line  bundle  on  a  projective  variety 
has  non-zero  degree,  then  it  is  topologically  (and  hence  algebraically)  non-trivial. 

Finally,  if  z  :  C  <->•  PQ  is  an  (algebraic)  embedding  of  a  curve  in  some  projective  space, 
we  define  the  degree  of  the  bundle  OcO)  =  z*Op«  (1)  as  the  degree  of  the  curve  C  (in 
P£).  We  note  that  [C]  :=  z*(//c)  e  //2(Pc,  Z)  is  called  tt\e  fundamental  class  of  C  in 
P£,  and  by  definition  deg  C  =  (ci(0eO)),  Mc>  =  (ci(Op£(l)),  [C]V  Geometrically,  one 
intersects  C  with  a  generic  hyperplane,  which  intersects  C  away  from  its  singular  locus  in 
a  finite  set  of  points,  and  then  counts  these  points  of  intersection  with  their  multiplicity. 

More  generally,  a  complex  projective  variety  X  c  PQ  of  complex  dimension  m  has  a 
unique  orientation  class  /.JLX  €  H2m(X,  Z).  Its  image  in  /^(PQ,  Z)  is  denoted  [X],  and 

the  degree  degX  of  X  is  defined  as  /(c  1(0^(1  )))'",  [X]V  It  is  known  that  if  X  =  V(F) 
for  a  homogeneous  polynomial  F  of  degree  d,  then  deg  X  =  d. 
We  need  the  following  remark  later  on. 

Remark5.2.  Iff  :  C  ->  £>  is  a  regular  isomorphism  of  complex  projective  curves  C  and  D, 
both  of  pure  dimension  i,and  if  T  is  a  complex  line  bundle  on  D,  then  deg  /*JF  =  degJF. 
This  is  because  jf*(/zc)  =  MD,  so  that 


Now  we  can  compute  the  degrees  of  all  the  line  bundles  introduced. 

Lemma  5.3,  The  degrees  of  the  various  line  bundles  above  are  as  follows: 

(i)  degOc(l) 

(ii)  degOD(l)  = 
(iii)  deg£*  =  8 
(iv)  deg  home  (£3,  A*)  =  deg  (£*3  0  Oc(-2))  =  12. 


/.  We  denote  the  image  of  orientation  class  /^r  of  the  curve  F  (see  Definition  3.3  for 
the  definition  of  F)  in  #2(^0  x  PQ,  Z)  by  [F].  By  the  part  (iv)  of  Lemma  3.4,  we  have  that 
the  homology  class  [F]  is  the  same  as  the  homology  class  of  the  intersection  cycle  defined 


396  Vishwambhar  Pati 

by  the  four  divisors  £>/  :=  (£/(u,  0  =  0)  inside  #2  (PC  x  PC*  Z)-  By  the  generalized 
Bezout  theorem  in  PjL  x  Pji,  the  homology  class  of  the  last-mentioned  intersection  cycle 
is  the  homology  class  Poincare-dual  to  the  cup  product 

d  :=  d\  U  di  U  d^  U  c/4, 

where  d/  is  the  first  Chern  class  of  the  the  line  bundle  L\  corresponding  to  D/,  for  i  = 
1,2,  3,  4  (see  [4],  p.  237,  Ex.  2). 
Since  each  5/(u,  f)  is  separately  linear  in  u,  r,  the  line  bundle  defined  by  the  divisor 

DI  is  the  bundle  rc*O^  (1)  ®  x^Opz  (1),  where  ;TI,    7r2  are  the  projections  to  Pi  and 

c  **      c 

P£  respectively.  If  we  denote  the  hyperplane  classes  which  are  the  generators  of  the 
cohomologies  //2(P^,  Z)  and  //2(P^,  Z)  by  jc  and  y  respectively,  we  have 


Then  we  have,  from  the  cohomology  ring  structures  of  PQ  and  P£  that  ;c  U  x  U  jt  U  x  = 
'  U  >'  U  y  =  0.  Hence  the  cohomology  class  in  //8(Pjl  x  P£,  Z)  given  by  the  cup-product 
is 

d  :=  rfi  U  J2  U  d3  Ud4  =  (Jrfto  4-  TrjOO)4  -  4^r*U3)7r2*(j)  +  6jr*(;t2)7r2*(>'2), 


where  jc3  =  x  U  jc  U  z  .  .  .  etc.  By  part  (ii)  of  Lemma  3.4,  the  map  n\  :  F  ->  C  is  an 
isomorphism,  so  applying  the  Remark  5.2  to  it,  we  have 

-     deg7r*0c(l) 


+  67r*(jc2)7r*(y2))  ,  [P3,  x  P2] 


=     6,  (9) 

where  we  have  used  the  Poincare  duality  cap-product  relation  [F]  =  [P£  x  P|]  fi  d 
mentioned  above,  and  that  TrfOc3)  U  7r^(j2)  is  the  generator  of  H[0(P3C  x  P2,  Z),  so 
evaluates  to  1  on  the  orientation  class  [P^  x  P^"|,  and  ;c4  =  0.  This  proves  (i). 

The  proof  of  (ii)  is  similar,  we  just  replace  C  by  D,  and  n\  by  7T2,  and  n*(x)  by  <(>')  in 
the  equalities  of  (9)  above,  and  get  4  (as  one  should  expect,  since  D  is  defined  by  a  degree 
4  homogeneous  polynomial  in  P^).  This  proves  (ii). 

For  (iii),  we  use  the  identity  of  Lemma  4.5  that  £2  =  Oc(-2)  ®  g*e>D(l),  and  the 
Remark  5.2  applied  to  the  isomorphism  of  curves  g  :  C  ->  D  (part  (iii)  of  Lemma  3.4) 
to  conclude  that  deg  £2  =  deg  D  -  2deg  C  =  4  -  1  2  =  -8,  by  (i)  and  (ii)  above,  so  that 
deg£*  =  8. 

For  (iv),  we  have  by  (vi)  of  Lemma  4.4  that  homc(£3,  A*)  ~  £*3  ®  C?c(-2),  so  that 
its  degree  is  3deg£*  -  2degC  =  24  -  12  =  12  by  (i)  and  (iii)  above. 

This  proves  the  lemma.  D 

From  (iv)  of  the  lemma  above,  we  have  the  following. 


Unitary  tridiagonalization  in  M(4,  C)  397 

COROLLARY  5.4 

The  line  bundle  home  (£3,  A*)  is  a  non-trivial  line  bundle. 

6.  Proof  of  the  main  theorem 

Proof  of  Theorem  1.1.  By  the  third  and  fourth  exact  sequences  in  (i)  of  Lemma  4.4,  we 
have  a  bundle  morphism  s  of  line  bundles  on  C  defined  as  the  composite: 

AnnW3  =  £3  -4  AnnW2  -+  A*  =  AnnW2/AnnVV3 


which  vanishes  at  v  e  C  if  and  only  if  the  fibre  AnnH^  is  equal  to  the  fibre  AnnWa  (U 
inside  AnnW2,u-  At  such  a  point  i>  e  C,  we  will  have  AnnH^y  =  AnnVV^,  so  that 
W3(u)  =  W3,u'=  W(u)  +  AW(v)  =  VV3,u  =  W(v)  +  A*W(v)  =  W3(y). 

Now,  this  morphi  sm  s  is  a  global  section  of  the  bundle  homc(£3,  A*),  which  is  not  a 
trivial  bundle  by  Corollary  5.4  of  the  last  section.  Thus  there  does  exist  a  v  e  C,  satisfying 
,$•(1;)  =  0,  and  consequently  the  flag 

0  C  W\  :=  Wi  ,„      =     Cv  C  W2  :=  W2,u  =  W(v)  =  C-span{i;,  Av,  A*u} 
C  W3     :=     W3(u)  =  W(v)  +  AW(u) 
C  W4=  V  =C4 


satisfies  the  requirements  of  (ii)  of  Lemma  2.1,  (as  noted  after  Remark  4.2)  and  the  main 
theorem  1  .  1  follows.  D 

Remark  6.  1  .  Note  that  since  dim  C  =  1  ,  the  set  of  points  v  e  C  such  that  s(v)  =  0,  where 
s  is  the  section  above,  will  be  a  finite  set.  Then  the  set  of  flags  that  satisfy  (ii)  of  Lemma 
2.1  which  tridiagonalize  A  of  the  kind  considered  above  (viz.  A  satisfying  the  assumptions 
of  3.1),  will  only  be  finitely  many  (at  most  12  in  number!). 

Acknowledgments 

The  author  is  grateful  to  Bhaskar  Bagchi  for  posing  the  problem,  and  to  B  V  Rajarama 
Bhat  and  J  Holbrook  for  pointing  the  relevant  literature.  The  author  is  also  deeply  grateful 
to  the  referee,  whose  valuable  comments  have  led  to  the  elimination  of  grave  errors,  and  a 
substantial  streamlining  of  this  paper. 

References 

[1]  Fong  C  K  and  Wu  P  Y,  Band  Diagonal  Operators,  Linear  Algebra  Appl.  248  (1996)  195-204 
[2]  Hartshorne  R,  Algebraic  Geometry,  Springer  GTM  52  (1977) 

[3]  Longstaff  WE,  On  tridiagonalisation  of  matrices,  Linear  Algebra  Appi  109(1988)  153-163 
[4]  Shafarevich  I  R,  Basic  Algebraic  Geometry,  2nd  Edition  (Springer  Verlag)  (1994)  vol.  1 


Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  1 1 1,  No.  4,  November  2001,  pp.  399-405. 
©  Printed  in  India 


On  Ricci  curvature  of  C-totally  real  submanifolds  in 
Sasakian  space  forms 

LIU  XIMIN 

Department  of  Applied  Mathematics,  Dalian  University  of  Technology,  Dalian 

116  024,  China 

Present  address:  Department  of  Mathematical  Sciences,  Rutgers  University,  Camden, 

New  Jersey  08 102,  USA 

E-mail:  xmliu@dlut.edu.cn;  xmliu@camden.rutgers.edu 

MS  received  26  February  2001 

Abstract.  Let  M'1  be  a  Riemannian  n-manifold.  Denote  by  S(p)  and  Ric(/?)  the 
Ricci  tensor  and  the  maximum  Ricci  curvature  on  M",  respectively.  In  this  paper 
we  prove  that  every  C-totally  real  submanifold  of  a  Sasakian  space  form  M2m+l  (c) 

satisfies  5  <  (("~1)4(c"l"3)  -f  fJ^H2)g,  where  H2  and  g  are  the  square  mean  curvature 
function  and  metric  tensor  on  M11,  respectively.  The  equality  holds  identically  if  and 
only  if  either  M"  is  totally  geodesic  submanifold  orn  =  2  and  Mn  is  totally  umbilical 
submanifold.  Also  we  show  that  if  a  C-totally  real  submanifold  Mn  of  M2n+](c) 
satisfies  Ric  =  (/t~1)4(c'+3)  -f  '-^-H2  identically,  then  it  is  minimal. 

Keywords.     Ricci  curvature;  C-totally  real  submanifold;  Sasakian  space  form. 

1.  Introduction 

Let  M n  be  a  Riemannian  rc-manifold  isometrically  immersed  in  a  Riemannian  m-manifold 
Mm  (c)  of  constant  sectional  curvature  c.  Denote  by  g ,  R  and  h  the  metric  tensor,  Riemann 
curvature  tensor  and  the  second  fundamental  form  of  Mn,  respectively.  Then  the  mean 
curvature  vector  H  of  M72  is  given  by  H  =  -trace h.  The  Ricci  tensor  5  and  the  scalar 
curvature  p  at  a  point  p  e  Mn  are  given  by  S(X,  Y)  =  ££_|  {/?(£/,  ^^'  e'^  anc*  ^  ^ 
5I/=I  ^(*/»  e/)»  respectively,  where  {£],...,  en]  is  an  orthonormal  basis  of  the  tangent 
space  TpMn.  A  submanifold  M"  is  called  totally  umbilical  if  /i,  H  andg  satisfy  h(X,  Y)  = 
g(X,  Y)H  forX,  Y  tangent  to  Mn. 
The  equation  of  Gauss  for  the  submanifold  Mn  is  given  by 


)    =    c(g(X,W)g(Y,Z)-g(X,Z)g(Y,W)) 

+  g(h(X9  W),  h(Y,  Z))  -  g(h(X,  Z),  h(Y,  W))9      (1) 

where  X,  7,  Z,  W  e  TMn.  From  (1)  we  have 

p=n(n  -  l)c  +  n2H2-\h\2,  (2) 

where  \h\-  is  the  squared  norm  of  the  second  fundamental  form.  From  (2)  we  have 

P<n(n-  l)c  +  n2H2, 
with  equality  holding  identically  if  and  only  if  M77  is  totally  geodesic. 

399 


400  Liu  Ximin 

Let  Ric(p)  denote  the  maximum  Ricci  curvature  function  on  Mn  defined  by 
=  max{S(n,  u)\u  €  TM"  ,    p  e  Mn], 


where  T*Mn  =  [v  e  TpMn\(v,  v)  =  [}. 

In  [3],  Chen  proves  that  there  exists  a  basic  inequality  on  Ricci  tensor  S  for  any  subman- 
ifold M'7  in  Af'"(c),  i.e. 

/  n2     A 

S<  l(n  -\)c+—H2\g,  (3) 

with  the  equality  holding  if  and  only  if  either  Mn  is  a  totally  geodesic  submanifold  or  n  =  2 
and  Mn  is  a  totally  umbilical  submanifold.  And  in  [4],  Chen  proves  that  every  isotropic 
submanifold  Mn  in  a  complex  space  form  Mm(4c)  satisfies  Ric  <  (n  -  l)c  +  j-#2, 
and  every  Lagrangian  submanifold  of  a  complex  space  form  satisfying  the  equality  case 
identically  is  a  minimal  submanifold.  In  the  present  paper,  we  would  like  to  extend  the 
above  results  to  the  C  -totally  real  submanifolds  of  a  Sasakian  space  form,  namely,  we 
prove  that  every  C  -totally  real  submanifold  of  a  Sasakian  space  form  M2m+}(c)  satisfies 
S  <  ((/?"1)4(6'+3)  +  ^#2)s,  and  the  equality  holds  identically  if  and  only  if  either  Mn  is 
totally  geodesic  submanifold  or  n  =  2  and  Mn  is  totally  umbilical  submanifold.  Also  we 
show  that  if  a  C  -totally  real  submanifold  Mn  of  a  Sasakian  space  form  Af2"+1(c)  satisfies 
Ri£  =  («-0(c+3)  +  njLn2  identically,  then  it  is  minimal. 

2.  Preliminary 

Let  M2"1*1  be  an  odd  dimensional  Riemannian  manifold  with  metric  g.  Let0  be  a  (1,1)- 
tensor  field,  £  a  vector  field,  and  r\  a  1-form  on  M2/"+l,  such  that 


02X  =  - 

£(0X,  07)  =  g(X,  Y)  -  ri(X)r)(Y)  ,    rj(X)  =  g(X,  £). 

If,  in  addition,  drj(Xt  Y)  =  g((/)X,  7),  for  all  vector  fields  X,  Y  on  M2in+[,  then  M2m+{ 
is  said  to  have  a  contact  metric  structure  (0,  £,  77,  g),  and  M2m+I  is  called  a  contact  metric 
manifold.  If  moreover  the  structure  is  normal,  that  is  if  [0X,  07]  +  02[X,  7]  -0[X,  07]  - 
0[0X,  7]  =  -2d77(X,  7)£,  then  the  contact  metric  structure  is  called  a  Sasakian  structure 
(normal  contact  metric  structure)  and  M2/"+1  is  called  a  Sasakian  manifold.  For  more 
details  and  background,  see  the  standard  references  [I]  and  [8]. 

A  plane  section  a  in  TpM2/n+{  of  a  Sasakian  manifold  M2m+l  is  called  a  ^-section  if 
it  is  spanned  by  X  and  0X,  where  X  is  a  unit  tangent  vector  field  orthogonal  to  £.  The 
sectional  curvature  K(a)  with  respect  to  a  0-section  cr  is  called  a  0-sectional  curvature. 
If  a  Sasakian  manifold  M2m+l  has  constant  0-sectional  curvature  c,  A/2'"4"  !  is  called  a 
Sasakian  space  form  and  is  denoted  by  M2m+l(c), 

The  curvature  tensor  R  of  a  Sasakian  space  form  M2"1^1  (c)  is  given  by  [8] 

R(X,Y)Z    =     CJl 


-  i1(Y)rj(Z)X  +  g(X,  Z)r)(Ytf  -  gff, 


,  Z)0X  -  £(0X,  Z)07  -  2^(0X,  7)0Z), 


Sasakian  space  forms  401 

for  any  tangent  vector  fields  X,  7,  Z  to  M2m+l  (c). 

An  rc  -dimensional  submanifold  Mn  of  a  Sasakian  space  form  M2;"+1(c)  is  called  a 
C-totally  real  submanifold  of  M2'"~*~l(c)  if  £  is  a  normal  vector  field  on  A/".  A  direct 
consequence  of  this  definition  is  that  4>(T  Mn]  C  T^-Mn9  which  means  that  Mn  is  an 
anti-invariant  submanifold  of  M2m+  !  (c).  So  we  have  n  <  wz. 

The  Gauss  equation  implies  that 


4 

+  g(h(X,  W),  h(Y,  Z))  -  g(h(X,  Z),  /z(7,  W)),       (4) 

for  all  vector  fields  X,  7,  Z,  W  tangent  to  M",  where  /z  denotes  the  second  fundamental 
form  and  R  the  curvature  tensor  of  Mn  . 

Let  A  denote  the  shape  operator  on  Mn  in  M2m+1(c).  Then  A  is  related  to  the  second 
fundamental  form  h  by 

,7),  (5) 


where  a  is  a  normal  vector  field  on  Mn  . 
For  C-totally  real  submanifold  in  M2m+  1  (c),  we  also  have  (for  example,  see  [7]) 


,  7)  =  A0X7,    A^  =  0.  (6) 

g(h(X,  7),  4>Z)  =  g(h(X,  Z),  07).  (7) 

3.    Ricci  tensor  of  C  -totally  real  submanifolds 

We  will  need  the  following  algebraic  lemma  due  to  Chen  [2]. 

Lemma  3.1.  Lef  fl  j  ,  .  .  .  ,  an  ,  c  Z?e  w  +  1  (>i  ^  2J  real  numbers  such  that 
2 


i=l  /=l 

Then  2a\a^  >  c,  vwY/z  equality  holding  if  and  only  ifa\  +  ^2  =  ^3  =  •••  =  «/,. 

For  a  C-totally  real  submanifold  M"  of  M2m+l(c),  we  have 

Theorem  3.1.  TfM"  is  a  C-totally  real  submanifold  of  M2mJr{(c\  then  the  Ricci  tensor  of 
Mn  satisfies 

. 


and  the  equality  holds  identically  if  and  only  if  either  Mn  is  totally  geodesic  orn  =  2  and 
Mn  is  totally  umbilical 

Proof.  From  Gauss'  equation  (4),  we  have 

•  3)    .     o  „.,      ,,  l0 


402  Liu  Ximin 

Put*  =  p  -  '^^±21  _  ^2  Then  from  (,0) 


Let  L  be  a  linear  (n  -  l)-subspace  of  T/}M»,  p  e  M",  and  {e{ , . . . ,  e2n 
orthonormal  basis  such  that  ( 1 )  e , , . . . ,  en  are  tangent  toAf " ,  (2)  '*, ", . '  .  ~" 
if  H(p)  ^  0,  en+ 1  is  in  the  direction  of  the  mean  curvature  vector  at  p  ' 

Put*/  =  /i^+1,  /  =  1, . . . ,  n.  Then  from  (11)  we  get 


i  +  Efl/2  +  E^?y+1)2  + 


01) 

=  £}an 
^  and  (3) 


(12) 


r=n+2  i,j=\ 


Equation  (12)  is  equivalent  to 


. 

By  Lemma  3.1  we  know  that  if  (V?  .  5/)2  = 
ho,d,n8  lf,nd  on,,  if,,  +  ^'' 


^        •  , 
S  '  "* 


which  gives 


3) 


Using  Gauss'  equation  we  have 


~     E 


2/w+l 


2m +  1 


fn-\ 


+  2^](/z;2/7-f-])2 

2" 


From  (15)  and  (16)  we  have 


(16) 


2m +1 

E 


i  =  l 


.     (17) 


Sasakian  space  fo  rms  403 

So  we  have 

fo-l)(c  +  3)      n2__o       .. 

*n)  (18) 


4  4 

with  equality  holding  if  and  only  if 


for  1  <  j  <  n  —  1,  1  <  i  <  n  and  n  +  2  <  r  <  2m  +  1  and,  since  Lemma  3.1  states  that 
2a\ai  —  c  if  and  only  iftfj  -f-«2  =  #3,  we  also  have  /zj},"}"1  =  £]y~{  ^/^1-  Since  en  can  be 
any  unit  tangent  vector  of  Mn ,  then  (18)  implies  inequality  (9). 
If  the  equality  sign  case  of  (9)  holds  identically,  then  we  have 

n . .     ~  0       (I'^^T2-/'*^^)' 
ij  / 

h'j  =  0       (1  <  i,  7  <  n;  n  -f  2  <  r  <  2m  +  1), 

^/V1"1  ^  X]  ^l?*"1 '      XI /?**  =  °'       (n  "*"  2  ~  r  -  2m  +  l)'  (2°) 

If  A,/  =  /zJA+1(l  <  i  <  n),  we  find  £^/  A^  =  A/(l  <  /  <  n)  and,  since  the  matrix 

A^  =  (aff)  with  afj.    =  1  —  28ij  is  regular  for  n  ^  2  and  has  kernel  R(\,  1)  for  n  =  2, 
we  conclude  that  Mn  is  either  totally  geodesic  orw.  =  2  and  M"  is  totally  umbilical. 
The  converse  is  easy  to  prove.  This  completes  the  proof  of  Theorem  3.  h 

4.  Minimality  of  C-totally  real  submanifojds 

Theorem  4.1.  If  Mn  is  an  n-dimensional  C-totally  real  submanifold  in  a  Sasakian  space 
/0rmM2'z+l(c),  then 

Ric  < 1 H2.  (21) 

4  4 

If  Mn  satisfies  the  equality  case  of(2\)  identically,  then  Mn  is  minimal. 
Clearly  Theorem  4.1  follows  immediately  from  the  following  Lemma. 

Lemma  4.1.  lfMn  is  an  n-dimensional  totally  real  submanifold  in  a  Sasakian  space  form 
M2m+ !  (c),  then  we  have  (21).  If  a  C-totally  real  submanifold  Mn  in  M2m+l  (c)  satisfies  the 
equality  case  of  (2 1)  at  a  point  p,  then  the  mean  curvature  vector  H  at  p  is  perpendicular 
tocj>(TpMn). 

Proof.  Inequality  (21)  is  an  immediate  consequence  of  inequality  (9). 

Now  let  us  assume  that  Mn  is  a  C-totally  real  submanifold  of  M2/n+I  (c)  which  satisfies 
the  equality  sign  of  (2 1 )  at  a  point  p  €  Mn .  Without  loss  of  the  generality  we  may  choose 
an  orthonormal  basis  {e\,...,en}  of  TpMn  such  thatRic(p)  =  S(en,e}l).  From  the  proof 
of  Theorem  3. 1 ,  we  get 


404  Liu  Ximin 

where  hsr  denote  the  coefficients  of  the  second  fundamental  form  with  respect  to  the 
orthonormal  basis  {e\,  .  .  .  ,  en}  and  (en+[,  .  .  .  ,  <?2w-H  —  ?}• 

If  for  all  tangent  vectors  M,  u  and  u;  at  /?,  g(h(u,  i>),  0u,>)  =  0,  there  is  nothing  to  prove 
So  we  assume  that  this  is  not  the  case.  We  define  a  function  fp  by 


fp  :    ry         -+R:     v^  fp(v)  =  g(h(v,  v),  0v).  (23; 

Since  TlMn  is  a  compact  set,  there  exists  a  vector  v  e  TpMn  such  that  fp  attains  at 
absolute  maximum  at  u.  Then  /y;(i>)  >  0  and  g(h(v<  v),  0iy)  =  0  for  all  tu  perpendicula 
tou.  So  from  (5),  we  know  that  u  is  an  eigenvector  of  A^.  Choose  a  frame  {^i,  ei,  .  .  -  ,  ^ 
ofTpMn  suchthatei  =  uande/  be  an  eigenvector  of  A0e,  with  eigenvalue  A./  .  Thefunctioi 
//,  i  >  2,  defined  by  //(r)  =  /y,(cos  r  ej  4-  sin  ?  £2)  has  relative  maximum  at  t  =  0,  s< 
//'(O)  <  0.  This  will  lead  to  the  inequality  A]  >  2A./.  Since  A.I  >  0,  we  have 

A./  ^A-i,    A.I  >2A./,    /  >2.  (24 

Thus,  the  eigenspace  of  A^ei  with  eigenvalue  A.J  is  1  -dimensional. 

From  (22)  we  know  that  ^  is  a  common  eigenvector  for  all  shape  operators  at  p.  Oi 
the  other  hand,  we  have  e\  ^  ±en  since  otherwise,  from  (22)  and  A$eien  =  ^A^eie\  - 
±A0g,e/  =  ±Xi6i±.en  (i  =  2,  ____  ;i),  we  obtain  A./  =  0,  *  =  2,  .  .  .  ,  n\  and  hence  A,  i  =  < 
by  (22),  which  is  a  contradiction.  Consequently,  without  loss  of  generality  we  may  assumi 
e[  =e\,...,en  =  en. 

By  (6),  A^ene\  =  A^en  =  Xnen.  Comparing  this  with  (22)  we  obtain  A.n  =  0.  Thus 
by  applying  (22)  once  more,  we  get  A  i  H  -----  h  A.w_  i  =  Xn  =  0.  Therefore,  trace  A06J  =  C 

For  each  i  =  2,  .  .  .  ,  n,  we  have 

hnn'  =  £OV/<?;M  en)  =  ^(A^/;^,  (?/?)  =  /Z^. 

Hence,  by  applying  (22)  again,  we  get  /iJJ+'  =  0.  Combining  this  with  (22)  yield 
trace  A^.  =  0.  So  we  have  trace  A^x  =  0  for  any  X  e  TpMn.  Therefore,  we  con 
elude  that  the  mean  curvature  vector  at  p  is  perpendicular  io(/>(TpM"). 

Remark  4.1.  From  the  proof  of  Lemma  4.1  we  know  that  if  M"  is  a  C-totally  real  subman 
ifold  of  M2n+l(c)  satisfying 

+  3)+^ 
4 

then  M"  is  minimal  and  A<f>v  =  0  for  any  unit  tangent  vector  satisfying  S(v,  v)  =  Ric 
Thus,  by  (6)  we  have  A^xv  =  0.  Hence,  we  obtain  h(v,  X)  =  0  for  any  X  tangent  ti 
M"  and  any  u  satisfying  S(v,  v)  =  Ric.  Conversely,  if  Af  n  is  a  minimal  C-totally  rea 
submanifold  of  M2/i+  !  (c)  such  that  for  each  p  €  Mn  there  exists  a  unit  vector  u  e  T/;MJ 
such  that  h(v,  X)  =  0  for  all  X  €  T^Af",  then  it  satisfies  (25)  indentically. 
For  each  p  €  M",  the  kernel  of  the  second  fundamental  form  is  defined  by 

T>(p)  =  {Y  E  TpMn\h(X,  Y)  =  0,  VX  e  r/;M/?}.  (26 

From  the  above  discussion,  we  conclude  that  Mn  is  a  minimal  C-totally  real  submanifol< 
of  M2m+1(c)  satisfying  (25)  at  p  if  and  only  if  dim  V(p)  is  at  least  1  -dimensional. 
Following  the  same  argument  as  in  [4],  we  can  prove 

Theorem  4.2.  Let  Mn  be  a  minimal  C-totally  real  submanifold  ofM2n+  l  (c).  Then 


Sasakian  space  forms  405 

(1)  Mn  satisfies  (25)  at  a  point  p  if  and  only  ifdim'D(p)  >  1. 

(2)  If  the  dimension  ofD(p)  is  positive  constant  d,  then  T>  is  a  completely  integral  distri- 
bution and  Mn  is  d-ruled,  i.e.,  for  each  point  p  e  Mn,  Mn  contains  a  d -dimensional 
totally  geodesic  submanifold  N  ofM~n+l(c)  passing  through  p. 

(3)  A  ruled  minimal  C -totally  real  submanifold  Mn  ofM2n+l(c)  satisfies  (24)  identically 
if  and  only  if,  for  each  ruling  N  in  Mn,  the  normal  bundle  T^~Mn  restricted  to  N  is  a 
parallel  normal  subbundle  of  the  normal  bundle  T^N  along  N. 

Acknowledgements 

This  work  was  carried  out  during  the  author's  visit  to  Max-Planck-Institut  fur  Mathematik 
in  Bonn.  The  author  would  like  to  express  his  thanks  to  Professor  Yuri  Manin  for  the 
invitation  and  very  warm  hospitality.  This  work  is  partially  supported  by  the  National 
Natural  Science  Foundation  of  China. 

References 

[1]  Blair  D  E,  Contact  manifolds  in  Riemannian  geometry,  Lecture  Notes  in  Math.  509  (Berlin: 

Springer)  (1976) 
[2]  Chen  B  Y,  Some  pinching  and  classification  theorems  for  minimal  submanifolds,  Arch.  Math. 

60(1993)568-578 
[3]  Chen  B  Y,  Relations  between  Ricci  curvature  and  shape  operator  for  submanifolds  with  arbitrary 

codimension,  Glasgow  Math.  J.  41  (1999)  33-41 
[4]  Chen  B  Y,  On  Ricci  curvature  of  isotropic  and  Lagrangian  submanifolds  in  the  complex  space 

forms,  Arch.  Math.  74  (2000)  154-160 
[5]  Chen  B  Y,  Dillen  F,  Verstraelen  L  and  Vrancken  L,  Totally  real  submanifolds  of  CPn  satisfying 

a  basic  equality,  Arch.  Math.  63  (1994)  553-564 
[6]  Defever  F,  Mihai  I  and  Verstraelen  L,  Chen's  inequality  for  C-totally  real  submanifolds  of 

Sasakian  space  forms,  Boll  Un.  Mat.  Hal  B(7)ll  (1997)  365-374 
[7]  Dillen  F  and  Vrancken  L,  C-totally  real  submanifolds  of  Sasakian  space  forms,  J.  Math.  Pures 

Appi  69(1990)85-93 
[8]  Yano  K  and  Kon  M,  Structures  on  manifolds,  Ser.  Pure  Math.  3,  (Singapore:  World  Scientific) 

(1984) 


Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  Ill,  No.  4,  November  2001,  pp.  407-414. 
©  Printed  in  India 


A  variational  proof  for  the  existence  of  a  conformal  metric  with 
preassigned  negative  Gaussian  curvature  for  compact  Riemann 
surfaces  of  genus  >  1 

RUKMINI  DEY 

Harish  Chandra  Research  Institute,  Chhatnag  Road,  Jhusi,  Allahabad  21 1  019,  India 
E-mail:  rkmn@mri.ernet.in 

MS  received  20  October  2000;  revised  6  March  2001 

Abstract.  Given  a  smooth  function  £  <  0  we  prove  a  result  by  Berger,  Kazhdan  and 
others  that  in  every  conformal  class  there  exists  a  metric  which  attains  this  function  as 
its  Gaussian  curvature  for  a  compact  Riemann  surface  of  genus  g  >  I.  We  do  so  by 
minimizing  an  appropriate  functional  using  elementary  analysis.  In  particular  for  K 
a  negative  constant,  this  provides  an  elementary  proof  of  the  uniformization  theorem 
for  compact  Riemann  surfaces  of  genus  g  >  1 . 

Keywords.  Uniformization  theorem;  Riemann  surfaces;  prescribed  Gaussian 
curvature. 

1.  Introduction 

In  this  paper  we  present  a  variational  proof  of  a  result  by  Berger  [2],  Kazhdan  and  Warner 
[6]  and  others,  namely  given  an  arbitrary  smooth  function  K  <  0  we  show  that  in  every 
conformal  class  there  exists  a  metric  which  attains  this  function  as  its  Gaussian  curvature 
for  a  compact  Riemann  surface  of  genus  g  >  1.  In  particular,  this  result  includes  the 
uniformization  theorem  of  Poincare  [8]  when  K  is  a  negative  constant.  In  his  proof  Berger 
considers  the  critical  points  of  a  functional  subject  to  the  Gauss-Bonnet  condition.  He 
shows  that  the  functional  is  bounded  from  below  and  uses  the  Friedrich's  inequality  to 
complete  the  proof.  The  functional  we  choose  is  positive  definite  so  that  it  is  automatically 
bounded  from  below.  Our  proof  is  elementary,  using  Hodge  theory,  i.e.,  the  existence  of  the 
Green's  operator  for  the  Laplacian.  Our  proof  could  be  useful  for  analysing  the  appropriate 
condition  on  K  for  a  corresponding  result  for  genus  g  =  1  and  g  =  0  [6,  10,  3],  the  two 
other  cases  considered  by  Berger,  Kazdan  and  Warner.  Another  variational  proof  of  the 
uniformization  theorem  for  genus  g  >  1  can  be  found  in  a  gauge-theoretic  context  in  [5] 
which  uses  Uhlenbeck's  weak  compactness  theorem  for  connections  with  LP  bounds  on 
curvature  [9]. 

Let  M  be  a  compact  Riemann  surface  of  genus  g  >  1  and  let  ds2  =  hdz  ®  dz  be  a 
metric  on  M  normalized  such  that  the  total  area  of  M  is  1 .  Let  K  <  0.  We  minimize  the 
functional 


JM 

over  C°°(Af ,  R),  where  K  (a)  stands  for  the  Gaussian  curvature  of  the  metric  eads2,  and 
d/i  =  ^-hdz  A  dz  is  the  area  form  for  the  metric  ds2.  Using  Sobolev  embedding  theorem 

407 


408  Rukmini  Dey 

we  show  that  S(cr)  takes  its  absolute  minimum  on  C°°(M)  which  corresponds  to  a  metric 
on  Af  of  negative  curvature  K  . 

2.  The  main  theorem 

2.1 

All  notations  are  as  in  §  1. 

The  functional  5(cr)  =  fM(K(cr)  -  #)2e2<TdM  is  non-negative  on  C°°(M,  R),  so  that 
its  infimum 

S0  =  inf{S(a),  a  e  C°°(M,  (R)} 

exists  and  is  non-negative.  Let  {^JJJli  C  C°°(M,  R)  be  a  corresponding  minimizing 
sequence, 

lim  S(crn)  =  50. 

n-»oo 

Our  main  result  is  the  following 

Theorem  2.1  Let  M  be  a  compact  Riemann  surface  of  genus  g  >  1.  The  infimum  SQ  is 
attained  at  cr  e  C°°(M,  R),  j.e.  the  minimizing  sequence  {crn}  contains  a  subsequence  that 
converges  in  C°°(M,  R)  to  cr  e  C°°(M,  R)  and  S(cr)  =  0.  The  corresponding  metric 
e?hdz  ®  dl  w  r/z^  w/t/^w^  metric  on  M  of  negative  curvature  K. 

2.2  Uniform  bounds 

Since  {crn}  is  a  minimizing  sequence,  we  have  the  obvious  inequality 

5(or«)=  f  (Kn~K)2&2«»d^=  f 
JM  JM 

for  some  m  >  0,  where  we  denoted  by  Kn  the  Gaussian  curvature  K(an)  of  the  metric 
e?nh  and  by  ^o  that  of  the  metric  /z,  and  used  that 


Note.  Here  A^  =  4h~[(d2/3z3z)  stands  for  the  Laplacian  defined  by  the  metric  h  on  M. 

Lemma  2.2.  There  exist  constants  C\  and  C2  such  that,  uniformly  in  n, 

(a) 

(b) 


1 


•  By  Minkowski  inequality,  and  using  (2.1),  we  get 

1/2 

i    r   (          1  \" 

dM 


< 
J 


Compact  Riemann  surfaces  409 

so  that 

C2.  (2.2) 


/M 
Let  A'|  =  {  A/?cr,,  >  0}  and  A'i  =  {A/2^  <  Q}. 


A/?cr/,A:ea"dM=   /* 

.M'; 

>   minK         (A/z(jy?)ea" 
•M 


max  AT 


(min  K )  /A.  ( A,^,,  )eff"  dA  +  (max  ^ )  fA,_ 


where 


=  -r/7  /    l^iil^d/A,     by  Stoke's  theorem. 

^M 

Note  that  rw  <  max  AT  <  0.  Thus  by  (2.2),  we  get 

1  f  (A//o-,)2dM+  /  ^V^d/x-T,  /*   |azcr,,|V''dA6<C2.  (23) 

^  JM  JA/  JM 

Since  each  term  is  positive,  the  result  follows.  D 

2.3.  Pointwise  convergence  of  zero  mean-value  part 
Next,  for  a  e  C°°(M)  denote  by  m(a)  its  mean  value, 

w(cr)  =   /    crdM, 
JM 

and  by  a  =  a  —  m(«j)  denote  its  zero-mean  value  part.  For  the  minimizing  sequence  {crn} 
we  denote  the  corresponding  mean  values  by  m/z.  (Note:  We  had  normalized  the  volume 


Lemma  2.3.  The  mean-value  -zero  part  {^'//}^=1  o/r/ze  minimizing  sequence  {ffn}%L\  is 
uniformly  bounded  in  the  Sobolev  space  W2'2(Af). 

Proof.  By  Hodge  theory,  there  exists  an  operator  G  such  that  GA/Z  =  /  —  P,  where  /  is 
the  identity  operator  in  L2(M)  and  P  is  the  orthogonal  projection  onto  kernel  of  A/j.  We 
also  know  A/x  :  W2'2  ->  L2  boundedly  and  G  :  L2  ->•  W^2  is  a  bounded  operator. 

Now,  by  Lemma  2.2  {A/2orn}  are  bounded  uniformly  in  L2.  Thus,  {GA/jCr,2}  are  bounded 
uniformly  in  W2<2.  But  GAflan  =  (I  —  P)an  =  cfw. 

Now  we  can  formulate  the  main  result  of  this  subsection. 

PROPOSITION  2.4 

The  sequence  {&in  }^  {  contains  a  subsequence  {or/;j  }(^=l  with  the  following  properties. 


410  Rukmini  Dey 

(a)  The  sequences  {&in}%Li  and  {e07"4"'"'" }  converge  in  C°(M)  topology  to  continuous 
j unctions  a  andu  respectively.  Moreover,  &  e  W2'2(M). 

(b)  The  subsequence  {A/zcr/w}  converges  weakly  in  L2  to  f  ==  Ajjlstr<j — a  distribution 
Laplacianofv. 

(c)  Passing  to  this  subsequence  {&in},  the  following  limits  exist 


}mQS(ala)  =  SQ  = 

where 

lim  e07""1""1"  =  w. 

In  fact,  the  convergence  in  (b)  is  strong  in  L2. 

Proof.  Part  (a)  follows  from  the  Sobolev  embedding  theorem  and  Rellich  lemma  since, 
for  dim  Af  =  2,  the  space  W2'2(M)  is  compactly  embedded  into  C°(M)  (see,  e.g.  [1,  7]). 
Therefore  the  sequence  {<J/2},  which,  according  to  Lemma  2.3,  is  uniformly  bounded  in 
W2i2(Af ),  contains  a  convergent  subsequence  in  C°(Af ).  Passing  to  this  subsequence  {o/n} 
we  can  assume  that  there  exists  mean- value  zero  function  a  e  C°(M)  such  that 

lim  &in  =  or. 

Since  crn's  are  uniformly  bounded  in  a  Hilbert  space  W2<2(M ),  they  weakly  converge  to 
s  €  W2'2(M)  (after  passing  to  a  subsequence  if  necessary).  The  uniform  limit  coincides 
with  s  so  that  or  =  s  e  W2<2(M). 
Moreover,  since 


<  / 

JM 


then  by  part  (b)  of  Lemma  2.2  the  sequence  {mn }  is  bounded  above.  Passing  to  a  sub- 
sequence, if  necessary,  we  obtain  that  there  exists  u  e  C°(M)  such  that  in  the  C°(M) 
topology  lim^oo  e?in+m»  =  u.  The  function  u  may  be  identically  zero  if  mn  -»  -co. 

In  order  to  prove  (b),  set  fa  =  A/2or/n  and  observe  that,  according  to  part  (a)  of  Lemma 
2.2,  the  sequence  {fa}  is  bounded  in  L2.  Therefore,  passing  to  a  subsequence,  if  necessary, 
there  exists  /  e  L2(M)  such  that 

lim    /    fag  =       fg 

n-*°°  JM  JM 

for  all  g  6  L2(M)'.  In  particular,  considering  g  e  C°°(M),  this  implies  /  =  Afstra. 
In  order  to  prove  (c)  we  use  the  following  lemma. 

Lemma  2.5.  If  a  sequence  [fa]  converges  to  f  e  L2  in  the  weak  topology,  then 

^UrnJIV^II  >  ll/ll- 
Further  linWoo  \\fa\\  =  ||/||  iff  there  is  strong  convergence. 


Compact  Riemann  surfaces 

Proof.  The  lemma  follows  from  considering  the  following  inequality: 
Hm 


411 


D 

To  continue  with  the  proof  of  the  proposition,  suppose  lim/z _»oo  IIV'/ill  >  11/11-  Using  the 
definition  of  the  functional,  we  have 


-  K^+m»f-   I   tn( 

JM 

From  parts  (a)  and  (b)  it  follows  that  the  sequence  S(crf})  converges  to  SQ  and 


=      Hm  S(crn) 

/7-»OO 


=      lim   L\waf+  \\K0-Kuf-  I   f(K0-Ku)dn 

n-»oo4  JM 


-  I  f(K0  - 
JM 


We  will  show  that  this  inequality  contradicts  that  {crn}  was  a  minimizing  sequence,  i.e. 
we  can  construct  a  sequence  {r  +  m!n}  e  C°°(M)  such  that  S(r  +  m/;j)  gets  as  close  to 


Namely,  for  any  €  >  0  we  can  construct,  by  the  density  of  C°°  in  W2-2,  a  function 
r  e  C°°(Af)  approximating  cr  e  W2'2  such  that  ||  AAr  -  /||  <  €  and  ||4(u  -  w)||  <  €/2 


where  u  = 


we  have 


er+/;t/»  .  Since 


=   lim  S(r 

/?-»oo 


»o 
£, 

Now  setting  8  =  v^o  —  II  —  T/  +  ^o  —  Ku\\  >  $  and  choosing  €  <  8/2,  and  using 
V^r  <  II  ~  T/  +  ^o  —  K  u  II  +  €"  we  get  V37  <  V^o  —  f  — a  contradiction,  since  SQ  was 
the  infimum  of  the  functional. 

Thus,  lim^-^oo  ||A/zcf/7||  —  ||/||,  so  that,  in  fact,  by  Lemma  2.5,  the  convergence  is  in  the 
strong  L2  topology.  This  proves  part  (c). 

2.4  Convergence  and  the  non-degeneracy 

PROPOSITION  2.6 

The  minimizing  sequence  {&n}%L\  contains  a  subsequence  that  converges  in  C°(Af)  to  a 
function  cr  e  C°(M),  so  that  the  resulting  metric  eah  is  non-degenerate. 


412  RukminiDey 

Proof.  Since  <?„  =  a  -f-  ww,  by  Proposition  2.4  and  Lemma  2.3,  it  is  enough  to  show  that 
the  sequence  [mn]  is  bounded  below.  Supposing  the  contrary  and  passing,  if  necessary,  to 
a  subsequence,  we  can  assume  that 

lim  mn  =  -co, 

J1-— >OO 

so  that,  in  notations  of  Proposition  2.4,  u  =  0.  By  Proposition  2.4(c)  we  get 

f   f  I     distr.\2 

We  shall  show  that  this  contradicts  the  fact  that  50  is  the  infimum  of  the  functional  S  and 
that  {on}  is  a  minimizing  sequence.  First  we  have  the  following  lemma. 


Lemma  2.1.  Letb  =  KQ—^&^O  e  L2(M),  where  &n  ->  a  andmn  ->  —ooasn  -+  oo. 
Then 


f 

,/A 


=  0 
A/ 


jfor  a//  yS  6  C°°(M)  aw^/Z?  =  -L,  vv/zere  L  /j  a  positive  constant. 

Proof.  Consider  Gn(t)  =  5(crn  -f-  /^)  -  SQ  —  a  smooth  function  off  for  a  fixed  ft.  Then  by 
Proposition  2.4(c)  we  have 

G(f)     =      lim  Gn(t) 


=      f   ^o-^Af^a  +  ^VdM-  / 
JM  \         ^  /  J 


and  G(f)  is  a  smooth  function  off  for  fixed  ft.  Since  50  is  the  infimum  of  5,  we  have  that 
G(0  >  0  for  all  t  and  G(0)  =  0.  Therefore  it  follows  that 

dG 

=  0 


dr    r=0 


for  all  ft  e  C°°(M).  Straightforward  computation  yields 
dG 


r=0 

Therefore,  &  satisfies  the  Laplace  equation  A/7&  =  0  in  a  distributional  sense  and  from 
elliptic  regularity  it  follows  that  b  is  smooth.  Thus  b  is  harmonic  and  therefore  is  a  constant. 
Finally,  by  the  Gauss-Bonnet  theorem,  we  have  fM  bdf^  =  4;r(l  -  g)  and  recalling  that 
g  >  1  ,  we  conclude  that  b  =  4;r(l  —  #)  =  -L  <  0.  D 

To  complete  the  proof  of  the  proposition,  we  get  a  contradiction  as  follows.  By  Lemma 
2.7,  we  have  that  SQ  =  /A/(-L)2d/z  =  L2  is  the  infimum  of  the  functional.  Since 
L  >  0,  and  {mn}  -*  -oo,  we  consider  r  =  a  4-  mn  and  choose  n  large  enough  so 
that  -ATeT  <  L/2.  We  have 


1 

0  -  ~ 
M  \         2 


\2  /» 

T      d/x  =  /   (-L  - 

J  JM 


Then,  since  -L  -f  a  <  -L  -  ^er  <  -L/2,  where  a  >  0  is  the  infimum  of  -#er,  we 
have  (-L  -  #er)2  <  (L  -  a)2  so  that  5(r)  <  L2—  a  contradiction.  D 


Compact  Riemann  surfaces  413 

3.  Smoothness  and  uniqueness 

Here  we  complete  the  proof  of  the  main  Theorem  3.1. 

PROPOSITION  3.1 

The  minimizing  function  a  e  C°(M)  is  smooth  and  corresponds  to  the  unique  Kdhler 
metric  of  negative  curvature  K . 

Proof.  Let  b  =  (AT0  -  ^A^istrcr  -  ^ea)  e  I2(M);  according  to  Proposition  2.4(c)  and 
Proposition  2.6,  S0  =  fM  b2diJi.  Set  G(0  =  S(a  +  tft)  -  So,  where  ft  e  C°°(M). 
Repeating  arguments  in  the  proof  of  Lemma  2.7,  we  conclude  that  G(t)  for  fixed  ft  is 
smooth,  G(0)  =  0  and  G(t)  >  0  for  all  t.  Therefore, 

dG 
dt 

A  simple  calculation  yields 
dG 


Thus  b  G  L2(M)  satisfies,  in  a  distributional  sense,  the  following  equation 

-A/,*?  -  2Kefrb  =  0.  (3.1) 

First,  we  will  show  that  b  =  0  is  the  only  weak  L2  solution  to  eq.  (3.1).  Indeed,  by 
elliptic  regularity  b  is  smooth,  so  that  multiplying  (3.1)  by  b  and  integrating  over  M  using 
the  Stokes  formula,  we  get 

M  JM 

which  implies  that  b  =  0.  Thus  we  have  shown  that  SQ  =  0. 
Second,  equation  b  =  0  for  the  minimizing  function  a  e  C°(M)  reads 

I 

2 

Therefore,  A/f^'a  belongs  to  V\M)  so  that  cr  e  W2'/;forallp,  By  the  Sobolev  embedding 
theorem  it  follows  that  cr  e  C It0f  (M)  for  some  0  <  a  <  1 .  Therefore,  the  right  hand  side 
of  eq.  (3.2)  actually  belongs  to  the  space  C1'a(M),  and  therefore  a  e  C3*a(Af )  and  so  on. 
This  kind  of  bootstrapping  argument  shows  that  a  is  smooth  [7], 

The  equation  b  =  0  satisfied  by  a  now  translates  to  K(a)  =  K,  where  K(a)  is  the 
Gaussian  curvature  of  the  metric  Qahdz  ®  dz  ,  a  6  C°°(M). 

The  minimizing  function -a  is  unique:  here  is  the  standard  argument,  which  goes  back 
to  Poincare.  Let  rj  be  another  minimizing  function,  which  is  smooth  and  also  satisfies  eq. 
(3.2) 


414  Rukmini  De  y 

so  that 


Multiplying  this  equation  by  cr  —  rj  and  integrating  over  M  with  the  help  of  Stokes  formula, 
we  get 


-  I  df  A  *  df  =  I  -2K(a  -  ?7)(ea  -  e'?)dyu,, 
JM  JM 


where  we  set  f  =  cr  -  77.  Since  -2£(a  -  rf)(sa  -  e7])  >  0,  we  conclude  that  d£  =  0  and, 
in  fact,  £  =  0. 

The  proof  of  Theorem  3.  1  is  complete. 

Acknowledgement 

I  would  like  to  thank  Professor  L  Takhtajan  for  his  invaluable  help  in  a  previous  version 
of  this  paper.  I  would  also  like  to  thank  Professor  D  Geller  for  his  careful  reading  of  the 
manuscript.  I  did  this  work  while  I  was  at  SUNY  at  Stony  Brook,  New  York,  USA. 

References 

[1]  Aubin  T,  Nonlinear  Analysis  on  manifolds;  Monge-Ampere  equations 

[2]  BergerM  S,  Riemannian  structures  of  prescribed  Gaussian  curvature  for  compact  2-mani  folds, 

J.  Differ.  Geom.  (1971)  325-332 
[3]  Chen  W  and  Li  C,  A  necessary  and  sufficient  condition  for  the  Nirenberg  problem,  Comm. 

Pure.  Appi  Math.  48(6)  (  1  995)  657-667 
[4]  Farkas  H  and  Kra  I,  Riemann  surfaces 
[5]  Hitchin  N  J,  Self-duality  equations  over  a  Riemann  surface,  Proc.  London  Math.  Soc.  55(3) 

(1987)59-126 
[6]  Kazdan  J  and  Warner  F  W,  Curvature  functions  for  compact  2-manifolds,  Ann.  Math.  99(2) 

(1974)  14_47 

[7]  Kazdan  J,  Applications  of  PDE  to  problems  in  geometry 

[8]  Poincare  H,  Les  fonctions  ruchsiennese  1'  equation  Aw  =  e",  /.  Math.  PuresAppl  4(5)  (1  898) 
[9]  Uhlenbeck  K,  Connections  with  Lp  bounds  on  curvature,  Comm.  Math.  Phys.  83  (1  982)  3  1  -42 
[10]  Xu  X  and  Yang  P  C,  Remarks  on  prescribing  Gauss  curvature,  Trans.  AMS  336(2)  (1993) 
831-840 


Proc.  Indian  Acacl.  Sci.  (Math.  Sci.),  Vol.  1 1 1,  No.  4,  November  2001,  pp.  415-437. 
©  Printed  in  India 


Homogeneous  operators  and  projective  representations  of  the 
Mobius  group:  A  survey 

BHASKAR  BAGCHI  and  GADADHAR  MISRA 

Theoretical  Statistics  and  Mathematics  Division,  Indian  Statistical  Institute, 
Bangalore  560  059,  India 

MS  received  31  January  2001 

Abstract.  This  paper  surveys  the  existing  literature  on  homogeneous  operators  and 
their  relationships  with  projective  representations  of  PSL(2,  R)  and  other  Lie  groups. 
It  also  includes  a  list  of  open  problems  in  this  area. 

Keywords.  Projective  representations;  homogeneous  operators;  reproducing  ker- 
nels; Sz-Nagy-Foias  characteristic  functions. 

1.  Preliminaries 

This  paper  is  a  survey  of  the  known  results  on  homogeneous  operators.  A  small  proportion 
of  these  results  are  as  yet  available  only  in  preprint  form.  A  miniscule  proportion  may  even 
be  new.  The  paper  ends  with  a  list  of  thirteen  open  problems  suggesting  possible  directions 
for  future  work  in  this  area.  This  list  is  not  purported  to  be  exhaustive,  of  course! 

All  Hilbert  spaces  in  this  paper  are  separable  Hilbert  spaces  over  the  field  of  complex 
numbers.  All  operators  are  bounded  linear  operators  between  Hilbert  spaces.  If  W,  /C  are 
two  Hilbert  spaces,  B(H,  /C)  will  denote  the  Banach  space-of  all  operators  from  W  to  /C, 
equipped  with  the  usual  operator  norm.  If  H  =  /C,  this  will  be  abridged  to  B(W-  The 
group  of  all  unitary  operators  in  B(H)  will  be  denoted  by  UCH).  When  equipped  with  any 
of  the  usual  operator  topology  UCH)  becomes  a  topological  group.  All  these  topologies 
induce  the  same  Borel  structure  on  U(H).  We  shall  view  UCH)  as  a  Borel  group  with  this 
structure. 

Z,  IR  and  C  will  denote  the  integers,  the  real  numbers  and  the  complex  numbers,  respec- 
tively. D  and  T  will  denote  the  open  unit  disc  and  the  unit  circle  in  C,  respectively,  and  B 
will  denote  the  closure  of  D  in  C.  Mob  will  denote  the  Mobius  group  of  all  biholomorphic 
automorphisms  of  O.  Recall  that  Mob  =  {<pa.p  '  a  £  ~D~»  P  e  D},  where 

~~  eD.  (1.1) 


1  -  pz 

For  @  e  O,  (pft  :=  <P-i,/$  is  the  unique  involution  (element  of  order  2)  in  Mob  which 
interchanges  0  and  ft.  Mob  is  topologized  via  the  obvious  identification  with  T  x  O.  With 
this  topology,  Mob  becomes  a  topological  group.  Abstractly,  it  is  isomorphic  toPSL(2,R) 
andtoPSt/(l,  1). 

The  following  definition  from  [6]  has  its  origin  in  the  papers  [21]  and  [22]  by  the  second 
named  author. 

415 


4 1 6  Bhaskar  Bagchi  and  Gadadhar  Misra 

DEFINITION  1.1 

An  operator  T  is  called  homogeneous  if(p(T)  is  unitarily  equivalent  to  T  for  all  (p  in  Mob 
which  are  analytic  on  the  spectrum  of  T. 

It  was  shown  in  Lemma  2.2  of  [6]  that 

Theorem  1.1.  The  spectrum  of  any  homogeneous  operator  T  is  either T  or  D.  Hence  <p(T) 
actually  makes  sense  (and  is  unitarily  equivalent  to  T)  for  all  elements  (p  of  Mob. 

Let  *  denote  the  involution  (i.e.  automorphism  of  order  two)  of  Mob  defined  by 

<p*(z)  =  ^Ciy,   z  e  O,  (p  e  Mob.  (1 .2) 

Thus  (p*  p  =  <PQ  a  for  (a,  P)  e  T  x  O.  It  is  known  that  essentially  (i.e.  up  to  multiplication 
by  arbitrary  inner  automorphisms),  *  is  the  only  outer  automorphism  of  Mob.  It  also 
satisfies  <p*(z)  =  (p(z~~])~{  for  z  €  T.  It  follows  that  for  any  operator  T  whose  spectrum 
is  contained  in  O,  we  have 

^(r*)^*(7y,  <p(T~l)  =  <p*(Trl,  0.3) 

the  latter  in  case  T  is  invertible,  of  course.  It  follows  immediately  from  (1.3)  that  the  adjoint 
T*  ~  as  well  as  the  inverse  T~{  in  case  T  is  invertible  -  of  a  homogeneous  operator  T  is 
again  homogeneous. 

Clearly  a  direct  sum  (more  generally,  direct  integral)  of  homogeneous  operators  is  again 
homogeneous. 

2.  Characteristic  functions 

Recall  that  an  operator  T  is  called  a  contraction  if  ||7*||  <  1,  and  it  is  called  completely 
non-unitary  (cnu)  if  T  has  no  non-trivial  invariant  subspace  M,  such  that  the  restriction  of 
T  to  M  is  unitary  .  T  is  called  a  pure  contraction  if  ||  Tx  \\  <  \\x  \\  for  all  non-zero  vectors 
x.  To  any  cnu  contraction  T  on  a  Hilbert  space,  Sz-Nagy  and  Foias  associate  in  [25]  a  pure 
contraction  valued  analytic  function  0T  on  O,  called  the  characteristic  function  of  T. 

Reading  through  [25]  one  may  get  the  impression  that  the  characteristic  function  is  only 
contraction  valued  and  its  value  at  0  is  a  pure  contraction.  However,  if  0  is  a  contraction 
valued  analytic  function  on  ID  and  the  value  of  0  at  some  point  is  pure,  its  value  at  all  points 
must  be  pure  contractions.  This  is  immediate  on  applying  the  strong  maximum  modulus 
principle  to  the  function  z  ->  0(z)x,  where  x  is  an  arbitrary  but  fixed  non-zero  vector. 

Two  pure  contraction  valued  analytic  functions  0£-  :  O  — »•  #(/C,;,  £/),  i  =  1,  2  are  said 
to  coincide  if  there  exist  two  unitary  operators  TJ  :  JC\  ->  £2,  T2  :  £>\  ->•  £2  such  that 
02(z)T|  =  r26>}  (z)  for  all  z  €  0.  The  theory  of  Sz-Nagy  and  Foias  shows  that  (i)  two  cnu 
contractions  are  unitarily  equivalent  if  and  only  if  their  characteristic  functions  coincide, 
(ii)  any  pure  contraction  valued  analytic  function  is  the  characteristic  function  of  some  cnu 
contraction.  In  general,  the  model  for  the  operator  associated  with  a  given  function  0  is 
difficult  to  describe.  However,  if  0  is  an  inner  function  (i.e.,  0  is  isometry- valued  on  the 
boundary  of  D),  the  description  of  the  Sz-Nagy  and  Foias  model  simplifies  as  follows: 

Theorem  2.1.  Let  0  :  D  ->  B(/C,  £)  be  a  pure  contraction  valued  inner  analytic  function. 
Let  M  denote  the  invariant  subspace  of  #2(B)  <g>  C  corresponding  to  0  in  the  sense  of 


Homogeneous  operators  and  projective  representations  4  1  7 

Beurling's  theorem.  That  is,  M  =  {z  M>  6(z)f(z)  :  f  €  #2(D)  ®  /C}.  77z*/x  61  coincides 
with  the  characteristic  function  of  the  compression  of  multiplication  by  z  to  the  subspace 


From  the  general  theory  of  Sz-Nagy  and  Foias  outlined  above,  it  follows  that  if  T  is  a  cnu 
contraction  with  characteristic  function  0  then,  letting  T[fi]  denote  the  cnu  contraction  with 
characteristic  function  /z0  for  0  <  M  <  1  ,  we  find  that  [T[fji]  :  0  <  IJL  <  1  }  is  a  continuum 
of  mutually  unitarily  inequivalent  cnu  contractions  (provided  0  is  not  the  identically  zero 
function,  of  course).  In  general,  it  is  difficult  to  describe  these  operators  explicitly  in  terms 
ofT  alone.  But,  in  [7],  we  succeeded  in  obtaining  such  a  description  in  case  6  is  an  inner 
function  (equivalently,  when  T  is  in  the  class  C  o,  i.e.,  T*nx  —  >  0  as  n  .  -»  oo  for  every 
vector  x)  -  so  that  T  has  the  description  in  terms  of  0  given  in  Theorem  2.1  .  Namely,  for 
a  suitable  Hilbert  space  £,  T  may  be  identified  with  the  compression  of  M  to  M.-*-,  where 
M  :  H^  :=  //2(O)  ®  £  —  >  H£  is  multiplication  by  the  co-ordinate  function  and  M.  is  the 

invariant  subspace  for  M  corresponding  to  the  inner  function^.  Let  M  —  I      ll  J  be 

the  block  matrix  representation  of  M  corresponding  to  the  decomposition  H^  =  M^t&M. 
(Thus,  in  particular,  T  =  M\  \  and  M2?  is  the  restriction  of  M  to  A4.)  Finally,  let  JC  denote 
the  co-kernel  of  Mo2,  N  :  H^  ->  H^  be  multiplication  by  the  co-ordinate  function  and  let 
E  :  HJ^  ~*  M  be  defined  by  £/  =  /(O)  e  JC.  In  terms  of  these  notations,  we  have 

Theorem  2.2.  Let  T  be  a  cnu  contraction  in  the  class  C  Q  with  characteristic  function  0. 
Let  }Ji  be  a  scalar  in  the  range  0  <  ii  <  1  and  put  8  =  -/I  —  /^2.  Then,  with  respect  to  the 
decomposition  .M1  ®  M  0  ^  0//te  domain,  the  operator  T[>]  :  H£  0  //^  ®  /^  -»  //^. 
/za^1  ^/ie  block  matrix  representation 


In  Theorem  2.9  of  [6],  it  was  noted  that 

Theorem  2.3.  A  pure  contraction  valued  analytic  function  0  on  O  is  the  characteristic  func- 
tion of  a  homogeneous  cnu  contraction  if  and  only  if  9  o  (p  coincides  with  0  for  every  (p  in 
Mob. 

From  this  theorem,  it  is  immediate  that  whenever  T  is  a  homogeneous  cnu  contraction, 
so  are  the  operators  T[IJL]  given  by  Theorem  2.2.  Some  interesting  examples  of  this  phe- 
nomenon were  worked  out  in  [7].  See  §6  for  these  examples. 

As  an  interesting  particular  case  of  Theorem  2.3,  one  finds  that  any  cnu  contraction 
with  a  constant  characteristic  function  is  necessarily  homogeneous.  These  operators  are 
discussed  in  [1 1]  and  [6].  Generalizing  a  result  in  [6],  Kerchy  shows  in  [19]  that 

Theorem  2.4.  Let  9  be  the  characteristic  function  of  a  homogeneous  cnu  contraction.  If 
0(0)  is  a  compact  operator  then  0  must  be  a  constant  function. 

(Actually  Kerchy  proves  the  same  theorem  with  the  weaker  hypothesis  that  all  the  points 
in  the  spectrum  of  (9(0)  are  isolated  from  below.) 


4  1  8  Bhaskar  Bagchi  and  Gadadhar  Misra 

Sketch  of  Proof.  Let  0  :  0  ->  B(JC,  C]  be  the  characteristic  function  of  a  homogeneous 
operator.  Assume  C  :=  0(0)  is  compact.  Replacing  6  by  a  coincident  analytic  function  if 
neceesary,  we  may  assume  without  loss  of  generality  that  JC  =  C  and  C  >  0.  By  Theorem 
2.3  there  exists  unitaries  Uz,  Vz  such  that  0(z)  =  UZCVZ,  z  €  O.  Let  A.J  >  A2  >  •  •  -  be 
the  non-zero  eigenvalues  of  the  compact  positive  operator  C.  At  this  point  Kerchy  shows 
that  (as  a  consequence  of  the  maximum  modulus  principle  for  Hilbert  space  valued  analytic 
functions)  the  eigenspace  K\  corresponding  to  the  eigenvalue  A.J  is  a  common  reducing 
subspace  for  Uz,  K,  z  e  0  (as  well  as  for  C  of  course)  and  hence  for  0(z),  z  €  0.  So  we 
can  write  0(z)  =  6{(z)  0  02(z)  where  0j  is  an  analytic  function  into  B(/Ci).  Since  0{  is  a 
unitary  valued  analytic  function,  it  must  be  a  constant.  Repeating  the  same  argument  with 

02,  one  concludes  by  induction  on  n  that  the  eigenspace  JCn  corresponding  to  the  eigenvalue 
^n  is  reducing  for0(z),  z  €  0,  and  the  projection  of  9  to  each  JCn  is  a  constant  function. 
Since  the  same  is  obviously  true  of  the  zero  eigenvalue,  we  are  done. 

3.  Representations  and  multipliers 

Let  G  be  a  locally  compact  second  countable  topological  group.  Then  a  measurable  function 
n  :  G  -+  U(H]  is  called  &  protective  representation  of  G  on  the  Hilbert  space  ft  if  there 
is  a  function  (necessarily  Borcl)  m  :  G  x  G  -~>  T  such  that 


7T(1)  =  7,  n(g{g2)  =  m(g{,  g2)n(g\)n(g2)  (3.1) 


for  all  g\  ,  #2  in  G.  (More  precisely,  such  a  function  n  is  called  a  prqjective  unitary  repre- 
sentation of  G;  however,  we  shall  often  drop  the  adjective  unitary  since  all  representations 
considered  in  this  paper  are  unitary.)  The  projective  representation  n  is  called  an  ordi- 
nary representation  (and  we  drop  the  adjective  'projective')  if  m  is  the  constant  function 
1.  The  function  m  associated  with  the  projective  representation  n  via  (3.1)  is  called  the 
multiplierof  TT.  The  ordinary  representation  n  of  G  which  sends  every  element  of  G  to 
the  identity  operator  on  a  one  dimensional  Hilbert  space  is  called  the  identity  (or  trivial) 
representation  of  G.  It  is  surprising  that  although  projective  representations  have  been  with 
us  for  a  long  time  (particularly  in  the  Physics  literature),  no  suitable  notion  of  equivalence 
of  projective  representations  seems  to  be  available.  In  [7],  we  offered  the  following: 

DEFINITION  3.1 

Two  projective  representations  n  {  ,  7t2  of  G  on  the  Hilbert  spaces  H  \  ,  HI  (respectively)  will 
be  called  equivalent  if  there  exists  a  unitary  operator  U  :  U\  ->  HI  and  a  function  (nec- 
essarily Borel)  /  :  G  ->  T  such  that  n2((p)U  =  f  (<p)U  n  \(<p)  for  all  <p  €  G. 

We  shall  identify  two  projective  representations  if  they  are  equivalent.  This  has  the  some 
what  unfortunate  consequence  that  any  two  one  dimensional  projective  representations  are 
identified.  But  this  is  of  no  importance  if  the  group  G  has  no  ordinary  one  dimensional 
representation  other  than  identity  representation  (as  is  the  case  for  all  semi-simple  Lie 
groups  G.)  In  fact,  the  above  notion  of  equivalence  (and  the  resulting  identifications)  saves 
us  from  the  following  disastrous  consequence  of  the  above  (commonly  accepted)  notion 
of  projective  representations:  Any  Borel  function  from  G  into  T  is  a  (one  dimensional) 
projective  representation  of  the  group  !  ! 


Homogeneous  operators  and  projective  representations  4  1  9 

3.  1  Multipliers  and  cohomology 

Notice  that  the  requirement  (3.1)  on  a  projective  representation  implies  that  its  associated 
multiplier  m  satisfies 


m((pt  1)  =  1  =  m(l,  <p),  m(<p\,  (P2)™(<P\<P2'  ^3)  =  m((Pl>  (P2<P3)m(<P2>  ^3)    (3.2) 


for  all  elements  <p,  q>\,  $2,  93  of  G.  Any  Borel  function  m  :  G  x  G  ->  T  satisfying  (3.2) 
is  called  a  multiplier  of  G.  The  set  of  all  multipliers  on  G  form  an  abelian  group  M(G)y 
called  the  multiplier  group  of  G.  If  m  e  M(G),  then  taking  W  =  L2(G)  (with  respect  to 
Haar  measure  on  G),  define  n  \  G  ->  £Y(W)  by 

(3.3) 

for<p,  V  inG,/  inL2(G).  Then  one  readily  verifies  that  n  is  a  projective  representation  of  G 
with  associated  multiplier  m.  Thus  each  element  of  M(G)  actually  occurs  as  the  multiplier 
associated  with  a  projective  representation.  A  multiplier  m  e  M(G)  is  called  exact  if 
there  is  a  Borel  function  /  :  G  ->  T  such  that  m(<plt  <p2)  =  (f(9\)f(<P'$)lf(<P\<P'2)  for 
<Pl,  <P2  in  G.  Equivalently,  m  is  exact  if  any  projective  representation  with  multiplier  m 
is  equivalent  to  an  ordinary  representation.  The  set  Mo(G)  of  all  exact  multipliers  on 
G  form  a  subgroup  of  M(G).  Two  multipliers  raj,  w2  are  said  to  be  equivalent  if  they 
belong  to  the  same  coset  of  A/o(G)  .  In  other  words,  m  \  and  ni2  are  equivalent  if  there  exist 
equivalent  projective  representations  n\  ,  n2  whose  multipliers  are  m  \  and  m2  respectively. 
The  quotient  M(G)  /M0(G)  is  denoted  by  #2(G,  T)  and  is  called  the  second  cohomology 
group  of  G  with  respect  to  the  trivial  action  of  G  on  T  (see  [24]  for  the  relevant  group 
cohomology  theory).  For  m  e  M(G),  [m]  e  #2(G,  T)  will  denote  the  cohomology  class 
containing  m,  i.e.,  [  ]  :  Af(G)  ->  H2(G,  T)  is  the  canonical  homomorphism. 

The  following  theorem  from  [8]  (also  see  [9])  provides  an  explicit  description  of  H2(G  ,  T) 
for  any  connected  semi-simple  Lie  group  G. 

Theorem  3.1.  Let  G  be  a  connectedjemi-simple  Lie  group.  Then  H2(G,  T)  is  naturally 
isomorphic  to  the  Pontryagin  dual  n  l  (G)  of  the  fundamental  group  n  l  (G)  ofG. 

Explicitly,  if  G  is  the  universal  cover  of  G  and  TT  :  G  ->  G  is  the  covering  map  (so  that 
the  fundamental  group  n  ]  (G)  is  naturally  identified  with  the  kernel  Z  of  n)  then  choose  a 
Borel  section  51  :  G  -»  G  for  the  covering  map  (i.e.,  s  is  a  Borel  function  such  that  n  o  s  is 
the  identity  on  G,  andj(l)  =  1).  For  x  €  Z,  define  mx  :  G  x  G  ->  I  by 

mx(x9y)  =  x(s(yr[s(xrls(xy)),   x,yeG.  (3.4) 

Then  the  main  theorem  in  [8]  shows  that  x  ^  tmxl  ^s  an  isomorphism  from  Z  onto 
H2(G,  T)  and  this  isomorphism  is  independent  of  the  choice  of  the  section  s. 

The  following  companion  theorem  from  [8]  shows  that  to  find  all  the  irreducible  projec- 
tive representations  of  a  group  G  satisfying  the  hypotheses  of  Theorem  3.1,  it  suffices  to 
find  the  ordinary  irreducible  representations  of  its  universal  cover  G.  Let  Z  be  the  kernel  of 
the  covering  map  from  G  onto  G.  Let  ft  be  an  ordinary  unitary  representation  of  G.  Then 
we  shall  say  that  ft  is  of  pure  type  if  there  is  a  character  x  of  Z  such  that  fi(z)  =  X  (z)I  for 
all  z  in  Z.  If  we  wish  to  emphasize  the  particular  character  which  occurs  here,  we  may  also 
say  that  /3  is  pure  of  type  x-  Notice  that,  if  ft  is  irreducible  then  (as  Z  is  central)  by  Schur's 


420  Bhaskar  Bagchi  and  Gadadhar  Misra 

Lemma  /J  is  necessarily  of  pure  type.  In  terms  of  this  definition,  the  second  theorem  in  [8] 
says 

Theorem  3.2.  Let  G  be  a  connected  semi-simple  Lie  group  and  let  G  be  its  universal 
cover.  Then  there  is  a  natural  bijection  between  (the  equivalence  classes  of)  projective 
unitary  representations  ofG  and  (the  equivalence  classes  of)  ordinary  unitary  representa- 
tions of  pure  type  ofG.  Under  this  bijection,  for  each  x  the  projective  representations  ofG 
with  multiplier  mx  correspond  to  the  representations  ofG  of  pure  type  x>  and  vice  versa. 
Further,  the  irreducible  projective  representations  ofG  correspond  to  the  irreducible  rep- 
resentations ofG,  and  vice  versa. 

Explicitly,  if/?  is  an  ordinary  representation  of  pure  type  x  ofG  then  define  fx  :  G  ->  T 
by/xM  =  XO"1  --TOTTM),  x  e  G.  Defined  on  G  by  fi(jc)  =  fx(x)p(x).  Then  a  is  a 
projective  representation  ofG  which  is  trivial  on  Z.  Therefore  there  is  a  well-defined  (and 
uniquely  determined)  projective  representation  a  of  G  such  that  a  =  a  o  n  .  The  multiplier 
associated  with  ex  is  mx  .  The  map  ft  H>  a  is  the  bijection  mentioned  in  Theorem  3.2. 

Finally,  as  was  pointed  out  in  [9],  any  projective  representation  (say  with  multiplier  w)  of  a 
connected  semi-simple  Lie  group  can  be  written  as  a  direct  integral  of  irreducible  projective 
representations  (all  with  the  same  multiplier  m)  of  the  group.  It  follows,  of  course,  that  any 
multiplier  of  such  a  group  arises  from  irreducible  projective  representations.  It  also  shows 
that,  in  order  to  have  a  description  of  all  the  projective  representations,  it  is  sufficient  to 
have  a  list  of  the  irreducible  ones  and  to  know  when  two  of  them  have  identical  multipliers. 
This  is  where  Theorems  3.1  and  3.2  come  in  handy. 

3.2  The  multipliers  on  Mob 

Notice  that  for  any  element  <p  of  the  Mobius  group,  <p'  is  a  non-  vanishing  analytic  function 
on  0  and  hence  has  a  continuous  logarithm  on  this  closed  disc.  Let  us  fix,  once  for 
all,  a  Borel  determination  of  these  logarithms.  More  precisely,  we  fix  a  Borel  function 
(z,  <p)  H>  log</(z)  from  D  x  Mob  into  C  such  that  logp'fe)  =  0  for  <p  =  id.  Now  define 

)  to  be  the  imaginary  part  of  log^'(z). 
Define  the  Borel  function  n  :  Mob  x  Mob  ->  Z  by 

n(«pj-  !,  <pil)  =  —  (arg(<p2<?i)7(0)  -  arg^(O)  - 
For  any  a  eJr  define  m^  :  Mob  x  Mob  -»  T  by 


The  following  proposition  is  a  special  case  of  Theorem  3.1.   Detailed  proofs  may  be 
found  in  [9]. 

PROPOSITION  3.1 

For  co  €  T,  /n<u  is  a  multiplier  of  Mob.  It  is  trivial  if  and  only  ifa>=  1.  Every  multiplier 
on  Mob  is  equivalent  to  m^for  a  uniquely  determined  co  in  T.  In  other  words,  0)  h-»  [m^] 
is  a  group  isomorphism  between  the  circle  group  T  and  the  second  cohomology  group 
#2(M6b,T). 


Homogeneous  operators  and  protective  representations  421 

3.3  The  protective  representations  of  the  Mobius  group 

Every  projective  representation  of  a  connected  semi-simple  Lie  group  is  a  direct  integral 
of  irreducible  projective  representations  (cf.  [9],  Theorem  3J).  Hence,  for  our  purposes, 
it  suffices  to  have  a  complete  list  of  these  irreducible  representations  of  Mob.  A  complete 
list  of  the  (ordinary)  irreducible  unitary  representations  of  the  universal  cover  of  Mob  was 
obtained  by  Bargmann  (see  [29]  for  instance).  Since  Mob  is  a  semi-simple  and  connected 
Lie  group,  one  may  manufacture  all  the  irreducible  projective  representations  of  Mob  (with 
Bargmann's  list  as  the  starting  point)  via  Theorem  3.2.  Following  [8]  and  [9],  we  proceed 
to  describe  the  result.  (Warning:  Our  parametrization  of  these  representations  differs 
somewhat  from  the  one  used  by  Bargmann  and  Sally.  We  have  changed  the  parametrization 
in  order  to  produce  a  unified  description.) 

For  n  €  Z,  let  fn  :  T  ->  T  be  defined  by  fn(z)  =  zn.  In  all  of  the  following  examples, 
the  Hilbert  space  T  is  spanned  by  an  orthogonal  set  {//,  :  n  e  /},  where  /  is  some  subset 
of  Z.  Thus  the  Hilbert  space  of  functions  is  specified  by  the  set  /  and  {||//z||,  n  €  /}.  (In 
each  case,  \\fn  \\  behaves  at  worst  like  a  polynomial  in  \n\  as  n  ->  oo,  so  that  this  really 
defines  a  space  of  function  on  T.)  For  (p  €  Mob  and  complex  parameters  A  and  //,,  define 
the  operator  Rx^L((p~~{)  on  T  by 

(Rx^(9~l)f)(z)  =  p'(0A/V(*)r(/(p(z)),   z  e  T,  /  e  F,  <p  e  Mob. 

Here  one  defines  <p'(z)x/2  as  exp  A./2  log  <p'(z)  using  the  previously  fixed  Borel  determina- 
tion of  these  logarithms. 

Of  course,  there  is  no  a  priori  guarantee  that  Rxtll((p~[)  is  a  unitary  (or  even  bounded) 
operator.  But,  when  it  is  unitary  for  every  <p  in  Mob,  it  is  easy  to  see  that  R^IJL  is  then  a 
projective  representation  of  Mob  with  associated  multiplier  mw,  where  co  =  ei;rA>.  Thus  the 
description  of  the  representation  is  complete  if  we  specify  /,  {\\fn  ||~,  n  e  /}  and  the  two 
parameters  A,  JJL.  It  turns  out  that  almost  all  the  irreducible  projective  representations  of 
Mob  have  this  form, 

In  terms  of  these  notations,  here  is  the  complete  list  of  the  irreducible  projective  unitary 
representations  of  Mob.  (However,  see  the  concluding  remark  of  this  section.) 

•  Principal  series  representations  P^St    -  1  <"A  <  1,  s  purely  imaginary.  Here  X  = 
A,  fM  =  i^i  +  s,  I  =Z,  \\fn  ||2  =  1  for  all  n  (so  the  space  is  L2(T)). 


•  Holomorphic  discrete  series  representations  D*:  Here  A  >  0,  fj,  =  0,  7  =  {n  e 
Z  :  n  >  0}  and  ||/n||2  =   r([!^)(A')  for  n  >  0.   For  each  /  in  the  representation 
space  there  is  an  /,  analytic  in  O,  such  that  /  is  the  non-tangential  boundary  value 
of  /.   By  the  identification  /  «•>  /,  the  representation  space  may  be  identified  with 
the  functional  Hilbert'  space  H(^  of  analytic  functions  on  D  with  reproducing  kernel 
(I  -  zw)~x,  z,  w  €  D. 

•  Anti-holomorphic  discrete  series  representations  D^,  X.  >  0:  D^  may  be  defined  as 
the  composition  of  D*  with  the  automorphism  *  of  eq.  (1.2):  D^(<p)  =  D*((p*),  (p  in 
Mob.  This  may  be  realized  on  a  functional  Hilbert  space  of  anti-holomorphic  functions 
on  D,  in  a  natural  way. 

•  Complementary  series  representation  CA,<T>    ~~1  <  X  <  1,  0<<J  <  ^(1  —  |A|):  Here 
A.  =  X,  IJL  —  j(l  -  X)  +  cr,  /  =  Z,  and 


422  Bhaskar  Bagchi  and  Gadadhar  Misra 

-  - 

= 


Jk=0  2  +  2  +  a 

where  one  takes  the  upper  or  lower  sign  according  as  n  is  positive  or  negative. 

Remark  3.1.  (a)  All  these  projective  representation  of  Mob  are  irreducible  with  the  sole 
exception  of  P\$  for  which  we  have  the  decomposition  P\$  =  D*  0  Z)  [~  .  (b)  The  multiplier 
associated  with  each  of  these  representations  is  m^  where  co  =  e~/7T;c  if  the  representation 
is  in  the  anti-holomorphic  discrete  series,  and  a)  =  e/7r^  otherwise.  It  follows  that  the 
multipliers  associated  with  two  representations  n\  and  TO  from  this  list  are  either  identical 
or  inequivalent.  Further,  if  neither  or  both  of  n\  and  Jti  are  from  the  anti-holomorphic 
discrete  series,  then  their  multipliers  are  identical  iff  their  A,  parameters  differ  by  an  even 
integer.  In  the  contrary  case  (i.e.,  if  exactly  one  ofn\  and  KI  is  from  the  anti-holomorphic 
discrete  series),  then  they  have  identical  multipliers  iff  their  A.  parameters  add  to  an  even 
integer.  This  is  Corollary  3.2  from  [9],  Using  this  information,  one  can  now  describe  all 
the  projective  representations  of  Mob  (at  least  in  principle). 

4.  Projective  representations  and  homogeneous  operators 

If  T  is  an  operator  on  a  Hilbert  space  7i  then  a  projective  representation  it  of  Mob  on  H  is 
said  to  be  associated  with  T  if  the  spectrum  of  T  is  contained  in  6  and 

<p(T)  =  n(<p)*Tn(<p)  (4.1) 

for  all  elements  <p  of  Mob.  Clearly,  if  T  has  an  associated  representation  then  T  is  homo- 
geneous. In  the  converse  direction,  we  have 

Theorem  4.1.  IfT  is  an  irreducible  homogeneous  operator  then  T  has  a  projective  repre- 
sentation of  Mob  associated  with  it.  This  projective  representation  is  unique  up  to  equiva- 
lence. 

We  sketch  a  proof  of  Theorem  4.1  below.  Thedetailsof  the  proof  may  be  found  in  [9].  The 
existence  part  of  this  theorem  was  first  proved  in  [23]  using  a  powerful  selection  theorem. 
This  result  is  the  prime  reason  for  our  interest  in  projective  unitary  representations  of  Mob.  It 
is  also  the  basic  tool  in  the  classification  program  for  the  irreducible  homogeneous  operators 
which  is  now  in  progress. 

Sketch  of  Proof.  Notice  that  the  scalar  unitaries  in  U(H)  form  a  copy  of  the  circle  group  T 
in  U(H).  There  exist  Borel  transversals  E  to  this  subgroup,  i.e.,  Borel  subsets  E  ofU(H) 
which  meet  every  coset  of  T  in  a  singleton.  Fix  one  such  (in  the  Proof  of  Theorem  2.2  in 
[9],  we  present  an  explicit  construction  of  such  a  transversal).  For  each  element  p  of  Mob, 
let  Ey  denote  the  set  of  all  unitaries  U  in  U(H)  such  that  U*TU  =  <p(T).  Since  T  is  an 
irreducible  homogeneous  operator,  Schur's  Lemma  implies  that  each  E9  is  a  coset  of  T  in 
U(H).  Defines  :  Mob  ->  U(H)  by 


Homogeneous  operators  and  protective  representations  423 

It  is  easy  to  see  that  TT,  thus  defined,  is  indeed  a  projective  representation  associated  with  T. 
Another  appeal  to  Schur's  Lemma  shows  that  any  representation  associated  with  T  must 
be  equivalent  to  TT.  This  completes  the  proof. 

For  any  projective  representation  TT  of  Mob,  let  TT#  denote  the  projective  representation  of 
Mob  obtained  by  composing  TT  with  the  automorphism  *  of  Mob  (cf.  (1.2)).  That  is, 

7r*(<p)  :=  7r(0>*),   <p  e  Mob.  (4.2) 

Clearly,  if  m  is  the  multiplier  of  TT,  then  m  is  the  multiplier  of  TT#.  Also,  from  (1.3)  it  is 
more  or  less  immediate  that  if  TT  is  associated  with  a  homogeneous  operator  T  then  TT#  is 
associated  with  the  adjoint  T*  of  T.  If,  further,  T  is  invertible,  then  JT#  is  associated  with 
T~l  also. 

4.1  Classification  of  irreducible  homogeneous  operators 

Recall  that  an  operator  T  on  a  Hilbert  space  T~L  is  said  to  be  a  block  shift  if  there  are  non  trivial 
subspaces  Vn  (indexed  by  all  integers,  all  non-negative  integers  or  all  non-positive  integers 
-  accordingly  T  is  called  a  bilateral,  forward  unilateral  or  backward  unilateral  block  shift) 
such  that  H  is  the  orthogonal  direct  sum  of  these  subspaces  and  we  have  T(Vn)  c  Vn+\ 
for  each  index  n  (where,  in  the  case  of  a  backward  block  shift,  we  take  V\  =  {0}).  In  [9] 
we  present  a  proof  (due  to  Ordower)  of  the  somewhat  surprising  fact  that  in  case  T  is  an 
irreducible  block  shift,  these  subspaces  Vn  (which  are  called  the  blocks  of  T)  are  uniquely 
determined  by  T.  This  result  lends  substance  to  the  following  theorem. 

For  any  connected  semi-simple  Lie  group  G  takes  a  maximal  compact  subgroup  IK  of  G 
(it  is  unique  up  to  conjugation).  Let  IK  denote,  as  usual,  the  set  of  all  irreducible  (ordinary) 
unitary  representation  of  K  (modulo  equivalence).  Let  us  say  that  a  projective  representa- 
tion IT  of  G  is  normalized  \f  n\^  is  an  ordinary  representation  of  IK.  (If  H2(K,  J)  is  trivial, 
then  it  is  easy  to  see  that  every  projective  representation  of  G  is  equivalent  to  a  normalized 
representation).  If  TT  is  normalized,  then,  for  any  x  £  IK,  let  Vx  denote  the  subspace  of  Hn 
(the  space  on  which  TT  acts)  given  by 

Vx  =  {v  e  Hn  :  n(k)v  =  x(k)v  Vk  e  K}. 

Clearly  Ti^  is  the  orthogonal  direct  sum  of  the  subspaces  Vx ,  x  £  ^.  The  subspace  Vx  is 
called  the  K-isotypic  subspace  of  l~in  of  type  x  - 

In  particular,  for  the  group  G  =  Mob,  we  may  take  (K  to  be  the  copy  {(pa,Q  :  a  6  T}  of  the 
circle  group  T.  (IK  may  be  identified  with  T  via  a  H>  <pa$-)  For  n  as  above  and  n  €  Z,  let 
V/i(7r)  denote  the  [K-isotypic  subspace  corresponding  to  the  character  Xn  :  I  ^  z~n>  z  ^  ~^- 
With  these  notations,  we  have  the  following  theorem  from  [9]. 

Theorem  4.2.  Any  irreducible  homogeneous  operator  is  a  block  shift.  Indeed,  ifT  is  such 
an  operator,  and  n  is  a  normalized  projective  representation  associated  with  T  then  the 
blocks  ofT  are  precisely  the  non-trivial  K-isotypic  subspaces  ofn. 

(Note  that  if  T  is  an  irreducible  homogeneous  operator,  then  by  Theorem  4.1  there  is  a 
representation  TT  associated  with  T.  Since  such  a  representation  is  determined  only  up  to 
equivalence,  we  may  replace  n  by  a  normalized  representation  equivalent  to  it.  Then  the 
above  theorem  applies.) 


424  Bhaskar  Bagchi  and  Gadadhar  Misra 

A  block  shift  is  called  a  weighted  shift  if  its  blocks  are  one-dimensional.  In  [9]  we  define 
a  simple  representation  of  Mob  to  be  a  normalized  representation  TT  such  that  (i)  the  set 
T(TT)  :=  {n  6  Z  :  Vn(n)  ^  {0}}  is  connected  (in  an  obvious  sense)  and  (ii)  for  each 
ft  e  T(JT),  Vn(n)  is  one  dimensional.  If  T  is  an  irreducible  homogeneous  weighted  shift, 
then,  by  the  uniqueness  of  its  blocks  and  by  Theorem  4.2,  it  follows  that  any  normalized 
representation  TT  associated  with  T  is  necessarily  simple.  Using  the  list  of  irreducible 
projective  representations  of  Mob  given  in  the  previous  section  (along  with  Remark  3.  l(b) 
following  this  list)  one  can  determine  all  the  simple  representations  of  Mob.  This  is  done 
in  Theorem  3.3  of  [9].  Namely,  we  have 

Theorem  4.3.  Up  to  equivalence,  the  only  simple  projective  unitary  representations  of  Mob 
are  its  irreducible  representations  along  with  the  representations  D£  ©  ^7-^  »   0  <  X  <  2. 

Since  the  representations  associated  with  irreducible  homogeneous  shifts  are  simple,  to 
complete  a  classification  of  these  operators,  it  now  suffices  to  take  each  of  the  representa- 
tions TT  of  Theorem  4.3  and  determine  all  the  homogeneous  operators  T  associated  with 
jr.  Given  that  Theorem  4.2  pinpoints  the  way  in  which  such  an  operator  T  must  act  on 
the  space  of  TT,  it  is  now  a  simple  matter  to  complete  the  classification  of  these  operators 
(at  least  it  is  simple  in  principle  -  finding  the  optimum  path  to  this  goal  turns  out  to  be  a 
challenging  task!).  To  complete  a  classification  of  all  homogeneous  weighted  shifts  (with 
non-zero  weights  -  permitting  zero  weights  would  introduce  uninteresting  complications), 
one  still  needs  to  find  the  reducible  homogeneous  shifts.  Notice  that  the  technique  outlined 
here  fails  in  the  reducible  case  since  Theorem  4.1  does  not  apply.  However,  in  Theorem 
2.1  of  [9],  we  were  able  to  show  that  there  is  a  unique  reducible  homogeneous  shift  with 
non-zero  weights,  namely  the  unweighted  bilateral  shift  B.  Indeed,  if  T  is  a  reducible  shift 
(with  non-zero  weights)  such  that  the  spectral  radius  of  T  is  =  1  ,  then  it  can  be  shown  that 
Tk  =  Bk  for  some  positive  integer  k,  and  hence  Tk  is  unitary.  But  Lemma  2.  1  in  [9]  shows 
that  if  T  is  a  homogeneous  operator  such  that  Tk  is  unitary,  then  T  itself  must  be  unitary. 
Clearly,  B  is  the  only  unitary  weighted  shift.  This  shows  that  B  is  the  only  reducible  ho- 
mogeneous weighted  shift  with  non-zero  weights.  When  all  this  is  put  together,  we  have 
the  main  theorem  of  [9]. 

Theorem  4.4.  Up  to  unitary  equivalence,  the  only  homogeneous  weighted  shifts  are  the 
known  ones  (namely,  the  first  five  series  of  examples  from  the  list  in  §6). 

Yet  another  link  between  homogeneous  operators  and  projective  representations  of  Mob 
occurs  in  [10].  Beginning  with  Theorem  2.3,  in  [10]  we  prove  a  product  formula,  involv- 
ing a  pair  of  projective  representations,  for  the  characteristic  function  of  any  irreducible 
homogeneous  contraction.  Namely  we  have 

Theorem  4.5.  IfT  is  an  irreducible  homogeneous  contraction  then  its  characteristic  func- 
tion 0  :  O  -*  B(/C,  £)  is  given  by 


where  n  and  a  are  two  projective  representations  of  Mob  (on  the  Hilbert  spaces  C  and  JC 
respectively)  with  a  common  multiplier.  Further,  C  :  JC  ->  C  is  a  pure  contraction  which 
intertwines  a  |  ^  and  n  \  ^. 


Homogeneous  operators  and  project  ive  representations  425 

Conversely,  whenevern,  a  are  projective  representations  of  Mob  with  a  common  multi- 
plier and  C  is  a  purely  contractive  intertwine  r  between  <?  |  ^  and  n  \  ^  such  that  the  function 
0  defined  by  Q(z)  —  n((pz)*Ca((pz)  is  analytic  on  0,  then  0  is  the  characteristic  function 
of  a  homogeneous  cnu  contraction  (not  necessarily  irreducible). 

(Here  tpz  is  the  involution  in  Mob  which  interchanges  0  and  z.  Also,  IK  =  {cp  e 
Mob  :  <£>(0)  =  0}  is  the  standard  maximal  compact  subgroup  of  Mob.) 


Sketch  of  Proof.  Let  9  be  the  characteristic  function  of  an  irreducible  homogeneous  cnu 
contraction  T  .  For  any  (p  in  Mob  look  at  the  set 

E9  :=  {(U,  V)  :  U*0(w)V  =  0(<p~l(w))  Viy  e  D}  C  U(C}  xU(K). 

By  Theorem  2.3,  E9  is  non-empty  for  each  <p.  By  Theorem  3.4  in  [25],  for  (U,  V)  e  E^ 
there  is  a  unitary  operator  r(C7,  V)  such  that  (i)  r(/7,  V)*Tr(U,  V)  -  (p(T)  and  (ii)  the 
restriction  of  r(U,  V)  to  £  and  /C  equal  U  and  V  respectively.  Therefore,  irreducibility  of 
T  implies  that,  for  (£7,  V),  (£/',  V7)  in  E^  r(U',  V')M^,  V)  is  a  scalar  unitary.  Hence 
EP  is  a  coset  of  the  subgroup  S  (isomorphic  to  the  torus  T2)  of  U(£)  x  U(JC]  consisting 
of  pairs  of  scalar  unitaries.  As  in  the  proof  of  Theorem  4.1,  it  follows  that  there  are 
projective  unitary  representations  n  and  a  with  a  common  multiplier  (on  the  spaces  C  and 
/C  respectively)  such  that  (n((p),  cr((p))  e  E^  for  all  (p  in  Mob.  So  we  have 

n(<p)*6(w)a((p)  =  0((p~l(w)),    w  e  D,  <p  e  Mob.  (4.3) 

Now,  choose  (p  =  (pz  and  evaluate  both  sides  of  (4.3)  at  w  =  0  to  find  the  claimed  formula 
for  9  with  C  =  0(0).  Also,  taking  w  =  0  and  cp  e  K  in  (4.3),  one  sees  that  C  intertwines 


For  the  converse,  let  0(z)  :=  7t((pz)*Ca(<pz)  be  an  analytic  function.  Since  C  =  0(0) 
is  a  pure  contraction  and  0(z)  coincides  with  0(0)  for  all  z,  0  is  pure  contraction  valued. 
Hence  0  is  the  characteristic  function  of  a  cnu  contraction  T.  For  <p  e  Mob  and  w  G  0, 
write  <pw<p  =  k<pz  where/:  €  IK  and  z  =  (<Pw<P)~l(0)  =  <p~[(w).  Then  we  have 


—     7r(k(pz)*Ccr(k(pz) 
=     7r(<pz)*n(k)*C 
=     n((pz)*Ca(<pz) 


(Here,  for  the  second  and  fourth  equality  we  have  used  the  assumption  that  n  and  a 
are  projective  representations  with  a  common  multiplier.  For  the  penultimate  equality, 
the  assumption  that  C  intertwines  cr\^  and  n\^  has  been  used.)  Thus  0  satisfies  (4.3). 
Therefore  0  o  (p  coincides  with  0  for  all  cp  in  Mob.  Hence  Theorem  2.3  implies  that  T  is 
homogeneous. 

5.  Some  constructions  of  homogeneous  operators 

Let  us  say  that  a  projective  representation  n  of  Mob  is  a  multiplier  representation  if  it  is 
concretely  realized  as  follows,  n  acts  on  a  Hilbert  space  H  of  E  -  valued  functions  on 


426  Bhaskar  Bagchi  and  Gadadhar  Misra 

Q,  where  Q  is  either  D  or  T  and  £  is  a  Hilbert  space.  The  action  of  n  on  H  is  given  by 
(n((p)f)(z)  =  c(<pi  z)f((p~lz)  for  z  e  &,  f  e  K,  <P  e  Mob.  Here  c  is  a  suitable  Borel 
function  from  Mob  x  Q  into  the  Borel  group  of  invertible  operators  on  E. 

Theorem  5.1.  Let  H  be  a  Hilbert  space  of  functions  on  £2  such  that  the  operator  T  on  H 
given  by 

=  */(*),  xeQ,  /€«, 


is  bounded.  Suppose  there  is  a  multiplier  representation  TC  of  Mob  on  T.  Then  T  is  homo- 
geneous and  n  is  associated  with  T. 

This  easy  but  basic  construction  is  from  Proposition  2.3  of  [6].  To  apply  this  theorem, 
we  only  need  a  good  supply  of  what  we  have  called  multiplier  representations  of  Mob. 
Notice  that  all  the  irreducible  projective  representations  of  Mob  (as  concretely  presented 
in  the  previous  section)  are  multiplier  representations. 

A  second  construction  goes  as  follows.  It  is  contained  in  Proposition  2.4  of  [6]. 

Theorem  5.2.  Let  T  be  a  homogeneous  operator  on  a.  Hilbert  space  *H  with  associated 
representation  TC.  Let  JC  be  a  sub  space  ofH  which  is  invariant  or  co-invariant  under  both 
T  and  n.  Then  the  compression  ofT  to  JC  is  homogeneous.  Further,  the  restriction  ofn  to 
1C  is  associated  with  this  operator. 

A  third  construction  (as  yet  unreported)  goes  as  follows: 

Theorem  5.3.  Let  n  be  a  projective  representation  of  Mob  associated  with  two  homoge- 
neous operators  T\  and  TI  on  a  Hilbert  space  7i.  Let  T  denote  the  operator  onJit&'H 
given  by 

/r,    T,-r2 


Then  T  is  homogeneous  with  associated  representation    n  ®  jr. 
Sketch  of  proof.  For  (p  in  Mob,  one  verifies  that 


Hence  it  is  clear  that  n  ®  n  is  associated  with  T. 

6.  Examples  of  homogeneous  operators 

It  would  be  tragic  if  we  built  up  a  huge  theory  of  homogeneous  operators  only  to  find  at 

the  end  that  there  are  very  few  of  them.  Here  are  some  examples  to  show  that  this  is  not 

going  to  happen. 

•  The  principal  series  example.  The  unweighted  bilateral  shift  B  (i.e.,  the  bilateral  shift 
with  weight  sequence  wn  =  1,  n  =  0,  ±1,  .  .  .)  is  homogeneous.  To  see  this,  apply 
Theorem  5.1  to  any  of  the  principal  series  representations  of  Mob.  By  construction,  all 
the  principal  series  representations  are  associated  with  B. 


Homogeneous  operators  and  protective  representations  427 


rhe  discrete  series  examples.  For  any  real  number  A.  >  0,  the  unilateral  shift  M^  >  with 
veight  sequence  J  ^—^-,  n  =  0,  1,  2, ...  is  homogeneous.  To  see  this,  apply  Theorem 
5.1  to  the  discrete  series  representation  D£ . 

•A,  >  1,  M^  is  a  cnu  contraction.  For  A,  =  1,  its  characteristic  function  is  the  (constant) 
ction  0  -  not  very  interesting!  But  for  A.  >  1  we  proved  the  following  formula  for  the 
iracteristic  function  of  M^  (cf.  [7]). 


eorem  6.1.  For  A.  >  1,  the  characteristic  function  of  M^  coincides  with  the  function 
%iven  by 

0X(Z)  =  (X(A  -  l))"1/2Dt-i(^z)*8*^+i(^)»    *  €  D, 
ere  8*  «•  die  adjoint  of  the  differentiation  operator  3  :  H^"^ 
is  theorem  is,  of  course,  an  instance  of  the  product  formula  in  Theorem  4.5. 


The  anti-holomorphic  discrete  series  examples.    These  are  the  adjoints  M^*  of  the 
operators  in  the  previous  family.  The  associated  representation  is  D^. 

vas  shown  in  [22]  that 


eorem  6.2.  Up  to  unitary  equivalence,  the  operators  M^*,  A.  >  0  are  the  only  homo- 
leous  operators  in  the  Cowen-Douglas  class  Z?i(O). 

Fhis  theorem  was  independently  re-discovered  by  Wilkins  in  ([33],  Theorem  4.  1  ). 
The  complementary  series  examples.  For  any  two  real  numbers  a  and  b  in  the  open  unit 
interval  (0,  1),  the  bilateral  shift  Kaj}  with  weight  sequence  Jjjqzf  ,  n  =  0,  ±1  ,  ±2, 
is  homogeneous.  To  see  this  in  case  0  <  a  <  b  <  1,  apply  Theorem  5.1  to  the 
complementary  series  representation  C\,a  with  A.  =  a  -f  b  —  1  and  or  =  (b  —  a.)/2.  If 
i  =  b  then  Kaj  =  B  is  homogeneous.  If  0  <  Z?  <  <7.  <  1  then  Kaj7  is  the  adjoint 
inverse  of  the  homogeneous  operator  Kb,a>  anc*  hence  is  homogeneous. 
The  constant  characteristic  examples.  For  any  real  number  A.  >  0,  the  bilateral  shift  B), 
with  weight  sequence  .  .  .  ,  1,  1,  1,  A.,  1,  1,  1,  .  .  .  ,  (A.  in  the  zeroth  slot,  1  elsewhere)  is 
homogeneous.  Indeed,  if  0  <  A.  <  1  then  BA,  is  a  cnu  contraction  with  constant  charac- 
teristic function  —A.;  hence  it  is  homogeneous.  Of  course,  B\  =  B  is  also  homogeneous. 
If  A.  >  1,  BX  is  the  inverse  of  the  homogeneous  operator  BIL  with  ju,  =  AT1,  hence  it 
is  homogeneous.  (In  [6]  we  presented  an  unnecessarily  convoluted  argument  to  show 
that  BX  is  homogeneous  for  X  >  1  as  well.)  It  was  shown  in  [6]  that  the  representation 
Df  ®  D]~  is  associated  with  each  of  the  operators  BX,  A,  >  0.  (Recall  that  this  is  the 
only  reducible  representation  in  the  principal  series!) 

In  [6]  we  show  that  apart  from  the  unweighted  unilateral  shift  and  its  adjoint,  the  operators 
.,  A.  >  0  are  the  only  irreducible  contractions  with  a  constant  characteristic  function. 
fact, 

icorem  6.3.  The  only  cnu  contractions  with  a  constant  characteristic  function  are  the 
-ect  integrals  of  the  operators  M(1),  M(l)*  and  BX,  X  >  0. 


428  Bhaskar  Bagchi  and  Gadadhar  Misra 

Since  all  the  constant  characteristic  examples  are  associated  with  a  common  represen- 
tation, one  might  expect  that  the  construction  in  Theorem  5.3  could  be  applied  to  any 
two  of  them  to  yield  a  plethora  of  new  examples  of  homogeneous  operators.  Unfortu- 
nately, this  is  not  the  case.  Indeed,  it  is  not  difficult  to  verify  that  for  A.  ^  /z,  the  operator 

in"       ^  o     '    I  is  unitai%  equivalent  to  Ba  0  B&  where  a  and  8  are  the  eigenvalues  of 

\     U  ^/A  / 


Notice  that  the  examples  of  homogeneous  operators  given  so  far  are  all  weighted  shifts. 
By  Theorem  4.4,  these  are  the  only  homogeneous  weighted  shifts  with  non-zero  weights. 
Wilkins  was  the  first  to  come  up  with  examples  of  (irreducible)  homogeneous  operators 
which  are  not  scalar  shifts. 

•  The  generalized  Wilkins  examples.  Recall  that  for  any  real  number  A.  >  0,  7i(x)  denotes 
the  Hilbert  space  of  analytic  functions  on  D  with  reproducing  kernel  (z,  w)  H>  (1  - 
zw)~~k.  (It  is  the  Hilbert  space  on  which  the  holomorphic  discrete  series  representation 
D*  lives.)  For  any  two  real  numbers  X\  >  0,  A,  2  >  0,  and  any  positive  integer  £, 
view  the  tensor  product  7i(Xl}  ®  W(A-2)  as  a  space  of  analytic  functions  on  the  bidisc 
D  x  D.  Look  at  the  Hilbert  space  y^1*^)  c  ft(*i)  0  ft(*i)  defined  as  the  ortho- 
complement  of  the  subspace  consisting  of  the  functions  vanishing  to  order  k  on  the 
diagonal  A  =  {(z,z)  :  z  €  D}  c  Ox  ED.  Finally  define  the  generalized  Wilkins  operator 
W^1'*2*  as  the  compression  to  y^-**)  of  the  operator  M(x')<8>7  on  W(Xi}®W(X2).  The 
subspace  Vk  !l  2  is  co-invariant  under  the  homogeneous  operator  M(XI)  ®  /  as  well  as 
under  the  associated  representation  D^  ®  Z)£.  Therefore,  by  Theorem  5.2,  W^1'  ^  is 
a  homogeneous  operator.  For  k  -  1,  W^  ^2)  is  easily  seen  to  be  unitarily  equivalent  to 
M(Xl+X2),  see  [7]  and  [14],  for  instance.  But  for  k  >  2,  these  are  new  examples. 
The  operator  w£Xl'X2)  may  alternatively  be  described  as  multiplication  by  the  co-ordinate 
function  z  on  the  space  of  C*-valued  analytic  functions  on  D  with  reproducing  kernel 


(Here  3  and  8  denote  differentiation  with  respect  to  z  and  u),  respectively.)  Indeed  (with 
the  obvious  identification  of  A  and  D)  the  map  /  H>  (/,  /',...,  /(*~1})|  A  is  easily  seen 
to  be  a  unitary  between  y^1'^  and  this  reproducing  kernel  Hilbert  space  intertwining 
Wk  and  the  multiplication  operator  on  the  latter  space.  (This  is  a  particular  instance 
of  they'**  construction  discussed  in  [15].)  Using  this  description,  it  is  not  hard  to  verify  that 
the  adjoint  of  Wk  l1  2  is  an  operator  in  the  Cowen-Douglas  class  B*(D).  The  following 
is  (essentially)  one  of  the  main  results  in  [34]. 

Theorem  6.4.  Up  to  unitary  equivalence,  the  only  irreducible  homogeneous  operators  In 
the  Cowen-Douglas  class  B2(B)  are  the  adjoints  of  the  operators  W^]'X2\X.  i  >  0,  A.2  >  0. 

This  is  not  the  description  of  these  operators  given  in  [34].  But  it  can  be  shown  that 
Wilkin's  operator  T£Q  is  unitarily  equivalent  to  the  operator  WO(AI>  A>2)  with  A  =  A.  i  +  A.2+  I  , 
Q  =  (A}  -f  X2  +  I)/  (A.2  +  I).  Indeed,  though  his  reproducing  kernel  H^Q  looks  a  little 
different  from  the  kernel  (with  k  =  2)  displayed  above,  a  calculation  shows  that  these 
two  kernels  have  the  same  normalization  at  the  origin  (cf.  [12]),  so  that  the  corresponding 


Homogeneous  operators  and  projective  representations  429 

multiplication  operators  are  unitarily  equivalent.  However,  it  is  hard  to  see  how  Wilkins 
arrived  at  his  examples  T£  while  the  construction  of  the  operators  W^  ''  given  above 
has  a  clear  geometric  meaning,  particularly  in  view  of  Theorem  5.2.  But,  as  of  now,  we 
know  that  the  case  k  =  2  of  this  construction  provides  a  complete  list  of  the  irreducible 
homogeneous  operators  in  #2(B)  only  by  comparing  them  with  Wilkins'  list-  we  have  no 
independent  explanation  of  this  phenomenon. 

Theorem  6.1  has  the  following  generalization  to  some  of  the  operators  in  this  series. 
(Theorem  6.1  is  the  special  case  k  =  1  of  this  theorem.) 

Theorem  6.5.  For  k  =  1,2,...  and  real  numbers  A.  >  k,  the  characteristic  func- 
tion of  the  operator  W^  1  ~  coincides  with  the  inner  analytic  function  0^  :  O  -> 
:))  given  by 


#f  }(z)  =  cuD+_k(<pz)*dk*D++k(<pz),  z  e  D. 
Here  dk*  is  the  adjoint  of  the  k-times  differentiation  operator  dk  :  ft^"®  ->  H^+V  and 


Sketch  of  Proof.  It  is  easy  to  check  that  C  :=  ^9**  is  a  pure  contraction  intertwining  the 
restrictions  to  IK  of  D^+k  and  D*_k.  Since  we  already  know  (by  Theorem  6.  1)  that  Q%  is 
an  inner  analytic  function  for  k  =  I  ,  the  recurrence  formula 


_ 

K~T~  I  I  k  —  I     I 

(for  k  >  I,  X  >  k  4-  I,  with  the  interpretation  that  9^  denotes  the  constant  function  I) 
shows  that  9^  is  an  inner  analytic  function  on  D  for  A.  >  k,  k  =  1,2,  —  Hence  it  is 
the  characteristic  function  of  a  cnu  contraction  T  in  the  class  C.Q.  By  Theorem  2.1,  T 
is  the  compression  to  ML  of  the  multiplication  operator  on  H^  ®  H^~k\  where  M  is 
the  invariant  subspace  corresponding  to  this  inner  function.  But  one  can  verify  that  M  is 
the  subspace  consisting  of  the  functions  vanishing  to  order  k  on  the  diagonal.  Therefore 

T  =  <-X-*). 

•  Some  perturbations  of  the  discrete  series  examples.  Let  H  be  a  Hilbert  space  with 
orthonormal  basis  [fk  :  k  =  0,  1,  .  .  .}  U  {hk^  •  k  =  0,  ±1,  ±2,  .  .  .}.  For  any  three 
strictly  positive  real  numbers  A,  [L  and  8,  let  M(x)[/z,  8]  be  the  operator  on  H  given  by 


and 

,   for  k> 


An  application  of  Theorem  2.2  to  the  operators  M(X)  in  conjunction  with  an  analytic 
continuation  argument  shows  that  these  operators  are  homogeneous.  This  was  observed 
in  [7]. 


430  Bhaskar  Bagchi  and  Gadadhar  Misra 

»  The  normal  atom.  Define  the  operator  TV  on  L2(O)  by  (Nf)(z)  =  z/(z),  z  €  0,  /  G 
L2(0).  The  discrete  series  representation  D*  naturally  lifts  to  a  representation  of  Mob 
on  L2(O).  Applying  Theorem  5.1  to  this  representation  yields  the  homogeneity  of  N. 

Using  spectral  theory,  it  is  easy  to  see  that  the  operators  B  and  N  are  the  only  homogeneous 
normal  operators  of  multiplicity  one.  In  consequence,  we  have 


Theorem  6.6.  Every  normal  homogeneous  operator  is  a  direct  sum  of  (countably  many) 
copies  of  B  and  N. 

Let  us  define  an  atomic  homogeneous  operator  to  be  a  homogeneous  operator  which 
can  not  be  written  as  the  direct  sum  of  two  homogeneous  operators.  Trivially,  irreducible 
homogeneous  operators  are  atomic.  As  an  immediate  consequence  of  Theorem  6.6,  we 
have 

COROLLARY  6.1 

B  and  N  are  atomic  (but  reducible)  homogeneous  operators. 

N  is  a  cnu  contraction.  Its  characteristic  function  was  given  in  [7]. 

Theorem  6.7.  The  characteristic  function  ON  :  O  -»  #(L2(B))  of  the  operator  N  is  given 
by  the  formula 

(0N(z)f)(w)  =  -<pw(z)f(w),    z,  w  €  O,  /  €  L2(0). 

(Here,  as  before,  (pw  is  the  involution  in  Mob  which  interchanges  0  and  w.) 

The  usual  transition  formula  between  cartesian  and  polar  coordinates  shows  that  L2(D)  = 
L2(T)  ®  L2([0,  1],  rdr).  Since  B  may  be  represented  as  multiplication  by  the  coordinate 
function  on  L2(T),  it  follows  that  the  normal  atom  N  is  related  to  the  other  normal  atom 
BbyN  =  B®C  where  C  is  multiplication  by  the  coordinate  function  on  L2([0,  1],  rdr). 
Clearly  C  is  a  positive  contraction.  Let  [fn  :  n  >  0}  be  the  orthonormal  basis  of 
L2([0,  1],  rdr)  obtained  by  Gram-Schmidt  orthogonal izati on  of  the  sequence  {r  \-+ 
r11  :  n  >  0}.  (Except  for  scaling,  /„  is  given  in  terms  of  classical  Jacobi  polynomi- 
als by  x  H>  P7I(ai)(2j  -  1),  cf.  [31].)  Then  the  theory  of  orthogonal  polynomials  shows 
that  (with  respect  to  this  orthonormal  basis)  C  is  a  tri-diagonal  operator.  Thus  we  have 

Theorem  6.8.   Up  to  unitary  equivalence,  we  have  N  =  B  ®  C  where  the  positive  con- 
traction  C  is  given  on  a  Hilbert  space  with  orthonormal  basis  {fn  :  n  >  0}  by  the  formula 

Cfn  =  fljj/n-l  +  bnfn  +  «;i+l//i  +  l,     n  =  0,  1 ,  2,  .  .  . 

where. (f -\  =  0)  and  the  constants  an,  bn  are  given  by 

2(n+l)2 


Homogeneous  operators  and  projective  representations  43 1 

7.  Open  questions 

7.1  Classification 

The  primary  question  in  this  area  is,  of  course,  the  classification  of  homogeneous  operators 
up  to  unitary  equivalence.  Theorem  4.4  is  a  beginning  in  this  direction.  We  expect  that  the 
same  methodology  will  permit  us  to  classify  all  the  homogeneous  operators  in  the  Cowen- 

Douglas  classes  5jt(O),  k  —  1,2, Work  on  this  project  has  already  begun.   More 

generally,  though  there  seem  to  be  considerable  difficulties  involved,  it  is  conceivable  that 
extension  of  the  same  techniques  will  eventually  classify  all  irreducible  homogeneous  op- 
erators. But,  depending  as  it  does  on  Theorem  4.1,  this  technique  draws  a  blank  when  it 
comes  to  classifying  reducible  homogeneous  operators.  In  particular,  we  do  not  know  how 
to  approach  the  following  questions. 

Question  1 .  Is  every  homogeneous  operator  a  direct  integral  of  atomic  homogeneous 
operators? 

Question  2.  Are  B  and  N  the  only  atomic  homogeneous  operators  which  are  not  irreducible? 

We  have  seen  that  the  homogeneous  operator  N  can  be  written  as  N  =  B  ®  C.  In  this 
connection,  we  can  ask: 

Question  3.  Find  all  homogeneous  operators  of  the  form  B  ®  X.  More  generally,  find  all 
homogeneous  operators  which  have  a  homogeneous  operator  as  a  'tensor  factor'. 

Another  possible  approach  towards  the  classification  of  irreducible  homogeneous  con- 
tractions could  be  via  Theorem  4.5.  (Notice  that  any  irreducible  operator  is  automatically 
cnu.)  Namely,  given  any  two  projective  representations  n  and  a  of  Mob  having  a  common 
multiplier,  we  can  seek  to  determine  the  class  C(n,  cr)  of  all  operators  C  :  H,a  -*  Tin  such 
that  (i)  C  intertwines  o~\^  and  ;r  |^  and  (ii)  the  function  z  i->  7r(^)*Ccr(^)  is  analytic 
on  D.  Clearly  C(TT,  or)  is  a  subspace  of  B(?ia,Hn),  and  Theorem  4.5  says  that  any  pure 
contraction  in  this  subspace  yields  a  homogeneous  operator.  Further,  this  method  yields 
all  irreducible  homogeneous  contractions  as  one  runs  over  all  n  and  a.  This  approach 
is  almost  totally  unexplored.  We  have  only  observed  that,  up  to  multiplication  by  scalars, 
the  homogeneous  characteristic  functions  listed  in  Theorem  6.5  are  the  only  ones  in  which 
both  n  and  a  are  holomorphic  discrete  series  representations.  (But  the  trivial  operation  of 
multiplying  the  characteristic  function  by  scalars  correspond  to  a  highly  non-trivial  opera- 
tion at  the  level  of  the  operator.  This  operation  was  explored  in  [7].)  So  a  natural  question 
is: 

Question  4.  Determine  C(TT,  cr)  at  least  for  irreducible  projective  representations  n  and  a 
(with  a  common  multiplier). 

Note  that  Theorem  6.5  gives  the  product  formula  for  the  characteristic  function  of 
W^i,x2)  for  ^^  =  j  But  for  wa,A2)  to  be  a  contraction  it  is  sufficient  (though  not 

necessary)  to  have  k\  >  1.  So  on  a  more  modest  vein,  we  may  ask: 


432  Bhaskar  Bagchi  and  Gadadhar  Misra 

Question  5.  What  is  the  (explicit)  product  formula  for  the  characteristic  functions  of  the 
operators  Wf  J'AoJ  forA.}  >  1? 

Recall  that  a  cnu  contraction  T  is  said  to  be  in  the  class  C\\  if  for  every  nonzero  vector 
x,  linwoo  Tnx  ^  0  and  linWoo  T*nx  ^  0.  In  [19],  Kerchy  asks: 

Question  6.  Does  every  homogeneous  contraction  in  the  class  C\  \  have  a  constant  charac- 
teristic function? 

7.2  Mobius  bounded  and polynomially  bounded  operators 

Recall  from  [30]  that  a  Hilbert  space  operator  T  is  said  to  be  Mobius  bounded  if  the  family 
{(p(T)  :  (p  e  Mob}  is  uniformly  bounded  in  norm.  Clearly  homogeneous  operators  are 
Mobius  bounded,  but  the  converse  is  false.  In  [30],  Shields  proved: 

Theorem  7.1.  IfT  is  a  Mobius  bounded  operator  then  \\Tm  \\  =  O(m)  as  m  — *  oo.   • 


Sketch  of  proof.  Say  ||^(T)||  <  c  for  (p  e  Mob.  For  any  (p  e  Mob,  we  have  an  expansion 
(p(z)  =  Y^m-Q  amZm,  valid  in  the  closed  unit  disc.  Hence, 

r 
amTm  =  /   (p(ctT)ct-m  da, 


where  the  integral  is  with  respect  to  the  normalized  Haar  measure  on  T.  Therefore  we 
get  the  estimate  |am|||r'"||  <  c  for  all  m.  Choosing  (p  =  p^,  we  see  that  for  m  >  1, 
\am\  =  (1  -  r2)rm~l  where  r  =  \(t\.  The  optimal  choice  r  =  J(m  -  l)/(m  4-  1)  gives 
\am\  =  0(l/m)  and  hence  ||Tm||  =  0(m). 

On  the  basis  of  this  Theorem  and  some  examples,  we  may  pose: 

Conjecture.  For  any  Mobius  bounded  operator  T,  we  have  ||  Tm  \\  —  O(m 1/2)  as  m  ->  oo. 

In  [30],  Shields  already  asked  if  this  is  true.  This  question  has  remained  unanswered 
for  more  than  twenty  years.  One  possible  reason  for  its  intractability  may  be  the  dif- 
ficulty involved  in  finding  non-trivial  examples  of  Mobius  bounded  operators.  (Con- 
tractions are  Mobius  bounded  by  von  Neumann's  inequality,  but  these  trivially  satisfy 
Shield's  conjecture.)  As  already  mentioned,  non-contractive  homogeneous  operators  pro- 
vide non-trivial  examples.  For  the  homogeneous  operator  T  =  M (X)  with  k  <  I ,  we 

have  ||rn||  =  yr(r)(L(+X)1)  and  hence  (by  Sterling's  formula)  ||Tm|l  ~  cm(1-W2  with 
c  =  r(X)1/2.  Thus  the  above  conjecture,  if  true,  is  close  to  best  possible  (in  the  sense  that 
the  exponent  1  /2  in  this  conjecture  cannot  be  replaced  by  a  smaller  constant).  An  analogous 
calculation  with  the  complementary  series  examples  C(a,  b)  (with  0  <  a  ^  b  <  1)  leads 
to  a  similar  conclusion.  This  leads  us  to  ask: 

Question  7.  Is  the  conjecture  made  above  true  at  least  for  homogeneous  operators  T1 

(It  is  conceivable  that  the  operators  T^s  introduced  below  contain  counter  examples  to 
Shield's  conjecture  in  its  full  generality.) 


Homogeneous  operators  and  projectile  representations  433 

Recall  that  an  operator  T,  whose  spectrum  is  contained  in  6,  is  said  to  bz  polynomially 
bounded  if  there  is  a  constant  c  >  0  such  that  ||/?(r)||  <  c  for  all  polynomial  maps 
p  :  D  ->  O.  (von  Neumann's  inequality  says  that  this  holds  with  c  =  1  iff  T  is  a 
contraction.)  Clearly,  if  T  is  similar  to  a  contraction  then  T  is  polynomially  bounded. 
Halmos  asked  if  the  converse  is  true,  i.e.,  whether  every  polynomially  bounded  operator  is 
similar  to  a  contraction.  In  [28],  Pisier  constructed  a  counter-example  to  this  conjecture. 
(Also  see  [13]  for  a  streamlined  version  of  this  counter-example.)  However,  one  may 
still  hope  that  the  Halmos  conjecture  is  still  true  of  some  'nice'  classes  of  operators.  In 
particular,  we  ask 

Question  8 .  Is  every  polynomially  bounded  homogeneous  operator  similar  to  a  contraction? 
For  that  matter,  is  there  any  polynomially  bounded  (even  power  bounded)  homogeneous  op- 
erator which  is  not  a  contraction? 

Notice  that  the  discrete  series  examples  show  that  homogeneous  operators  (though 
Mb'bius  bounded)  need  not  even  be  power  bounded.  So  certainly  they  need  not  be  polyno- 
mially bounded. 

7.3  Invariant subspaces 

If  T  is  a  homogeneous  operator  with  associated  representation  TT,  then  for  each  invariant 
subspace  M.  of  T  and  each  (p  e  Mob,  n(y>)(M)  is  again  T-invariant.  Thus  Mob  acts 
on  the  lattice  of  T-invariant  subspaces  via  jr.  We  wonder  if  this  fact  can  be  exploited  to 
explore  the  structure  of  this  lattice.  Further,  if  T  is  a  cnu  contraction,  then  the  Sz-Nagy- 
Foias  theory  gives  a  natural  correspondence  between  the  invariant  subspaces  of  T  and  the 
'regular  factorizations'  of  its  characteristic  function  (cf.  [25]).  Since  we  have  nice  explicit 
formulae  for  the  characteristic  functions  of  the  homogeneous  contractions  M(A.),  A.  >  1, 
may  be  these  formulae  can  be  exploited  to  shed  light  on  the  structure  of  the  corresponding 
lattices. 

Recall  that  Beurling's  theorem  describes  the  lattice  of  invariant  subspaces  of  M^  in 
terms  of  inner  functions.  Recently,  it  was  found  ([18]  and  [1])  that  certain  partial  analogues 
of  this  theorem  are  valid  for  the  Bergman  shift  M^  as  well.  We  may  ask: 

Question  9.  Do  the  theorems  of  Hedenmalm  and  Aleman  et  al  generalize  to  the  family 
M^,  X  >  1  of  homogeneous  unilateral  shifts? 

7.4  Generalizations  of  homogeneity 

In  the  definition  of  homogeneous  operators,  one  may  replace  unitary  equivalence  by  simi- 
larity. Formally,  we  define  a  weakly  homogeneous  operator  to  be  an  operator  T  such  that 
(i)  the  spectrum  of  T  is  contained  in  6  and  (ii)  <p(T)  is  similar  to  T  for  every  <p  in  Mob.  Of 
course,  every  operator  which  is  similar  to  a  homogeneous  operator  is  weakly  homogeneous. 
In  [11]  it  was  asked  if  the  converse  is  true.  It  is  not  -  as  one  can  see  from  the  following 
examples: 

Example  1 .  Take  H  =  L2(T)  and,  for  any  real  number  in  the  range  - 1  <  A  <  1  and  any 
complex  numbers  with  lm(s)  >  0,  define  P^s  :  Mob  ->  B(H)  by 


434  Bhaskar  Bagchi  and  Gadadhar  Misra  . 


For  purely  imaginary  s,  these  are  just  the  principal  series  unitary  projective  representa- 
tions discussed  earlier.  For  s  outside  the  imaginary  axis,  P^iV  is  not  unitary  valued.  But, 
formally,  it  still  satisfies  the  condition  (3.1)  with  m  =  m^,    CD  =  e/7r\  In  consequence, 
PX,.V  is  an  invertible  operator  valued  function  on  Mob. 
For  X  and  s  as  above,  let  T^tS  denote  the  bilateral  shift  on  L2(T)  with  weight  sequence 

n  e  2. 

When  s  is  purely  imaginary,  these  weights  are  unimodular  and  hence  7X,.V  is  unitarily 
equivalent  to  the  unweighted  bilateral  shift  B.  In  [9]  it  is  shown  that,  in  this  case  the 
principal  series  representation  P\tS  is  associated  with  T^tS  as  well  as  to  B.  That  is,  we  have 

<p(Ti.s)  =  PiAvrlT^Pi,s(<p)  (7.1) 

for  purely  imaginary  s.  By  analytic  continuation,  it  follows  that  eq.  (7.1)  holds  for  all 
complex  numbers  s.  Thus  T^s  is  weakly  homogeneous  for  lm(s)  >  0.  It  is  easy  to  see 
that  H7-&H  >  lir^/oll  >  IrKJrSii  where  «  =  0  +  *)/2  +  s,  6  =  (I  +  A.)/2  -  J  and 
/o  is  the  constant  function  1.  Hence  by  Sterling's  formula,  we  get 

IIO  >  cm2Re(j) 

for  all  large  m  (and  some  constant  c  >  0).  If  T^x  were  similar  to  a  homogeneous  operator, 
it  would  be  Mobius  bounded  and  hence  by  Theorem  7.1  we  would  get  \\T"\\\  =  O(m) 
which  contradicts  the  above  estimate  when  Re(s)  >  1/2.  Therefore  we  have 

Theorem  7.2.  The  operators  T^s  is  weakly  homogeneous  for  all  A.,  s  as  above.  However, 
for  Re(sO  >  1/2,  Ms  operator  is  not  Mobius  bounded  and  hence  is  not  similar  to  any 
homogeneous  operator. 

Example  2  (due  to  Ordower).  For  any  homogeneous  operator  T,  say  on  the  Hilbert  space 

(T     I\ 

H,  let  T  denote  the  operator  I  J  .  For  any  cp  in  a  sufficiently  small  neighbourhood 

of  the  identity,  <p(f)  makes  sense  and  one  verifies  that  <p(f)  =  (  ^        ^  L?  V  If  U  is 

\    0        <p(T)  J 

a  unitary  on  H  such  that  <p(T)  =  17*717  then  an  easy  computation  shows  that  the  operator 
L  =  U(p'(T)  1/2e/V(?T1/2  satisfies  LfL"1  =  ^(7).  Thus<Xf)  is  similar  to  f  for  all 
<p  in  a  small  neighbourhood.  Therefore  an  obvious  extension  of  Theorem  1.1  shows  that 
f  is  weakly  homogeneous.  Since  \\<p(f  )||  >  11^(7)  ||  and  since  the  family  (p't  (p  €  Mob 
is  not  uniformly  bounded  on  the  spectrum  of  T,  it  follows  that  f  is  not  Mobius  bounded. 
Therefore  we  have 

Theorem  7.3.  For  any  homogeneous  operator  T,  the  operator  f  is  weakly  homoge- 
neous but  not  Mobius  bounded.  Therefore  this  operator  is  not  similar  to  any  homoge- 
neous operator. 

These  two  classes  of  examples  indicate  that  the  right  question  to  ask  is 


Homogeneous  operators  and  projective  representations  435 

Question  10.  Is  it  true  that  every  Mobius  bounded  weakly  homogeneous  operator  is  similar 
to  a  homogeneous  operator? 

For  purely  imaginary  s9  the  homogeneous  operators  T^s  and  B  share  the  common  as- 
sociated representation  P^y;  hence  one  may  apply  the  construction  in  Theorem  5.3  to  this 
pair.  We  now  ask 

Question  11.    Is  the  resulting  homogeneous  operator  atomic?    Is  it  irreducible?    More 
generally,  are  there  instances  where  Theorem  5.3  lead  to  atomic  homogeneous  operators? 

Another  direction  of  generalization  is  to  replace  the  group  Mob  by  some  subgroup  G. 
For  any  such  G,  one  might  say  that  an  operator  T  is  G-homogeneous  if  (p(T)  is  unitarily 
equivalent  to  T  for  all  'sufficiently  small'  (p  in  G.  (If  G  is  connected,  the  analogue  of 
Theorem  1.1  holds.)  The  case  G  =  IK  has  been  studied  under  the  name  of  'circularly  sym- 
metric operators'.  See,  for  instance,  [17]  and  [3].  Notice  that  if  5  is  a  circularly  symmetric 
operator  then  so  is  5  ®  T  for  any  operator  T  -  showing  that  this  is  a  rather  weak  notion 
and  no  satisfactory  classification  can  be  expected  when  the  group  G  is  so  small.  A  more 
interesting  possibility  is  to  take  G  to  be  a  Fuchsian  group.  (Recall  that  a  closed  subgroup 
of  Mob  is  said  to  be  Fuchsian  if  it  acts  discontinuously  on  D.)  Fuchsian  homogeneity  was 
briefly  studied  by  Wilkins  in  [33].  He  examines  the  nature  of  the  representations  (if  any) 
associated  with  such  an  operator. 

Another  interesting  generalization  is  to  introduce  a  notion  of  homogeneity  for  commuting 
tuples  of  operators.  Recall  that  a  bounded  domain  fil  in  Cd  is  said  to  be  a  bounded  symmetric 
domain  if,  for  each  co  e  £2,  there  is  a  bi-holomorphic  involution  of  Q  which  has  a)  as 
an  isolated  fixed  point.  Such  a  domain  is  called  irreducible  if  it  cannot  be  written  as  the 
cartesian  product  of  two  bounded  symmetric  domains.  The  irreducible  bounded  symmetric 
domains  are  completely  classified  modulo  biholomorphic  equivalence  (see  [2]  or  [16]  for 
instance)  -  they  include  the  unit  ball  Imjl  in  the  Banach  space  of  all  m  x  n  matrices 
(with  operator  norm).  Let  G^  denote  the  connected  component  of  the  identity  in  the 
group  of  all  bi-holomorphic  automorphisms  of  an  irreducible  bounded  symmetric  domain 
£2.  If  T  =  (!Ti, . . . ,  TO)  is  a  commuting  d -tuple  of  operators  then  one  may  say  that  T  is 
homogeneous  if,  for  all  'sufficiently  small*  <p  €  GQ,  <p(T)  is  (jointly)  unitarily  equivalent 
to  T.  (Of  course,  this  notion  depends  on  the  choice  of  £2  -  for  most  values  of  d  there  are 
several  choices  -  so,  to  be  precise,  one  ought  to  speak  of  ^-homogeneity).  Theorem  1.1 
generalizes  to  show  that,  in  this  setting,  the  Taylor  spectrum  of  T  is  contained  in  &  (and  is  a 
G^-invariant  closed  subset  thereof).  Also,  if  T  is  an  irreducible  homogeneous  tuple  (in  the 
sense  that  its  components  have  no  common  non-trivial  reducing  subspace),  then  Theorem 
4.1  generalizes  to  yield  a  projective  representation  of  G^  associated  with  it.  Therefore, 
many  of  the  techniques  employed  in  the  single  variable  case  have  their  several  variable 
counterparts.  But  these  are  yet  to  be  systematically  investigated.  One  difficulty  is  that  for 
d  >  2,  the  (projective)  representation  theory  of  G^  (which  is  a  semi-simple  Lie  group) 
is  not  as  well  understood  as  in  the  case  ft  =  D.  But  this  also  has  the  potential  advantage 
that  when  (and  if)  this  theory  of  homogeneous  operator  tuples  is  investigated  in  depth,  the 
operator  theory  is  likely  to  have  significant  impact  on  the  representation  theory. 

With  each  domain  ft  as  above  is  associated  a  kernel  BQ  (called  the  Bergman  kernel) 
which  is  the  reproducing  kernel  of  the  Hilbert  space  of  all  square  integrable  (with  respect 
to  Lebesgue  measure)  analytic  functions  on  ft.  The  Wallach  set  W  =  WQ  of  &  is  the  set 


436  Bhaskar  Bagchi  and  Gadadhar  Misra 

of  all  X  >  0  such  that  B^8  is  (a  non-negative  definite  kernel  and  hence)  the  reproducing 
kernel  of  a  Hilbert  space  W^(£2).  (Here  g  is  an  invariant  of  the  domain  £2  called  its  genus, 
cf.  [2].)  It  is  well-known  that  the  Wallach  set  W  can  be  written  as  a  disjoint  union  Wj  U  Wc 
where  the  'discrete'  part  Wj  is  a  finite  set  (consisting  of  r  points,  where  the  'rank'  r  of  Q 
is  the  number  of  orbits  into  which  the  topological  boundary  of  £2  is  broken  by  the  action 
of  GQ)  and  the  'continuous'  part  Wc  is  a  semi-  in  finite  interval. 

The  constant  functions  are  always  in  H^  (£2)  but,  for  A  e  Wj,  H^  (£2)  does  not  contain 
all  the  analytic  polynomial  functions  on  £2.  It  follows  that  for  A  e  W(i  multiplication  by 
the  co-ordinate  functions  do  not  define  bounded  operators  on  7i(x)(£2).  However,  it  was 
conjectured  in  [4]  that  for  A  e  Wc,  the  d-  tuple  M^  of  multiplication  by  the  d  co-ordinates 
is  bounded.  (In  [5],  this  conjecture  was  proved  in  the  cases  £2  =  /m,/7.  In  general,  it  is 
known  that  for  sufficiently  large  A  the  norm  on  H^(Q)  is  defined  by  a  finite  measure 
on  £2,  so  that  this  tuple  is  certainly  bounded  in  these  cases.)  Assuming  this  conjecture, 
the  operator  tuples  M^,  A  e  Wc,  constitute  examples  of  homogeneous  tuples  -  this  is  in 
consequence  of  the  obvious  extension  of  Theorem  5.1  to  tuples.  In  [4]  it  was  shown  that 
the  Taylor  spectrum  of  this  tuple  is  £2  and 


Theorem  7.4.  Up  to  unitary  equivalence,  the  adjoint  s  of  the  tuples  M^,  A.  €  Wc,  are  the 
only  homogeneous  tuples  in  the  Cow  en-Douglas  class  B\  (Q). 

For  what  values  of  A  e  Wc  is  the  tuple  Mw  sub-normal?  This  is  equivalent  to  asking  for 
the  values  of  A  for  which  the  norm  on  7i(A>)(£2)  is  defined  by  a  measure.  In  [4]  we  conjecture 
a  precise  answer.  Again,  the  special  case  £2  =  /,„,„  of  this  conjecture  was  proved  in  [5]. 

Regarding  homogeneous  tuples,  an  obvious  meta-question  to  be  asked  is 

Question  12.  Formulate  appropriate  generalizations  to  tuples  of  all  the  questions  we  asked 
before  of  single  homogeneous  operators  -  and  answer  them! 

A  tff-tuple  T  on  the  Hilbert  space  H  is  said  to  be  completely  contractive  with  respect  to 
£2  if  for  every  polynomial  map  P  :  £2  -*  /mjl,  P(T)  is  contractive  when  viewed  as  an 
operator  from  U  %  Cn  to  H  <g>  Cm.  T  is  called  contractive  with  respect  to  &  if  this  holds 
in  the  case  m  =  n  =  1  .  In  general  one  may  ask  whether  contractivity  implies  complete 
contractivity.  In  general  the  answer  is  'no'  for  all  d  >  5  [27].  However  one  has  a  positive 
answer  in  the  case  £2  =  D.  But  an  affirmative  answer  (for  special  classes  of  tuples)  would 
be  interesting  because  complete  contractivity  is  tantamount  to  existence  of  nice  dilations 
which  make  the  tuple  in  question  tractable.  For  instance,  we  have  an  affirmative  answer 
for  subnormal  tuples.  We  ask 

Question  13.  Is  every  contractive  homogeneous  tuple  completely  contractive? 

References 

[1  ]  Aleman  A,  Richter  S  and  Sundberg  C,  Beurling's  theorem  for  the  Bergman  space,  Acta  Math. 

177(1996)275-310 
[2]  Arazy  J,  A  survey  of  invariant  Hilbert  spaces  of  analytic  functions  on  bounded  symmetric 

domains,  Contemp.  Math.  185  (1995)  7-65 
[3]  Arveson  W,  Hadwin  D  W,  Hoover  T  B  and  Kymala  E  E,  Circular  operators,  Indiana  U.  Math. 

7.33(1984)583-595 


Homogeneous  operators  and  projective  representations  437 

(•]  Bagchi  B  and  Misra  G,  Homogeneous  operators  and  systems  of  imprimitivity,  Contemp.  Math. 

185(1995)67-76 
>]  Bagchi  B  and  Misra  G,  Homogeneous  tuples  of  multiplication  operators  on  twisted  Bergman 

spaces,  J.  Fund  Anal  136  (1996)  171-213 
>]  Bagchi  B  and  Misra  G,  Constant  characteristic  functions  and  homogeneous  operators,  J.  Op. 

Theory  37  (}991)  5 1-65 
r]  Bagchi  B  and  Misra  G,  Scalar  perturbations  of  the  Nagy-Foias  characteristic  function,  in: 

Operator  Theory:  Advances  and  Applications,  special  volume  dedicated  to  the  memory  of 

Bela  Sz-Nagy  (2001)  (to  appear) 
>]  Bagchi  B  and  Misra  G,  A  note  on  the  multipliers  and  projective  representations  of  semi-simple 

Lie  groups,  Special  Issue  on  Ergodic  Theory  and  Harmonic  Analysis,  Sankhya  A62  (2000) 

425-432 

>]  Bagchi  B  and  Misra  G,  The  homogeneous  shifts,  preprint 

)]  Bagchi  B  and  Misra  G,  A  product  formula  for  homogeneous  characteristic  functions,  preprint 
]  Clark  D  N  and  Misra  G,  On  some  homogeneous  contractions  and  unitary  representations  of 

SU(l,  I),/  Op.  77z*?ry30(1993)  109-122 
']  Curto  R  E  and  Salinas  N,  Generalized  Bergman  kernels  and  the  Cowen-Douglas  theory,  Am. 

J.  Math.  106  (3  984)  447-488 
\]  Davidson  K  and  Paulsen  V  I,  Polynomially  bounded  operators,  J.  Reine  Angew.  Math.  487 

(1997)  153-170 
[•]  Douglas  R  G  and  Misra  G,  Geometric  invariants  for  resolutions  of  Hilbert  modules,  Operator 

Theory:  Advances  and  Applications  104  (1998)  83-1 12 

>]  Douglas  R  G,  Misra  G  and  Varughese  C,  On  quotient  modules  -  the  case  of  arbitrary  multi- 
plicity, J.  Fund  Anal.  174  (2000)  364-398 
)]  Faraut  J  and  Koranyi  A,  Analysis  on  symmetric  cones  (New  York:  Oxford  Mathematical 

Monographs,  Oxford  University  Press)  (1994) 
7]  Geller  R,  Circularly  symmetric  normal  and  subnormal  operators,  J.  d' analyse  Math.  32  (1 977) 

93-1  17 
5]  Hedenmalm  H,  A  factorization  theorem  for  square  area-integrable  analytic  functions,  /.  Reine 

Angew.  Math.  422  (1991)  45-68 

)]  Kerchy  L,  On  Homogeneous  Contractions,  J.  Op.  Theory  41  (1999)  121-126 
)]  Mackey  G  W,  The  theory  of  unitary  group  representations  (Chicago  University  Press)  (1 976) 
1]  Misra  G,  Curvature  and  the  backward  shift  operators,  Proc.  Amer.  Math.  Soc.  91  (1984) 

105-107 
1]  Misra  G,  Curvature  and  discrete  series  representation  of  SL2(K)»  J-  Int.  Ec/ns  Op.  Theory  9 

(1986)452-459 
$]  Misra  G  and  Sastry  N  S  N,  Homogeneous  tuples  of  operators  and  holomorphic  discrete  series 

representation  of  some  classical  groups,  J.  Op.  Theory  24  (1990)  23-32 
1-]  Moore  C  C,  Extensions  and  low  dimensional  cohomology  theory  of  locally  compact  groups, 

1,  Trans.  Am.  Math.  Soc.  113  (1964)  40-63 
5]  Sz-Nagy  B  and  Foias  C,  Harmonic  Analysis  of  Operators  on  Hilbert  Spaces  (North  Holland) 

(1970) 
5]  Parthasarathy  K  R,  Multipliers  on  locally  compact  groups,  Lecture  Notes  in  Math.  (New  York: 

Springer  Verlag)  (1969)  vol.  93 
7]  Paulsen  V  I,  Representations  of  function  algebras,  abstract  operator  spaces  and  Banach  space 

geometry,/  Funct.  Anal.  109  (1992)  1 13-129 

3]  Pisier  G,  A  polynomially  bounded  operator  on  Hilbert  space  which  is  not  similar  to  a  con- 
traction, J.  Am.  Math.  Soc.  10  (1997)  351-369 
?]  Sally  P  J,  Analytic  continuation  of  the  irreducible  unitary  representations  of  the  universal 

covering  group  of  SL(2,  R),  Mem.  Am.  Math.  Soc.  (Providence)  (1967)  vol.  69 
3]  Shields  A  L,  On  Mobius  bounded  operators,  Acta  Sci.  Math.  40  (1978)  371-374 
1]  Szego  G,  Orthogonal  polynomials,  Amer.  Math.  Soc.  (Colloquium  Publication)  (1985) 

vol.  23 

2]  Varadarajan  V  S,  Geometry  of  quantum  theory  (New  York:  Springer  Verlag)  1985 
S]  Wilkins  D  R,  Operators,  Fuchsian  groups  and  automorphic  bundles,  Math.  Ann.  290  (1991) 

405-424 
1]  Wilkins  D  R,  Homogeneous  vector  bundles  and  Cowen-Douglas  operators,  Int.  J.  Math.  4 

(1993)503-520 


>c.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  Ill,  No.  4,  November  2001,  pp.  439-463 
Printed  in  India 


[ulti wavelet  packets  and  frame  packets  of  L2 


BISWARANJAN  BEHERA 

Department  of  Mathematics,  Indian  Institute  of  Technology,  Kanpur  208  016,  India 
Current  address:  Stat.-Math.  Unit,  Indian  Statistical  Institute,  203  B.T.  Road,  Kolkata 
700  035,  India 
E-mail:  biswa_v@isical.ac.in 

MS  received  28  March  2001 

Abstract.  The  orthonormal  basis  generated  by  a  wavelet  of  L2((R)  has  poor  fre- 
quency localization.  To  overcome  this  disadvantage  Coifman,  Meyer,  and  Wicker- 
hauser  constructed  wavelet  packets.  We  extend  this  concept  to  the  higher  dimensions 
where  we  consider  arbitrary  dilation  matrices.  The  resulting  basis  ofL2(Ud)  is  called 
the  multiwavelet  packet  basis.  The  concept  of  wavelet  frame  packet  is  also  general- 
ized to  this  setting.  Further,  we  show  how  to  construct  various  orthonormal  bases  of 
L2(IRJ)  from  the  multiwavelet  packets. 

Keywords.     Wavelet;  wavelet  packets;  frame  packets;  dilation  matrix. 
Introduction 

>nsider  an  orthonormal  wavelet  of  L2(IR).  At  the  ;th  resolution  level,  the  orthonormal 
sis  {fyjk  •  j>  &  €  Z}  generated  by  the  wavelet  has  a  frequency  localization  proportional 
2-'.  For  example,  if  the  wavelet  ^  is  band-limited  (i.e.,  $  is  compactly  supported),  then 
3  measure  of  the  support  of  (^  /&)  A  is  2;  times  the  measure  of  the  support  of  $,  since 


.  ~,     ;,  *  e  Z, 

lere 


>  when  j  is  large,  the  wavelet  bases  have  poor  frequency  localization.  Better  frequency 
calization  can  be  achieved  by  a  suitable  construction  starting  from  an  MRA  wavelet  basis. 
Let  [Vj  :  j  €  Z}  be  an  MRA  of  L2(R)  with  corresponding  scaling  function  <p  and 
ivelet  V<\  Let  Wj  be  the  corresponding  wavelet  subspaces:  Wj  =  W{i^jk  :  k  e  1}.  In 
e  construction  of  a  wavelet  from  an  MRA,  essentially  the  space  V\  was  split  into  two 
thogonal  components  V0  and  WQ.  Note  that  V\  is  the  closure  of  the  linear  span  of  the 
nctions  {2^(2  -  —k)  :  k  e  Z},  whereas  V0  and  WQ  are  respectively  the  closure  of  the 
an  of  {<p(.  -k)  :  k]  and  {^(-  -  *)  :  *}.  Since  (p(2  .  -*)  =  9  (2(-  -  |)),  we  see  that 
e  above  procedure  splits  the  half-integer  translates  of  a  function  into  integer  translates  of 
ro  functions. 

In  fact,  the  splitting  is  not  confined  to  V\  alone:  we  can  choose  to  split  Wjt  which  is  the 
an  of  {Vr(2->'  -  -k)  :  k}  =  {^  (lj(>  -  ^))  :  *},  to  get  two  functions  whose  2~(j~l]k 
mslates  will  span  the  same  space  Wj.  Repeating  the  splitting  procedure  ;  times,  we  get 

439 


440  Biswaranjan  Behera 

2>i  functions  whose  integer  translates  alone  span  the  space  W/  .  If  we  apply  this  to  each 
Wj,  then  the  resulting  basis  of  L2(R),  which  will  consist  of  integer  translates  of  a  count- 
able number  of  functions  (instead  of  all  dilations  and  translations  of  the  wavelet  V0>  will 
give  us  a  better  frequency  localization.  This  basis  is  called  'wavelet  packet  basis'.  The 
concept  of  wavelet  packet  was  introduced  by  Coifman,  Meyer  and  Wickerhauser  [6,  7]. 
For  a  nice  exposition  of  wavelet  packets  of  L-(R)  with  dilation  2,  see  [11]. 

The  concept  of  wavelet  packet  was  subsequently  generalized  to  Ud  by  taking  tensor  prod- 
ucts [5].  The  non-tensor  product  version  is  due  to  Shen  [16].  Other  notable  generalizations 
are  the  biorthogonal  wavelet  packets  [4],  non-orthogonal  version  of  wavelet  packets  [3], 
the  wavelet  frame  packets  [2]  on  R  for  dilation  2,  and  the  orthogonal,  biorthogonal  and 
frame  packets  on  Ud  by  Long  and  Chen  [13]  for  the  dyadic  dilation. 

In  this  article  we  generalize  these  concepts  to  R^  for  arbitrary  dilation  matrices  and  we 
will  not  restrict  ourselves  to  one  scaling  function:  we  consider  the  case  of  those  MRAs  for 
which  the  central  space  is  generated  by  several  scaling  functions. 

DEFINITION  1.1 

A  d  x  d  matrix  A  is  said  to  be  a  dilation  matrix  for  Rd  if 
(i)  A(1d)  c  1d  and 
(ii)  all  eigenvalues  A.  of  A  satisfy  |A,|  >  1. 

Property  (i)  implies  that  A  has  integer  entries  and  hence  |  det  A|  is  an  integer,  and  (ii)  says 
that  |  det  A  |  is  greater  than  1  .  Let  B  =  A1  ,  the  transpose  of  A  and  a  =  |  det  A  |  =  |  det  B  \  . 

Considering  J.d  as  an  additive  group,  we  see  that  A~Ld  is  a  normal  subgroup  ofZd.  So 
we  can  form  the  cosets  of  A1d  in  TLd  .  It  is  a  well-known  fact  that  the  number  of  distinct 
cosets  of  A1d  in  Hd  is  equal  to  a  =  \  det  A|  ([10,  17]).  A  subset  of  Id  which  consists  of 
exactly  one  element  from  each  of  the  a  cosets  of  AZJ  in  Hd  will  be  called  a  set  of  digits 
for  the  dilation  matrix  A.  Therefore,  if  K^  is  a  set  of  digits  for  A,  then  we  can  write 


where  {AZd  +M  :  JJL  e  KA]  are  pairwise  disjoint.  A  set  of  digits  for  A  need  not  be  a  set  of 
digits  for  its  transpose.  For  example,  for  the  dilation  matrix  M  =  (  )  of  R2,  the  set 

is  a  set  of  digits  for  M  but  not  for  Ml  .  It  is  easy  to  see  that  if  K  is  a  set  of 


|  f  n  )  ,  1 


digits  for  A,  then  so  is  K  —  /x,  where  //,  €  K.  Therefore,  we  can  assume,  without  loss  of 
generality,  that  0  e  K. 

The  notion  of  a  multiresolution  analysis  can  be  extended  to  L2(Rd)  by  replacing  the 
dyadic  dilation  by  a  dilation  matrix  and  allowing  the  resolution  spaces  to  be  spanned  by 
more  than  one  scaling  function. 

DEFINITION  1.2 

A  sequence  {Vj  :  j  e  Z}  of  closed  subspaces  of  L2(RJ)  will  be  called  a  multiresolution 
analysis  (MRA)  of  L2(RJ)  of  multiplicity  L  associated  with  the  dilation  matrix  A  if  the 
following  conditions  are  satisfied: 


Multiwavelet  and  frame  packets  of  L-(Rd)  44  1 

Ml)  Vj  c  Vj+{  for  all  j  eZ 

M2)  Uf€ZV)  is  dense  in  L2([RJ)  and  r\,-€2  V/  =  {0} 

M3)  /  €  V/  if  and  only  if  /(A-)  e  V/+1 

M4)  there  exist  L  functions  {<Pi  ,  <P2>  •  •  •  »  <?L}  in  VQ,  called  the  scaling  functions,  such  that 

the  system  of  functions  {<£>/(•  —  A')  :    1  <  /  <  L,  &  e  2d}  forms  an  orthononnal  basis 

for  V0. 
'he  concept  of  multiplicity  was  introduced  by  Herve  [12]  in  his  Ph.D.  thesis. 

Since  {<?/(•  —  k)  :  1  <  I  <  L,  k  e  Zd]  is  an  orthononnal  basis  of  V0,  it  follows  from 
roperty  (M3)  that  [aJ/2<pi(AJ  •  -k)  :  1  <  /  <  L,  &  e  Z^}  is  an  orthononnal  basis  of  V/. 
)bserve  that  if  /  E  L2((RJ),  then 


The  Fourier  transform  of  a  function  /  e  LJ(R^)  is  defined  by 


b  define  the  Fourier  transform  for  functions  of  L2(R^),  the  operator  J7  is  extended  from 
-1  H  L2^),  which  is  dense  in  L2(RJ)  in  the  L2-norm,  to  the  whole  of  L2(Ra).  For  this 
efmition  of  the  Fourier  transform,  Plancherel  theorem  takes  the  form 


First  of  all  we  will  prove  a  lemma,  the  splitting  lemma  (see  [8]),  which  is  essential  for  the 
onstruction  of  wavelet  packets.  We  need  the  following  facts  for  the  proof  of  the  splitting 


a)  Let  T^  =  [-JT,  n]d  and  /  €  Ll(Rd).  Since  Rd  =  U^/OP7  -f  2for),  we  can  write 
/(*)d*=  f  \  E  /(Jc  +  2A;r)W  (1) 

JJd  1A-6 


3)  Let  [sk  :  k  €  Zd]  e  l{(1d)  and  KB  be  a  set  of  digits  for  the  dilation  matrix  B.  As  Zd 
can  be  decomposed  as  Zd  =  (Jt^KB  (^^-d  +  M)»  we  can  W1"ite 

E  ^  =    E     E  -fy+5*.  (2) 

^€2^  l^KBke1d 

c)  Let  ^5  be  a  set  of  digits  for  B.  Define 


Since  ^B  is  a  set  of  digits  for  5,  the  set  go  satisfies  U^e2j(<20  4-  2te)  =  Rrf.  This 
fact,  together  with  |2ol  =  (2^)^,  implies  that  {2o  -f  Ikn  :  k  e  1d}  is  a  pairwise 
disjoint  collection  (see  Lemma  1  of  [10]).  Therefore, 


E  /(^  +  2te)d^,     for/eL1).  (3) 


function  /  is  said  to  be  2nZd  -periodic  if  f(x  +  2kn)  =  /(jc)  for  all  k  e  Zd  and  for 
.e.  ;c  e  Rd. 


442  Biswaranjan  Behera 

2.  The  splitting  lemma 

Let  [<pi  :  1  <  /  <  L}  be  functions  in  L2((RJ)  such  that  {<p/(-  -  k)  :  1  <  /  <  L,  k  €  Zd] 
is  an  orthonormal  system.  Let  V  —  Tp{al/2<pi(A  -  ~k)  :  /,&}.  For  1  <  /,  y  <  L  and 
0  <  r  <  «  —  1,  suppose  that  there  exist  sequences  {hrrk  :  k  e  Z.d]  e  /2(Z^).  Define 

//'(*)  =  E  E  V1/2^*-*>-  (4) 


Taking  Fourier  transform  of  both  sides 

L 


E 


//'(£)   =    E 

L 

where 

A//f)  =   E  a~l/2hrljke~l(^k},       1  <  /,  ;  <  L,  0  <  r  <  a  -  1,  (6) 


and  h\.  is  2jrZJ-periodic  and  is  in  L2(Jd).  Now,  for  0  <  r  <  a  -  1,  define  the  L  x  L 


. 
matrices 


By  denoting 

0(*)     =     (^iW,...,^U))r  (8) 

*(?)     -     (^i«),...,^«))r,  (9) 

we  can  write  (5)  as 

Fr(ft  =  Hr(B-lftQ(B-[ft,    0<r<a-l9  (10) 

where  Fr(x)  =  (/[«,  /{  W,  .  .  .  ,  /Lr(^))f  and  £(f  )  =  (/f  (f  ), 


The  following  well-known  lemma  characterizes  the  orthonormality  of  the  system 
[<Pi(-  ~  ^}  :  1  <  /  <  L,  k  e  Zd}.  We  give  a  proof  for  the  sake  of  completeness. 

Lemma  2.1  .  The  system  {9l(.-k):   \<l<L,k  el.d}  is  orthonormal  if  and  only  if 


Proof.  Suppose  that  the  system  {^(.  -  *)  :   1  <  /  <  L,  jfc  6  1d]  is  orthonormal.  Note  that 
(<Pj(-  ~  /?).  ^/('  -  ^))  =  (^./^/(-  -  fe  -  /?)))  for  1  <  ;,  /  <  L  and  p,  q  e  Zd  .  Now 


E  ^7(f  +  2*^)#/(?  +  2^7r)e/^^d^     by  (1). 

jd 


Multivvavetet  and  frame  packets  ofL2(Rd)  443 

Therefore,  the  2*2^  -periodic  function  Gjt  (£)  =    £  <Pj  (£  +  2for)£/(f  +  2Jbr)  has  Four- 


ier  coefficients  G  //(-p)  =  <5//6op,  p  €  2J  which  implies  that  G//  =  <5;-/  a.e.  By  reversing 
the  above  steps  we  can  prove  the  converse.  D 

Let  M*(£  )  be  the  conjugate  transpose  of  the  matrix  M(£)  and  IL  denote  the  identity 
matrix  of  order  L. 

Lemma  2.2.  (The  splitting  lemma)  Let  {<pi  :  1  <  /  <  L}  be  functions  in  L2(Rd)  such  that 
the  system  {al/2(f>j(A  -  —k)  :  1  <  7  <  L,  ^  €  Zd}  is  orthonormal  Let  V  be  its  closed 
linear  span.  Let  K  be  a  set  of  digits  for  B.  Also  let  f[  ,  -Hr  be  as  above.  Then 

{//"(•  -  *)  :  0  <  r  <  a  -  1,   1  <  /  <  L,  )t  €  1d] 
is  an  orthonormal  system  if  and  only  if 

^v/L)     0<r,j<a-  1.    (11) 


Moreover,  {//"(•  -A')  :  0  <  r  <  «  -  1,   1  <  /  <  L,  Jk  6  Zd}  is  an  orthonormal  has  is  of  V 
whenever  it  is  orthonormal. 

Proof.  For  1  <  /,  j  <  L,  0  <  r,  JT  <  a  -  1  and  p  e  ZJ,  we  have 

/;,//•(-  - 


=  7^  /    E  E  ^ 

(27T)U  jRrf  ,„_!„=  i       ' 


(by  (5)) 


L       L 

E  E  E 


(by  (D) 


E   ^• 


—- 

(27T)"  JT^  /i€/Cm== 

E  ^(S-1  (?  +  2/.7T) 


(by  (2)) 


L      L 

E  E  hrjn 


(27T)rf 

(by  Lemma  2.1) 


.        '      f  I  £   ^ 

(2n)a  Jjd  (IJL€Km=\ 


444  Biswaranjan  Behera 

Therefore, 

(/[.  //('  -  />))       =      SrsSjlSop 


E   E  firJm(B-lS  +  2B-liui)hflm(B-^  +  2B-tfjLn)     =     Sr,Sji  for  a.e.  £  6 

m~\ 


E    EhrjmG  +  2B-[iJur)h*[mG  +  2B-liJUt)     =     ««fy  for  a.e.  f  e 

/^  6  AT  w  =  1 


We  have  proved  the  first  part  of  the  lemma. 

Now  assume  that  {//"(•  -  k)  :  0  <  r  <  a  -  1,  1  <  /  <  L,  /:  €  ZJ}  is  an  orthonormal 
system.  We  want  to  show  that  this  is  an  orthonormal  basis  of  V.  Let  /  €  V.  So  there 
exists  [cjp  :  p  6  Z^}  6  /2(ZJ),  1  <  ;  <  L  such  that 

/W=  E    E  c^a^(^-P)- 

7=lp€2'y 

Assume  that  /  JL  //  (•  -  fc)  for  all  r,  /,  k. 

Claim,  f  =  0. 
For  all  r,  /,  k  such  that  0  <  r  <  a  -  1,  1  <  /  <  L,  k  e  Zd,  we  have 


0    =    (//r  (•-«,/}  =   //(•  -A), 


(by  (5)) 

L  _ 


L 


_ 

E   E  ^y(§-h2^7r)e-/<*'^+2^^e/^^+2^7r)df     (by  (3)) 


r        L      L  ( 

/      E  E    E  «TO   E 
•/Goni=lj/=lpgZ''  l^eZ'' 


Multiwavelet  and  frame  packets  of  L2(Ucl )  445 

ai/2     f      L  

=     7T~v/"  /      E    E  hrim($)cmi,s-l(k*BVQl(P*]&%     (by  Lemma  2.1) 

VZ7TJ     JQom=\p£Zd 
fl]/2     _ 


n}/2 


ce  \f>~liy  Q~'(k'B'}  :  k  e  ^1  is  an  orthonormal  basis  for  L2(B~lJd),  the  above  equa- 
sgive 


E    E    E  c^e/(*+2/?"l^^)A/rw(?  +  25-'^)  =  0  a.e.      for  all  r,  /. 

fi€K  m~\  psZ'l 

m  =  1,2,  ...,L,  define 


(12) 


i  have 

L    


=  0,    0<r<fl-l,    1  <  /  <  L.     (13) 

AT  m=I 

ations  (1 1)  are  equivalent  to  saying  that  for  0  <  r  <  <z  —  1,   1  <  /  <  L   and  for  i.e. 
(R^,  the  vectors 


mutually  orthogonal  and  each  has  norm  1  ,  considered  as  a  vector  in  the  ^Lrdimensional 
:e  CaL,  so  that  they  form  an  orthonormal  basis  for  CaL.  Equation  (13)  says  that  the 
:or 

:  I  <  m  <  L,  /z  e  K  (14) 


rthogonal  to  each  member  of  the  above  orthonormal  basis  of  CaL.  Hence,  the  vector 
le  expression  (14)  is  zero.  In  particular,  Cw(£)  =  0,  for  all  m,  1  <  m  <  L.  That  is, 
=  0,  1  <  m  <  L,  p  <zZd.  Therefore,  /  =  0.  This  ends  the  proof.  Q 

'he  splitting  lemma  can  be  used  to  decompose  an  arbitrary  Hilbert  space  into  mutually 
ogonal  subspaces,  as  in  [7].  We  will  use  the  following  corollary  later. 


446  Biswaranjan  Behera 

COROLLARY  2.3 


Let  [Eik  :   I  <  I  <  L,  k  €  Z^]  be  an  orthononnal  basis  of  a  separable  Hilbert  space  H. 
Let  Hr,  ^<r<a-\beas  above  and  satisfy  (11).  Define 


Then  {F[k  :  1  <  /  <  L,  k  €  Zd]  is  an  orthonormal  basis  for  its  closed  linear  spanHr  and 

ft  =  ®?~dw. 

Proof.  Let  <p  1,^2,..-.^  be  functions  in  L2(Ud)  such  that  {<pi  (•-&)  :  1  <  /  <  I,jfc  €  Z^} 
is  an  orthonormal  system.  Let  V  =  Jp{a]/2q>i(A  -  —k)  :  /,  k}.  Define  a  linear  operator  T 
from  the  Hilbert  space  V  to  ft  by  T(a1/2<p/(A  -  -fc))  =  £/tJfc.  Let  /^  are  as  in  (4).  Then, 
T(f[(-  —  k))  =  F/^.  Now  the  corollary  follows  from  the  splitting  lemma.  D 

3.  Construction  of  multiwavelet  packets 

Let  (Vj  :  j  €  Z}  be  an  MRA  of  L2(R'/)  of  multiplicity  L  associated  with  the  dilation 
matrix  A.  Let  {<p/  :  1  <  /  <  L}  be  the  scaling  functions.  Since  <p/,  1  <  /  <  L  are  in 
VQ  C  V\  and  (a[/2(pj(A  •  -k)  :  1  <  j  <  L,k  e  1d]  forms  an  orthonormal  basis  of  V[9 
there  exist  [hijk  :  *  €  Zrf}  6  /2(ZC/)  for  1  <  /,  ;  <  L  such  that 

wW=  E  E^/^^V/C^-^)- 


Taking  Fourier  transform,  we  get 


-     E  E 


B-l&,  (15) 

7  =  1 

where  /i/7(f  )  =  E*e2</  a'l^hijkt"i(^y  and  /i//  is  2jrZ^  -periodic  and  is  in  L2(T^).  Let 
be  the  L  x  L  matrix  defined  by 


1</J<£ 

We  will  call  HQ  the  low-pass  filter  matrix.  Rewriting  (15)  in  the  vector  notations  (8)  and 
(9),  we  have 


|f).  (16) 

Let  Wj  be  the  wavelet  subspaces,  the  orthogonal  complement  of  Vj  in  V)+i : 

w/  =  v,+iev,.. 

Properties  (Ml)  and  (M3)  of  Definition  1.2  now  imply  that 


Multiwavelet  and  frame  packets  ofL2(R(l)  447 

and 

/  €  Wj  <»  f(A-J-)  €  Wo-  (17) 

Moreover,  by  (M2),  L2(R'')  can  be  decomposed  into  orthogonal  direct  sums  as 

=          W,  (18) 


(19) 
By  Lemma  2.1  and  eq.  (15),  we  have  (for  1  <l,j  <L) 


=      E  I  E  hjm(B-](!;+2kTt))vm(B-}(t;+2kn))} 

k£Z(!   l/«=l  J 


Now,  using  (2),  we  have 


L     L 

E  E  E 

;i=l  /2=1 


where  KB  is  a  set  of  digits  for  B.  Using  Lemma  2. 1  again,  we  get 

L  . 


This  is  equivalent  to  saying  that 

=  /L     fora.e.f. 


Equation  (20)  is  also  equivalent  to  the  orthonormality  of  the  vectors 

:   1  <  ;  <  L,  /z  e  ATB          1  <  /  <  L,  f  €  TJ. 


=    E     Ehj,n(B-[!=+2B-}i^)h!m(B-lt=  +  2B-[iJ.n).  (20) 


These  L  orthonormal  vectors  in  the  flL-dimensional  space  CaL  can  be  completed,  by 
Gram-Schmidt  orthonormalization  process,  to  produce  an  orthonormal  basis  for  CaL  .  Let 
us  denote  the  new  vectors  by 

jtf  +  2B-  VTT)  :  I  <  j  <  L,  p  e  KB>       I  <l  <  L>  I  <r  <a  -  I,  !=  eJd  , 


and  extend  the  functions  hrtj  (1  <  r  <  fl  -  1,  !</,;'<  L)  27rZ^  -periodically  (see  [9] 
for  the  one-dimensional  dyadic  dilation).  Denoting  by  Hr(%],  1  <  r  <  a  —  1  the  L  x  L 
matrix 


448  Biswaranjan  Behera 

we  have 

£  +  2B~{  ^n)  =  £r,/L     for  a.e.  £. 


Now,  for  1  <  r  <  0  -  1 ,  1  <  /  <  L,  define 

L 
//"(?)=  Y,h1j(B~l^Vj(B~1^-  (21) 

Since  hrtj  are  2jrZrf-periodic,  there  exist  {hrljk  :  k  e  Z<7}  e  I2(1d)  such  that 


Now,  applying  the  splitting  lemma  to  Vj,  we  see  that  {//"(•  —  &)  :  0  <  r  <  <z  —  1,  1  < 
/  <  L,  k  €  Z^}  is  an  orthonormal  basis  for  Vi  .  We  use  the  convention  (pi  =  f®,  1  <  /  <  L 
with  /2/y  =  7i^  and  A/j^  =  /z^.  The  decomposition  V\  =  VQ  ®  Wo>  and  the  fact  that 
{//°(-  -  /:)  :  1  <  /  <  L,  /:  €  Z^}  is  an  orthonormal  basis  of  VQ,  imply  that 

.{//(-  -  k)  :  I  <  r  <  a  -  1,  1  <  /  <  L,  k  6  Z^} 
is  an  orthonormal  basis  for  WQ-  By  (17)  and  (18),  we  see  that 

{aj/2ffr(Aj  •  -AO  :  1  <  r  <  a  -  1,  1  <  /  <  L,  ;  €  Z,  k  e  Zd] 

is  an  orthonormal  basis  for  L2(R^).  This  basis  is  called  the  multiwavelet  basis  and  the 
functions  {//"  :  1  <  r  <  a  —  1,  1  <  /  <  L}  are  the  multiwavelets  associated  with 
the  MRA  (Vj  :  j  e  Z}  of  multiplicity  L.  For  0  <  r  <  a  —  1,  by  denoting  Fr(x)  = 
(/f  W,  /2rW»  •  -  •  ,  /£  W)r  and  Fr(§)  =  (/f  (f  ),  /{(f  ),  .  .  .  ,  /£(?)/,  we  can  write  (16) 


and  (21)  as 

^),      0  <  r  <  a  ~  1.  (22) 


This  equation  is  known  as  the  scaling  relation  satisfied  by  the  scaling  functions  (r  =  0) 
and  the  multiwavelets  (1  <  r  <  a  —  1). 

As  we  observed,  applying  splitting  lemma  to  the  space  V\  =  5p{a}/2<pi(A  •  —k)  : 
I  <  I  <  L,  *  e  ZJ},  we  get  the  functions  f[,  0  <  r  <a~-  1 ,  1  <  /  <  L.  Now,  for  any 
n  €  NO  =  N  U  {0},  we  define  //l,  1  <  /  <  L  recursively  as  follows.  Suppose  that  //", 
r  E  f^o,  1  <  /  <  L  are  defined  already.  Then  define 

//+«r(jt)  =  £   Ehijka{/2fj(Ax-ky,    0<j<a-l,  1</<L.    (23) 

Taking  Fourier  transform 

L 

(//       )A(^)  =  E^//(^""1f)(//')A(^"1?)-  (24) 

7=1 

In  vector  notation,  (24)  can  be  written  as 

5-1?).  (25) 


Multiwavelet  and  frame  packets  ofL2(Rd)  449 

ote  that  (23)  defines  fj1  for  every  non-negative  integers  and  every  /  such  that  1  <  /  <  L. 
bserve  that  //*  =  <pi,  1  <  I  <  L  are  the  scaling  functions  and  ff,  1  <  r  <  a  —  1,  1  < 
<  L  are  the  multiwavelets.  So  this  definition  is  consistent  with  the  scaling  relation  (22) 
Ltisfied  by  the  scaling  functions  and  the  multiwavelets. 


EFINITION3.1 

he  functions  [ff1  :  n  >  0,  1  <  /  <  L}  as  defined  above  will  be  called  the  basic  multi- 
avelet  packets  corresponding  to  the  MRA  {  V)  :  7  €  2}  of  L2(R^)  of  multiplicity  L 
jsociated  with  the  dilation  A. 

he  Fourier  transforms  of  the  multiwavelet  packets 

'ur  aim  is  to  find  an  expression  for  the  Fourier  transform  of  the  basic  multiwavelet  packets 
i  terms  of  the  Fourier  transform  of  the  scaling  functions.  For  an  integer  n  >  1,  we  consider 
le  unique  'a-adic  expansion'  (i.e.,  expansion  in  the  base  a): 

n  —  /xi  +  M2<2  -f  l-i^a2  H  -----  h  Hjaj~~]  ,  (26) 


rhere  0  <  /z/  <  <3  —  1  for  all/  =  1  ,  2,  .  .  .  ,  j  and  IJLJ  ^  0. 

If  n  can  be  expressed  as  in  (26)  then  we  will  say  n  has  a-adic  length  /.  We  claim  that  if 
has  length  /  and  has  expansion  (26),  then 

^(^-^(B-1?)^^-2^...^/^-^)^-^),  (27) 

D  that  (//*)A(f  )  is  the  /th  component  of  the  column  vector  in  the  right  hand  side  of  (27). 
/e  will  prove  the  claim  by  induction. 

From  (22)  we  see  that  the  claim  is  true  for  all  n  of  length  L  Assume  it  for  length  j.  Then 
n  integer  m  of  fl-adic  length  j  -f  1  is  of  the  form  m  =  /x  +  flw,  where  0  <  M  <  a  —  1  and 

has  length  7  .  Suppose  n  has  the  expansion  (26).  Then  from  (25)  and(27),  we  have 


ince  m  =  p  -f  an  =  /i  +  MI  a  +  /^2«2  H h  /f/07' ,  A>*  (?)  has  the  desired  form.  Hence, 

le  induction  is  complete. 
The  first  theorem  regarding  the  multiwavelet  packets  is  the  following. 

lieorem  3.2.  Let  [ff  :  n  >  0,  1  <  /  <  L}  be  the  basic  multiwavelet  packets  constructed 
bove.  Then 

(0  {fin(-  ~  k)  :  aj'  <n<  aj'+l  -  1,    1  <  /  <  L,    k  e1d}  is  an  orthonormal  basis  of 
Wh  j  >  0. 

'")  {//"(•  ~~  k)  :  0  <  w  <  aj '  —  1,     1  <  /  <  L,    Jt  6  Hd}  is  an  orthonormal  basis  of 
•")  {//"(-  -  A;)  :  n  >  0,    1  <  /  <  L,    A:  6  1d]  is  an  orthonormal  basis  ofL2(Rd). 


450  Biswaranjan  Behera 

Proof.  Since  {//*  :  1  <  n  <  a  -  1  ,  1  <  /  <  L]  are  the  multiwavelets,  their  Zd-  translates 
form  an  orthonormal  basis  for  W0.  So  (i)  is  verified  for  ;  =  0.  Assume  for  ;.  We 
will  prove  for  j  -f  1.  By  assumption,  the  functions  {//*(-  -  k)  :  aj'  <  n  <  a^[  -  1, 
1  <  I  <  L,  k  €  Hd\  is  an  orthonormal  basis  of  Wj.  Since  /  6  W)  «>  /(A-)  €  W)+i, 
the  system  of  functions 

{al/2fll(A'-k)  :aj  <n<aj+l-l,    I  <  I  <  L,    *  6  1d] 
is  an  orthonormal  basis  of  W/+  1  .  Let 

En  =  Jp{a[/2ff(A  -~k):  1  <  /  <  L,    *  €  ZJ}. 
Hence, 


Applying  the  splitting  lemma  to  Ent  we  get  the  functions 

*)     (P<r<fl-l,    1</<L)     (29) 

so  that  [g"'r  (-—k)  :  Q<r  <a—  I,    1  <  /  <  L,   /:eZJ}isan  orthonormal  basis  of  £",,. 
But  by  (23),  we  have 


This  fact,  together  with  (28),  shows  that 

[fir+an(-  ~  k)  :  0  <  r  <  a  -  1,    1  <  /  <  L,    kz1d,  aj  <n<  aj+l  -  1} 
=     {/"(--A:):^1  <n<fl-/+2-l,    1  <  /  <  L,    ^c  e  Z^} 

is  an  orthonormal  basis  of  W/+i.  So  (i)  is  proved.  Item  (ii)  follows  from  the  observation 
that  Vj  =  V0  ©  W0  0  :  •  •  0  Wj-  1  and  (iii)  follows  from  the  fact  that  UV)  =  L2(Rd).     D 


4.  Construction  of  orthonormal  bases  from  the  multiwavelet  packets 

We  now  take  all  dilations  by  the  matrix  A  and  all  ZJ-translations  of  the  basic  multiwavelet 
packet  functions. 

DEFINITION  4.1 

Letf//1  :  n  >  0,   1  <  /  <  L}  be  the  basic  multiwavelet  packets.  The  collection  of  functions 
p  =  {aJ'^ff(AJ  -  -k)  :  n  >  0,    1  <  /  <  L,  ;  e  Z,    k  e  1d] 

will  be  called  the  '  'general  multiwavelet  packets'  associated  with  the  MRA  {Vj}  of  L2(IR£/) 
of  multiplicity  L. 

Remark  4.2.  Obviously  the  collection  P  is  overcomplete  in  L2(Rd).  For  example 


Multiwavelet  and  frame  packets  of  L-(Rd )  45 1 

(i)  The  subcollection  with  j  =  0,  n  >  0,   1  <  /  <  L,  k  €  Zd  gives  us  the  basic  multi- 
wavelet  packet  basis  constructed  in  the  previous  section, 
(ii)  The  subcollection  with  n  =  1,  2, ...,«-  1;    1  <  /  <  L,  ;  €  Z,  fc  €  Zrf  is  the  usual 

multiwavelet  basis. 

So  it  will  be  interesting  to  find  out  other  subcollections  of  P  which  form  orthonormal 
bases  for  L2(R^). 

For  w  >  0  and  j  €  Z,  define  the  subspaces 

Unj  =  sp(aj/2ff(Aj  •  -*)  :   1  <  /  <  L,    /:  €  2^}.  (30) 

Observe  that 


^=V/     and 
Hence,  the  orthogonal  decomposition  V)+i  =  V)  0  Wy  can  be  written  as 


r=0 

We  can  generalize  this  fact  to  other  values  of  n. 

PROPOSITION  4.3 

Forn  >  0  and  j  €  Z,  we  have 


Proof.  By  definition 


Let 

A-'+l  -  -*),      for  1  <  /  <  L, 


Then  {E/,/;  :   1  <  /  <  L,  k  e  Hd}  is  an  orthonormal  basis  of  the  Hilbert  space  Uj+l.  For 
0  <  r  <  a  -  1  ,  let 

*£*<*)  =  E    E  Ui.mj-AkE»jW.      l<l<L9keZ', 


and 

W  =  Jp{F[k  :   1  <  /  <  L, 
Then,  by  Corollary  2.3  we  have 


r=0 


452 
Now 


Biswaranjan  Behera 


E 


E 


L 
E     E 


E 

m= 


E 


-*),     by  (23). 


Therefore, 


and 


r=0 


Using  Proposition  4.3  we  can  get  various  decompositions  of  the  wavelet  subspa 
Wy,  j  >  0,  which  in  turn  will  give  rise  to  various  orthonormal  bases  of  L1' 

Theorem  4.4.  Let  j  >  0.  Then,  we  have 

a-\ 


Wj     = 


if/; 


r=l 


=     0  ^    /<; 


where  Uf  is  defined  in  (30). 

Proof.  Since  W)  =  ©"I,1  C/j,  we  can  apply  Proposition  4.3  repeatedly  to  get  (32). 

Theorem  4.4  can  be  used  to  construct  many  orthonormal  bases  of  L2(Ud).  We  have 
following  orthogonal  decomposition  (see  (19)): 

L2(Rd)  =  V0  ®  WQ  0  Wi  0  W2  0  •  •  - 


Multiwavelet  and  frame  packets  ofL2(Rd)  453 

For  each  j  >  0,  we  can  choose  any  of  the  decompositions  of  W/  described  in  (32).  For 
example,  if  we  do  not  want  to  decompose  any  Wj  ,  then  we  have  the  usual  multiwavelet 
decomposition.  On  the  other  hand,  if  we  prefer  the  last  decomposition  in  (32)  for  each 
Wj,  then  we  get  the  multiwavelet  packet  decomposition.  There  are  other  decompositions 
as  well.  Observe  that  in  (32),  the  lower  index  of  £/*  's  are  decreased  by  1  in  each  succ- 
essive step.  If  we  keep  some  of  these  spaces  fixed  and  choose  to  decompose  others  by  using 
(31),  then  we  get  decompositions  of  Wj  which  do  not  appear  in  (32).  So  there  is  certain 
interplay  between  the  indices  n  €  NO  and  j  €  Z. 

Let  5  be  a  subset  of  NO  x  2,  where  NO  =  M  U  {0}.  Our  aim  is  to  characterize  those  S 
for  which  the  collection 

iff(AJ  •  -*)  :   1  <  /  <  L,   *  €  ZJ,  (ws  jf)  e 

will  be  an  orthonormal  basis  of  L2(Rd).  In  other  words,  we  want  to  find  out  those  subsets 
S  of  NO  x  Z  for  which 


(nJ)eS 
By  using  (31)  repeatedly,  we  have 


r=0 

tf(rt-H)—  1  u(n+\)  —  1 

=     0   ";-,= 


Llv=0 


Let  /„,;  =  {r  e  NO  •'  ajn  <r<  aj(n  -f  1)  -  1}.  Hence, 


That  is, 


But  we  have  already  proved  in  Theorem  3.2  that 


r0 

Thus,  for  (33)  to  be  true,  it  is  necessary  and  sufficient  that  [Inj  :  (n,  j)  e  S}  is  a  partition 
of  N0.  We  say  {A/  :  /  €  /}  is  a  partition  of  N0  if  A/  C  MO.  ^/  's  are  pairwise  disjoint,  and 
U/€/  A/  =  NQ.  We  summarize  the  above  discussion  in  the  following  theorem. 

Theorem  4.5,  Let  {//'  :  n  >  0,  \  <l  <L]  be  the  basic  multiwavelet  packets  and 
S  C  NO  x  Z.  Then  the  collection  of  functions 


ftn(Aj  •  -*)  :   !</<!,    *  €  Zrf,  (n,  y)  € 
w  an  orthonormal  basis  ofL2(Rd)  if  and  only  if{Inj  :  (n,  ;)  €  5}  is  a  partition  ofMQ. 


454  Biswaranjan  Behera 

5.  Wavelet  frame  packets 

Let  H  be  a  separable  Hilbert  space.  A  sequence  [xk  :  k  e  Z}  of  H  is  said  to  be  a  frame  for 
ft  if  there  exist  constants  C\  and  Ci,  0  <  Cj  <  C2  <  oo  such  that  for  all  jc  e  7i, 

CiW2<  EK*.**>I22C2||*||2.  (36) 

*eZ 

The  largest  C]  and  the  smallest  Ci  for  which  (36)  holds  are  called  the  frame  bounds. 

Suppose  that  <D  =  [<p{,  <p2,  .  ..  ,<pN}  C  L2(RJ)  such  that  [<pl(-  -  k)  :  1  <  I  <  N, 
k  €  Z^}  is  a  frame  for  its  closed  linear  span  S(4>).  Let  ^  !  ,  i/r2  ____  ,  I/SN  be  elements  in 
5(<E>)  so  that  each  $J  is  a  linear  combination  of  <pl(-  —  k)\  I  <l  <  L,  k  e  ~Ld  .  A  natural 
question  to  ask  is  the  following:  when  can  we  say  that  {^-7(-  —  k)  :  1  <  j  <  N,  k  e  TLd} 
is  also  a  frame  for  5  (<$)? 

If  ^J  E  5(d>),  then  there  exists  [pjik  :  k  6  ZJ}  in  /2(ZJ)  such  that 

Vr''(*)=  E  E 


In  terms  of  Fourier  transform 

f/-(£)     =     E  E 


E^7«)^tf)     0<7<^),  (37) 

/=! 


where  P-/(|)  =   E  P7*e~'(U)-  Let  p(^)  be  the  N  x  ^  matrix 


Let5  and  T  be  two  positive  definite  matrices  of  order  N.  WesayS  <  Tif  (jc,  Sx)  <  U, 
for  all  x  e  RN.  The  following  lemma  is  the  generalization  of  Lemma  3.  1  in  [2]. 

Lemma  5.1.  Let  <pl,isl  for  I  <  I  <  N,  and  P(f  )  be  as  above.  Suppose  that  there  exist 
constants  C\  and  €2,  0  <  Ci  <  Ci  < 


C27     for  a.e.  $  €  T.  (38) 

Then,  for  all  f  €  L2(R^),  we  have 

Ci  E  E  |{/,  ^(-  -  ^))|2  <  E  E  |(/,  *'(•  -  ^))|2  <  c2  E  E  |(/,  ^(-  -  *))|2  -  (39) 


Let  A  be  a  dilation  matrix,  B  =  A1  and  a  =  |  det  A\  =  |  det  B\.  Let 

^={«o,  «!,...,  afl-,}  (40) 

and 

^5  =  Wo,^N...,^i}  (41) 


Multiwavelet  and  frame  packets  ofL2(R(])  455 

e  fixed  sets  of  digits  for  A  and  B  respectively.  For  0  <  r,  s  <  a  —  1  and   1  <  /,  j  <  L, 
efine  for  a.e.  £, 


P,*.«r/.  (42) 

et 


nd 

£(£)  =  (£"(f))0          _  •  (44) 

o  £(£)  is  block  matrix  with  a  blocks  in  each  row  and  each  column,  and  each  block  is 
square  matrix  of  order  L,  so  that  £(£)  is  a  square  matrix  of  order  aL.   We  have  the 
allowing  lemma  which  will  be  useful  for  the  splitting  trick  for  frames. 

emma  5.2.  (i)  //v  e  KA,  then    £  e-/2;r(/rVv)  _  ^ 

us  KB 
i)  The  matrix  £(£),  defined  in  (44),  is  unitary. 

'roof.  Item  (i)  is  the  orthogonal  relation  for  the  characters  of  the  finite  group  Zd/BZd  (see 
14]).  Observe  that  the  mapping 

/^  ~r  o iL    \—>  e     ""   »  ',       V  G  A/^ 

;  a  character  of  the  (finite)  coset  group  Zd/BZd.  If  v  =0  (i.e.,  if  v  e  AZd),  then 
lere  is  nothing  to  prove.  Suppose  that  v  ^  0,  then  there  exists  a  //  6  ^^  such  that 
~i.7T(  /^  ,u)  ^  |  Since  AT^  is  a  set  of  digits  for  B,  so  is  £#  —  p! .  Hence, 


low 


'herefore, 

£   e-/ar(*->M,y)  =  o,      since 


b  prove  (ii),  observe  that  the  (r,  s)\h  block  of  the  matrix  £(£)£*(£  )  is 


r=o 
'he  (/,  j)th  entry  in  this  block  is 

fl-l    L 

r=0/»=i 


456  Biswaranjan  Behera 

=     E'  E  8lma-V2 
r=0m=l 


L 

=      E  &imSjmSrS,     (by  (0  of  the  lemma) 

m=l 


This  proves  that  E(f  )£*(£)  =  /.  Similarly,  £*(£)£(£)  =  /.  Therefore,  E(f  )  is  a  unitary 
matrix.  D 

6.  Splitting  lemma  for  frame  packets 

Let  {(pi  :  1  <  /  <  L]  be  functions  in  L2(IR'/)  such  that  {<?/(•  -  k)  :  1  <  /  <  L,  fc  €  Zd]  is  a 
frame  for  its  closed  linear  span  V.  For  0  <  r  <  a  —  \  and  1  <  /  <  L,  suppose  that  there 
exist  sequences  {fcj^  :  k  eZd}e  I2(2d).  Define  /7r  as  in  (4)  and  (5).  That  is, 


=  E 


E 


Let  //r(f  )  be  the  matrix  defined  in  (7).  Let  KA  and  AT#  be  respectively  fixed  sets  of  digits 
for  A  and  B  as  in  (40)  and  (41).  Let  H  (?)  be  the  matrix 

n  (47) 

0</Vf<fl—  1 

is  a  block  matrix  with  a  blocks  in  each  row  and  each  column,  and  each  block  is  of 
order  L  so  that  H(£)  is  a  square  matrix  of  order  aL.  Assume  that  there  exist  constants  C[ 
and  Ci,  0  <  Cj  <  C?  <  oo  such  that 

C\I  <  #*(£)#(£)  <  C2I    for  a.e.  f  e  Trf.  (48) 

We  can  write  f[  as  % 


E  E'  E  *i.j«,+M"l/2Vj(Ax  -  «*  -  A*),  by  (2) 


where 

^}(jc)  =  a1/2^;(Ajc  -  a,),      0  <  s  <  a  -  1.  (49) 

Taking  Fourier  transform,  we  obtain 


Multiwavelet  and  frame  packets  ofL2(Rd)  457 

here  ,[/<£)  =  £t6Z,  ^.M,^^'*''  Define 

*r'<«  =  (rf/«>),  *.>*!.  (50) 

id 


(52) 
'here  £(£)  is  defined  in  (42)-(44). 

'roof  of  the  claim.  The  (r,  j)th  block  of  the  matrix  P(B£)E(f  )  is  the  matrix 


"he  (/,  j)th  entry  in  this  block  is  equal  to 

1  E  ^ 


* 


E  *?.^ 


,  the  (/,  j)th  entry  in  the  (r,  j)th  block  of  H(%)  is 
^(f  +  2B-1Ajr)=«-1/l  E  ^e- 

" 


E  hi,j.ai 

f=0^€2^ 

=  ^"l/2£E  E  A^^ 
r=0^ez^ 

So  the  claim  is  proved.  In  particular,  we  have 


Since  E(% )  is  unitary  by  Lemma  5.2,  #*(£)#(£)  and  P*(B£)P(B£)  are  similar  matrices. 
Let  X(£)  and  A(f )  respectively  be  the  minimal  and  maximal  eigenvalues  of  the  positive 
definite  matrix  #*(£)#(£),  and  let  X  =  inf  X(f )  and  A  =  sup  A(£ ).  (It  is  clear  from  (52) 

that  X(?)  and  A(£)  are  2nZd -periodic  functions.)  Suppose  0  <  X  <  A  <  oo.  Then  we 
have,  by  (48)  (in  the  sense  of  positive  definite  matrices), 

X/  <  H*(f)H(g)  <  A/     for  a.e.  §  6  Tr/ 


458  Biswaranjan  Behera 

which  is  equivalent  to 

U  <  P*(£)P(f  )  <  A/     for  a.e.  £  €  Jd. 
Then  by  Lemma  5.  1  ,  for  all  g  €  L2(fRJ),  we  have 


where  ^  is  defined  in  (49).  Since 

E  E  i(«v/2w(A--*))i2==  "EE  E  |(«^/(1)(--*))l2.       (55) 

" 


which  follows  from  (49),  inequality  (54)  can  be  written  as 

-«)|2   <    E'E 


E  |(^flI/2.-        < 


<     AE  E   |(*,fl1/2«(A--*))|2.   (56) 

/=  1  A-€Z^ 

This  is  the  splitting  trick  for  frames:  the  A~~  ^-translates  of  the  L  dilated  func- 
tions (pi(A-),  1  <  /  <  L,  are  'decomposed'  into  Z^  -translates  of  the  aL  functions  f£, 
0<s  <a-  1,  1  <l  <  L. 

We  now  apply  the  splitting  trick  to  the  functions  (ff  :  1  <  /  <  L}  for  each  s,  0  <  ^  < 
a  —  1  to  obtain 


E   |(*,fll/2//'(A--t)}|2,(57) 


where  //'r,  0  <  r  <  a  -  1  are  defined  as  in  (46)  (f£  now  replaces  (pi): 

ff'M  =  E  E  hfoaWf^Ax  -  *);    0  <  5  <  fl  -  1,  1  <  /  <  L.      (58) 


Summing  (57)  over  0  <  5  <  a  —  1,  we  have 

''       K*.  #•"(•-  *>)l2 


lV=0  /=  1  jte2rf  s—Q  r=0  /= 


Using  (56),  we  obtain 

*2i;  E  i(*,« 


.      (59) 


Multiwavelet  and  frame  packets  ofL2(Rd ) 

Now  as  in  the  case  of  orthonormal  wavelet  packets,  we  can  define  /«,  for  each  n  >  0  and 
1  <  /  <  L  (see  (23)  and  (27)).  In  order  to  ensure  that  //'  are  in  L2(RJ)  it  is  sufficient 
to  assume  that  all  the  entries  in.  the  matrix  H(£),  defined  in  (47),  are  bounded  functions 
Comparing  (58)  and  (23),  we  see  that 


So  (59)  can  be  written  as 

o  A    _     ,,         ->/^  „  .,2  a2~[  L 

•  -*)}|     <    E  E 

L 


By  induction,  we  get  for  each  7  >  1 , 


<  "E 


E 


--k).     (60) 
We  summarize  the  above  discussion  in  the  following  theorem. 

Note,  if/1  :  n  >  0,   1  <  /  <  L}  will  be  called  the  wavelet  frame  packets. 

Theorem  6.1.  Let{<pt  :  I  <  I  <  L}  c  L2(Ud)  be  such  that  {<pi(--k)  :  1  <  I  <  L,k  €  Zd] 
is  a  frame  for  its  closed  linear  span  VQ,  with  frame  bounds  C\  andCi.  LetH(%),  Hr(t-),l. 
and  A  be  as  above.  A  ssume  that  all  entries  ofHr(t;+2B~{  #VTT)  are  bounded  measurable 
functions  such  that  0  <  X  <  A  <  oo.  Let  {//l  :  n  >  0,  1  <  /  <  L}  be  the  wave  let  frame 
packets  and  let  Vj  =  {/  :  /(A"-7'-)  €  V0}.  Then  for  all  j  >  0,  the  system  of  functions 

{//"(•  -  *)  :  0  <  n  <  ^  -  1,  1  <  /  <  I,  it  6  Z^} 
w  a  frame  ofVj  with  frame  bounds  WC\  and  A-^Co. 


/  Since  {^/(.  -  k)  :   1  <  Z  <  L,  A:  €  Z^}  is  a  frame  of  VQ  with  frame  bounds  Ci  and 
€2  ,  it  is  clear  that  for  all  7 

[aj/2<pi(AJ  •  -*)  :   1  <  /  <  L,  Jt  6  Zrf} 
is  a  frame  of  V)  with  the  same  bounds.  So  from  (60),  we  have 

*.JCiM2<*i:E  E   \(g,ff(--V)?<*JC2M2    forallgeVj.    (61) 
n=0  /=UeZJ 

D 

In  Theorem  3.2  we  proved  that  the  basic  multi  wavelet  packets  form  an  orthonormal  basis 
for  L2(Ud)  =  UV).  An  analogous  result  holds  for  the  wavelet  frame  packets  if  the  matrix 
/f  (?),  defined  in  (47),  is  unitary. 


460  Biswaranjan  Behera 


Before  proving  this  result  let  us  observe  how  the  space  Uj>oV)  looks  like.  Let  VQ  = 
sp{<pt(-  -  k)  :  1  <  /  <  L,  k  €  Zd),  Vj  =  [f  :  f(A'J.)  e  V0}  and  Vj  C  Vj+{.  Let 
W  =  UV}.  Then  it  is  easy  to  check  that  /  €  W  =»/(•-  A'-'Jfc)  €  W  for  all  j  €  Z  and 
k  e  Zrf.  We  claim  that  elements  of  the  form  A~Jk  are  dense  in  [RJ.  For  AT  =  {A:  i,  #2.  ••-,&«} 
a  set  of  digits  for  A,  define  the  set 


2  = 

In  the  above  representation  of  x,  efs  need  not  be  distinct.  We  have 


where  C  is  a  constant  and  0  <  a.  <  1  (see  [17],  Chapter  5).  Therefore,  the  series  that  defines 
x  is  convergent.  Forjc  =  (jci,  jc2,  .  .  .,^)  e  Rd,  \\x\\  =  (ki|2H-|jc2|24--  -  •  +  |jt£/|2)i  The 
set  Q  satisfies  the  following  properties  (see  [10]): 

(0  fi  =  U?=!A-|(fi+A:/) 

GO  ^62(2  +  *)  =  I*' 

(iii)  Q  is  compact. 

Let  €  >  0  and  y  6  Q.    We  first  show  that  there  exist  ;   6  Z  and  k  e  Z.d  such  that 

||  y  -  A~^||  <  £.  From  (i)  we  have 


•] 


=     U  U  ( 

1=1  W=l 


Hence,  for  any  ;  >  1  and  any  y  e  Q,  there  exist  yj  €  Q  and  /i,  /2,  ...,//  €  #  such  that 

y  =  A~Syj  -f  A^'/j  +  A-W"l)/y-i  +  •  •  •  +  A"1/!. 

Therefore, 

=     ||A-'x/ll 


<  C'a-7     (as  2  is  compact) 

<  6,     choosing  j  suitably. 

Now  if  y  6  Rd,  then  by  (ii)  y  =  yQ  -f  p  for  some  >'o  e  Q  and  p  e  Zd.  For  >'o  €  2,  there 
exist  ;  >  0  and  k  e  1d  such  that  ||>-0  -  A~Jk\\  <  e.  That  is, 


So  the  claim  is  proved. 


Miiltiwavelet  and  frame  packets  ofL2(Ud)  46  1 

We  have  proved  that  W  is  invariant  under  translations  by  A~^k  and  these  elements  are 
dense  in  Ud.  Therefore,  W  is  a  closed  translation  invariant  subspace  of  L2((RJ).  Hence, 
W  =  L|(RJ)  for  some  E  C  Ud  (see  [15]),  where 

L2E(Rd)  =  {/  €  L2(Rrf)  :  supp  /  C  E}. 

Now  let 

L 


Claim.  E  =  £0  a.e.  _ 

To  prove  the  claim  we  will  follow  [1],  Theorem  4.3.  Since  <pi(Aj  •)  e  V/  C  W,  the 

function  (<p/(A'.))A  =  ^i(B'J-)  e  W  ={/:/€  W}.  Therefore,  £->(supp<£/)  = 
supp  (jj7#/(fl~~</"-))  C  £  for  all  7  >  0  and  1  <  /  <  L,  which  implies  that  £Q  C  E.  Let 
£i  =£\E0.Wehave 

),  (62) 


for  some  2n  Z.d  -periodic  functions  m/  6  L2(TJ).  Hence,  (62)  implies  that  /  =^0  on  E\  for 
all  /  e  Vj  and  hencejfor  all  /  e  UV)  =  W.  Taking  closure,  we  obtain  that  /  =  0  on  E\ 
for  all  /  €  W.  But  W  is  the  set  of  all  functions  whose  Fourier  transform  is  supported  in 
E.  Since  E\  C  £,  we  get  that  E\  =  0  a.e.  Therefore,  E  =  EQ  a.e.  D 


Theorem  6.2.  L^r  {^/(-  -  k)  :    1  <  /  <  L,  ^  €  ZJ}  C  L2(R^)  te  a  frame  for  its  closed 
linear  span  VQ,  with  frame  bounds  C\  and  €2  and  let  VQ  C  Vi,  where  Vj  =  {/  :  /(A"-7'-)  6 
isunitatyfora.e.  f.  Then{ff(—k)  :  n  >  0,  1  <  /  <  L,  *  €  ZJ} 


w  a  frame  for  the  space  U/>o  V/  w/f/z  r/z^  same  frame  bounds. 

More  generally,  let  S  =  {(n,  7)  €  MO  x  ^}  ^^  ^^A  r/zar  U(/i,/)6S  ^J'  ^  a  partition  of 
NO-  7%en  rftg  collection  of  functions  [aJ^f^AJ  •  -fc)  :  1  <  /  <  L,  (/x,  j)  €  5,  *  €  2^}  w 
a  frame  for  Uj>o  V/  w///z  ^/ze  5«m^  bounds  C\  and  €2  • 


Proof  Since  #(£)  is  unitary,  X  =  A  =  1  so  that  the  inequalities  in  (60)  are  equalities,  and 
from  (61)  we  have 

E   |M"(--*))|2<C2ll*ll2    forallgeV).        (63) 


Now  let  ft  e  U/>oV/.  Then  there  exists  hj  e  Vj  such  that  A;  ->  A  as  7  -»  oo.  Fix  7,  then 
for  7  <  /,  we  have  from  (63) 


E 


|(*y,//ll(--*)}|2<C2l|A/||2. 


Letting  7'  —  >  oo  first  and  then  j  —  >  oo,  we  have  for  all  A  6  U/>oVy 

EE   |{A,//I(--*))|2<C2||A||2.  (64) 


462  Biswaranjan  Behera 

To  get  the  reverse  inequality  we  again  use  (63): 

M2  *  "EE  E  \(hj,f>n(- 


"' 


=  EE 


Therefore, 

^-,l/2nt      it  /    v>    T~^    \ — v     \li  7       x  n  /  i  \\|2 

c\'  \\hj\\   <   {  E  E  E  \(hj-h<fi('-kn\ 


,       by  (64). 
Taking  7  -»  oo,  we  get 

c,IIMI2<  EE  E  l(A>//f(--*))|2 


for  all  h  €  (JVj.  So  the  first  part  is  proved. 

Now  let  U*}  =  sp{aJ/2f/l(AJ  • -k)  :    1  <l  <L,    he  Zd}.  Then  we  can  prove  as  in 
the  orthogonal  case  (see  (35))  that 


where  0  is  just  a  direct  sum  not  necessarily  orthogonal,  and  Inj  =  [r  e  NQ  :  a-^n  <  r  < 
a-i(n  +  1)  —  1}.  Now,  since  H(%)  is  unitary,  we  have  A  =  A  =  1  and  hence  (57)  is  an 
equality.  Therefore, 

E  E  \\t,aWf?(A-k)}?=  EE  E  \(g,fr+r(--k))f. 

l~\kzZd  r=0/=ljfe€Z«' 

From  this  we  get 

E  E  i(g,a2/2//'(A2.-fc))|2  =  "EE'E  E 


=  fl  't"  E  E  l(*.//r(--*))l2- 

Similarly, 

E  E.  |(«-  «;/2//"(AJ'  •  -^))|2   = "  "E    E  E  \(g>  f{(-  -  *))|2 

=    E   E  E   \(S,f[(--k)f.     (65) 


Multiwavelet  and  frame  packets  of  L~(R({)  463 

From  the  first  part  of  the  theorem,  we  have  for  all  /  e  UV) 

C,||/||2<  E  E  E  !(/,//^~*))|2<c2||/l!2. 

«>0  /=!  kzl^ 

But,  the  set  5  is  such  that  U(/7  ^s  Inj  =  NO-  Therefore, 

c,imi2<    E    E  E  E  \(f.f[(--V)f<C2\\f\\2. 

(n,j)eSr€lnjl=\ke2d 

Using  (65),  we  get 

Ciii/n2<    E   E  E  \(f,«j/2fin(AJ'-V)f<C2\\f\\2 


for  all  /  €  UV/.  This  completes  the  proof  of  the  theorem.  D 

Acknowledgements 

The  author  is  grateful  to  Prof.  Shobha  Madan  for  many  useful  suggestions  and  discussions. 
The  author  was  supported  by  a  grant  from  The  National  Board  for  Higher  Mathematics, 
Govt.  of  India. 

References 

[1  ]  deBoor  C,  DeVore  R  and  Ron  A,  On  the  construction  of  multivariate  (pre) wavelets,  Construc- 
tive Approximation  9  ( \  993)  1 23-1 66 
[2]  Chen  D,  On  the  splitting  trick  and  wavelet  frame  packets,  SIAM  J.  Math.  Anal  31(4)  (2000) 

726-739 
[3]  Chui  C  R  and  Li  C,  Non-orthogonal  wavelet  packets,  SIAM  7.  Math.  Anal.  24(3)  (1993) 

712-738 
[4]  Cohen  A  and  Daubechies  I,  On  the  instability  of  arbitrary  biorthogonal  wavelet  packets,  SIAM 

J.  Math.  Anal.  24(5)  (1993)  1340-1354 

[5]  Coif  man  R  and  Meyer  Y,  Orthonormal  wave  packet  bases,  preprint  (Yale  University)  (1 989) 
[6]  Coifman  R,  Meyer  Y  and  Wickerhauser  M  V,  Wavelet  analysis  and  signal  procesing,  in: 
Wavelets  and  Their  Applications  (eds)  M  B  Ruskai  etal  (Boston:  Jones  and  Bartlett)  (1 992) 
153-178 

[7]  Coi  fman  R,  Meyer  Y  and  Wickerhauser  M  V,  Size  properties  of  wavelet  packets,  in:  Wavelets 
and  Their  Applications  (eds)  M  B  Ruskai  etal  (Boston:  Jones  and  Bartlett)  (1992)  453-470 
[8]  Daubechies  I,  Ten  Lectures  on  Wavelets  (CBS-NSF  Regional  Conferences  in  Applied  Math- 
ematics, Philadelphia:  SIAM)  (1992)  vol.  61 
[9]  Goodman  TNT,  Lee  S  L  and  Tang  W  S,  Wavelets  in  wandering  subspaces,  Trans.  Am,  Math. 

Soc.  338(2)  (1993)  639-654 
[10]  Grochenig  K  and  Madych  W  R,  Multiresolution  analysis,  Haar  bases,  and  self-similar  tilings 

of  R",IEEE  Trans.  Inform.  Theory  38(2)  (1992)  556-568 

[11]  Hernandez  E  and  Weiss  G,  A  First  Course  on  Wavelets  (Boca  Raton:  CRC  Press)  (1996) 
[12]  Herve  L,  These,  Laboratoire  de  Probabilites,  Universite  de  Rennes-I  (1992) 
[  1 3]  Long  R  and  Chen  W,  Wavelet  basis  packets  and  wavelet  frame  packets,  J.  Fourier  Anal.  Appl. 

3(3)  (1997)  239-256 

[14]  Rudin  W,  Fourier  Analysis  on  Groups  (New  York:  John  Wiley  and  Sons)  (1962) 
[15]  Rudin  W,  Real  and  Complex  Analysis  (New  York:  McGraw-Hill)  (1966) 
[16]  Shen  Z,  Nontensor  product  wavelet  packets  in  Lt(W),  SIAM  J.  Math.  Anal.  26(4)  (1995) 

1061-1074 

[17]  Wojtaszczyk  P,  A  Mathematical  Introduction  to  Wavelets  (Cambridge,  UK:  Cambridge  Uni- 
versity Press)  (1997) 


Proc.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  11 1,  No.  4,  November  2001,  pp.  465-470. 
©  Printed  in  India 


A  variational  principle  for  vector  equilibrium  problems 

KRKAZMI 

Department  of  Mathematics,  Aligarh  Muslim  University,  Aligarh  202  002,  India 
MS  received  1  December  2000;  revised  17  May  2001 

Abstract.     A  variational  principle  is  described  and  analysed  for  the  solutions  of 
vector  equilibrium  problems. 

Keywords.    Vector  equlibrium  problem;  variational  principle;  P-convexity;  P  —  $- 
monotonicity. 

1.  Introduction 

Throughout  this  paper,  X  is  a  real  topological  vector  space;  K  c  X  be  a  nonempty,  closed 
and  convex  set;  (F,  P)  be  a  real  ordered  topological  vector  space  with  a  partial  order  </> 
induced  by  a  solid,  pointed,  closed  and  convex  cone  P  with  apex  at  origin,  thus 

x  <P  y   <£=>  y  -  jc  e  P    Vx,  y  €  Y. 

If  intP  denotes  the  topological  interior  of  the  cone  P ,  then  weak  ordering,  say  &mtp  (or 
^int/3)'  on  Y  is  defined  by 

x  ^imP  y  °r  y  ^int/>  x  4=*  y  -  x  $  intP    V*,  y  e  Y. 

Let  /  :  X  x  X— >  Y  be  a  mapping  with  /(jc,  x)  =  0  Vjt  e  X,  then  vector  equilibrium 
problem  (for  short,  VEP(/,  K))  is  to  find  x  e  K  such  that 

f(x,y)£miPQ,    Vx.yeK. 

VEP(/,  K)  has  been  studied  by  Kazmi  [K2].  VEP(/,  K)  includes  as  special  cases,  vector 
optimization  problems,  vector  variational  inequalities,  vector  variational-like  inequalities, 
vector  complementarity  problems,  etc.,  see  Kazmi  [K2]  and  the  references  therein. 

If  Y  =  R,  P  =  R+,  then  VEP(/,  K)  reduces  to  the  scalar  equilibrium  problem  [B-O1, 
B-O2]  of  finding  x  €  K  such  that 

/C*,y)>0,    VyeK. 

In  this  paper,  we  shall  describe  and  analyse  a  variational  principle  for  the  solutions  of 
VEP(/,  K). 

The  construction  of  variationai  principles  is  of  interest  both  theoretically  and  in  practice. 
Conceptually,  it  is  of  significance  to  know  that  there  is  a  mapping  defined  on  X  which  is 
optimized  precisely  at  the  solutions  of  VEP(/,  K).  In  practice,  it  is  of  importance  because 
it  allows  one  to  use  the  highly  developed  theory  of  numerical  optimization  to  numerically 
approximate,  and  compute  solutions  of  these  problems. 

465 


466  K  R  Kazmi 

More  precisely,  following  the  terminology  of  Auchmuty  [A],  we  say  that  a  variational 
principle  holds  for  VEP(/,  K),  if  there  exists  a  mapping  F  :  K  —  >  Y  depending  on 
the  data  of  VEP(/,  K)  but  not  on  its  solution  set,  such  that  the  solution  set  of  VEP(/,  K) 
coincides  with  the  solution  set  of  the  vector  maximization  problem  (for  short,  VMP(/,  K)) 

maxjnt/>  F(x),     subject  to  x  e  K. 

If  f(x,y)  =  (<t>'(x),  rj(y,x))t  where  r\  :  K  x  K  —  *  X  is  a  continuous  function  and 
4>  :  K  —  >  Y  is  Frechet  (or  linear  Gateaux)  differentiable  and  P-convex  mapping,  the  x  is 
a  solution  of  VEP(/,  K)  if  and  only  if  jc  is  a  solution  of 

mining  <t>(x)>      subject  to  x  e  K, 

see  Kazmi  [Kl].  For  related  work,  see  [K3,  K-A]. 
Thus,  setting  F  =  —  $,  a  variational  principle  for  VEP(/,  K)  holds. 
Now,  consider  the  case: 


where  g,  h  :  K  x  K  —  >  Y  are  nonlinear  mappings,  then  VEP(/,  K)  becomes: 
(VEP(£  +  h,  K)),  find  jc  e  K  such  that  g(xt  y)  +  h(x,  y)£lniP  0,   Vy  e  K. 

We  shall  use  the  following  concepts  and  result: 
The  mapping  g  is  called  P-monotone  if  and  only  if 

g(x9  y)  <P  -g(y,  x),    VJT,  y  e  K. 

A  mapping  T  :  X  —  >  Y  is  called  jP-convex  if  and  only  if  for  each  pair  x,  y  e  K  and 
^  6  [0,1], 

7(Xjc  +  (1  -  X)v)  <P  XT(x)  +  (1  -  X)T(y). 

Note  that  if  g(x,  y)  =  <0  (jc),  y—x)  where  <f>  :  X  —  >  Y  be  P-convex  and  linear  Gateaux 
differentiable,  then  g  is  P-monotone  since  0  (•)  is  P-monotone. 

Lemma  1  [C].  Let  (Y,  P)  be  an  ordered  topological  vector  space  w/Y/z  a  solid,  pointed, 
closed  and  convex  cone  P.  Then  YJC,  y  e  X,  we  have 

y<Px  and   y  £m[P  0   imply  x  £int/>  0. 

Finally,  in  order  to  formulate  our  variational  principle  we  introduce  a  perturbation  mapping 
VT(-  ,  •)  :  K  x  K  —  >  Y  which  satisfies  for  all  jc,  y  e  K: 

(i)  0</>  V(*,y), 

(ii)  V(*,*)  =  0, 

(iti)  VC*,  ty  +  (1  -  A.)x)  =  o(X);     1  E  [0,  1]. 

Let  us  indicate  some  possible  choices  for  i/r(«  ,  •)'•  K  x  K  —  >  Y  satisfying  properties 
(iM«0  above.  Clearly,  the  choice  ^(-  ,  •)  =  0  is  always  possible.  Next  let  </>(•  ,  •): 
K  x  K  —  >  Y  be  P-convex  in  the  second  argument,  and  YJC  e  K,  let  (/>(-  ,  •)  be  Gateaux 
differentiable  at  x  with  Gateaux  differential  </>'(*,  •)  €  L(X,  Y)  where  L(X,  7)  is  a  space 
of  all  linear  bounded  functional  from  X  to  Y.  Set 

')  -  *(*,  ^)  -  (<t>\x,x)9  y-x)9 


Vector  equilibrium  problems  467 

n  VT(.  ,  .)  satisfies  properties  (i)-(iii).  In  particular  if  0(«):  K  —  >  Y  be  P-convex,  and 
teaux  differentiable,  then  we  may  choose 

tf  (*.  y)  =  0(30  -  0(*)  -  <0'(*),  >'  -  x). 

ally,  if  VG  *  •)•  K  x  #  —  *  R  |J{oo},  where  K  is  a  subset  of  normed  linear  space  X, 
n  we  may  choose  VU,  >0  =  a||;y  -  *||2,  fora  >  0,  which  satisfies  (i)-(iii). 
,  we  define  a  mapping  G  :  K  —  >  Y  by  means  of 


GO)  :=  inf{-g(y,  x)  +  A(*,  >>)  +  ^U,  7)  :  y  e  K},  (I) 

I  we  associate  to  VEP(g  4-  /i,  A')  the  following  vector  maximization  problem: 
VMP(#  -f  A,  V,  /Q  :  maxintp{G(;t)  :  x  e  K}. 

remark  that  the  mapping  G(-)  generalizes  the  gap  function  used  in  connection  with 
iational  inequalities,  see  Herker  and  Pang  [H-P],  and  the  references  therein. 
"rom  —g(x,  A-)  H-  h(x,  x)  -f  ty(x,  x)  =  0  follows 

G(JC)  <P  0    V^:  €  K.  (2) 

also  define  the  following  concept. 
_et  \js  satisfy  (i)-(iii),  the  mapping  g  is  called  P  —  -^-monotone  if  and  only  if 

g(x,  y)  <P  MX,  >')  -  g(y,  x),     Vx,  y  €  K. 
ls(x,  >')  =  0,  Vjc,  y  €  K,  then  P  —  ^-monotone  mapping  becomes  P-monotone. 

Results 

st  we  prove  the  following  results: 

eorem  2.  Let  the  following  assumptions  hold: 

*  The  mapping  g  satisfies:    g(x,  x)  =  0,  Vx  €  K;  g  is  P-monotone;  Vx,  y  e  K,  the 

mapping  A  e  [0,  1]  —  >  g(X.y  -f  (1  —  tyx,  y)  is  continuous  at  0+;  g  is  P  -convex  in 

the  second  argument. 

i  The  mapping  h  satisfies:  h(x,x)  =0,  V.x  e  K\h  is  P  -convex  in  the  second  argument. 
en  VEP(g  +  h,  K)  and  the  problem  of  finding  x  e  K  such  that 


pO,  (3) 

lh  have  the  same  solution  set. 

lof.   Let  x  be  a  solution  of  VEP(#  +  h,K),  that  is, 

g(x,y)+h(x,y)£miP09  VyeK.  (4) 

ice  g  is  P-monotone,  YJC,  y  e  K,  we  have 

(g(x,  y)  +  h(x,  >'))  ~  (-£(}',*)  +  h(x,  y))  <P  0,  (5) 

>m  Lemma  1  ,  eqs  (4)  and  (5),  it  follows  that 


468  K  R  Kazmi 

or, 

-g(y,x)+h(x,y)eW:=Y\(-inlP),     VyzK.  (6) 

Since  ifr(x,  y)  e  P,  we  have 

-g(y,x)  +  h(x,y)  +  t(x,y)€W  +  PcW9     Vy  e  AT, 
which  implies 

inf{-£(v,x)+/z(*,y)  +  1/^,30   :  >'  €  K}£iniP  0, 
that  is, 

G(*)&nt/>  0. 

Conversly,  let*  be  a  solution  of  problem  (3).  Then  by  the  definition  of  G(-),  we  have 
-*(?,  x)  +  h(x,  y)  +  VT(JC,  y);£intp  0,    V>'  e  K. 

Fix  y  €  K  arbitrarily,  let*x  :=  ky  +  (1  —  X)*,  A.  €]0,  1],  jc^  6  K  as  K  is  convex,  and 
hence  the  above  inequality  becomes 


0.  (7) 

Since  g  is  P-convex  in  the  second  argument,  we  have 

0      = 

<p 

-(l-k)g(x^x)     <P 

By  using  preceding  inequality  and  the  properties  of  cone  P,  we  have 


<P  Igfa,  y) 

<P 


using  the  properties  of  h  and  i//". 

Since  (1  -  A)  >  0,  after  dividing  the  preceding  inequality  by  (1  -  X)  >  0,  we  have,  from 
Lemma  1,  (7)  and  the  resultant  inequality, 


+  *A(x,  y)  +  o(A)^intP  0  e  W. 


(L  —  A) 

After  dividing  the  preceding  inclusion  by  A  >  0,  letting  A  4,  0  and  hence  ^  —  *  x  e 
and  then  by  hemicontinuity  of  g  and  closedness  of  W,  we  have 


POt    Vy  e  K. 
This  completes  the  proof. 

Theorem  3.  Let  all  the  assumptions  of  Theorem  2  except  P  -monotonicity  of  g  hold.  Let 
g  be  P  -  ^-monotone,  then  VEP(g  +  /i,  K)  and  problem  (3)  both  have  the  same  solution 
set. 


Vector  equilibrium  problems  .         469 

Proof.  Let  x  be  a  solution  of  VEP(£  +  h,K),  that  is, 

g(x,y)  +  h(x,ytetopO>    vve*.  (8) 

Since  £  is  P  ~  ^  -monotone,  Vx,  >>  e  tf,  we  have 

(g(*,30  +  M*,)0)-  (-£(}%*)  +M*,  7)  +  lK*.>0)  <p  0.  (9) 

From  Lemma  1  ,  (8)  and  (9),  it  follows  that 

-g(y,  x)  +  h(x,  y)  +  ^  (x,  }')^intP  0,     Vy  e  K, 
which  implies 

mf{~g(y,x)  +  h(x,y)  +  il'(x,y)  :  y  e  #};£int/>  0. 

Converse  part  of  theorem  is  just  same  as  the  converse  part  of  Theorem  2.  This  completes 

the  proof. 

Now,  on  combining  Theorem  2  (or  Theorem  3)  with  inequality  (2),  we  have  the  following 

variational  principle  for  VEP(g  +  ht  K). 

Theorem  4.  Let  the  assumptions  of  Theorem  2  and  inequality  (2)  hold,  x  is  a  solution  of 
VEP(g  +  h^K)  ifand  oniy  ifG(x)  =  0.  If  the  solution  set  of  VEP(g  +h,K)  is  nonempty, 
then  the  solution  sets  of  VEP(g  +  h,  K)  and  VMP(g  +  h,1r,K)  coincide. 

Proof.   If  x  is  a  solution  of  VEP(£  +  h,K)  then,  by  Theorem  2, 

G(*)£nt/>  0. 
From  (2), 

G(x)  <P  0. 

These  above  inequalities  imply  that  G(x)  =  0.  Next,  if  G(x)  =  0  then  by  definition  of 
G(-),  we  have 

0^intP0<P    -g(y,x)+h(x,y)  +  ilr(x,y),    VyzK. 
By  Lemma  1,  it  follows  that 

-g(y,  ^  +  h(x,  30  +  ^(x,  y)£intP  0  Vy  e  K. 


Follow  the  same  lines  of  converse  part  of  Theorem  2,  we  can  have  that  x  is  a  solution  of 
VEP(g+ht  K).  This  proves  the  first  part  of  the  theorem.  IfxisasolutionofVEP(£+fc,  *0, 
then  GO)  =  0,  and  from  inequality  (2)  follows  that*  is  a  solution  of  VMP(g  +  h,  ^i  ^)- 
Then  all  solutions  of  VMP(£  +  h,  &,  K)  must  satisfy  G(*)  =  0,  and  therefore  are  in  the 
solution  set  of  VEP(£  +  h,K).  This  completes  the  proof. 

We  remark  that  the  variational  principle  described  in  this  paper  is  a  generalization  of 
variational  principles  described  by  Blum  and  Oettli  [B-O1,  B-02],  and  Auchmuty  [A]. 


470  K  R  Kazmi 

References 

[A]  Auchmuty  G,  Variational  principles  for  variation al  inequalities,  Numer.  Fund  Anal.  Optim. 

10(1989)863-874 

[B-01]  Blum  E  and  Oettli  W,  Variational  principles  for  equilibrium  problems,  parametric  opti- 
mization and  related  topics,  III  (Giistrow  1991),  in:  Approximation  and  Optimization 
(eds)  J  Guddat,  H  Th  Jongen,  B  Kummer  and  F  Nozieka  (Lang,  Frankfurt  am  Main)  3 
(1993)79-88 

[B-O2]  Blum  E  and  Oettli  W,  From  optimization  and  variational  inequalities  to  equilibrium  prob- 
lems, Math.  Stud,  63  (1994)  123-145 
[C]  Chen  G-Y,  Existence  of  solution  of  vector  variational  inequality:    An  extension  of  the 

Hartmann-Stampacchia  theorem,  J.  Optim.  Theory  Appl  74  (1992)  445-456 
[C-C]  Chen  G-Y  and  Craven  B  D,  Existence  and  continuity  of  solutions  for  vector  optimization, 

J.  Optim.  Theory  Appl  81  (1994)  459-468 

[H-P]  Marker  P  T  and  Pang  J-S,  Finite-dimensional  variational  inequality  and  nonlinear  comple- 
mentarity problems:  A  survey  of  theory,  algorithms  and  applications,  Math.  Prog.  B48 
(1990)  161-220 
[Kl]  Kazmi  K  R,  Some  remarks  on  vector  optimization  problems,  J.-Optim.  Theory  Appl  96 

(1998)133-138 
[K2]  Kazmi  K  R,  On  vector  equilibrium  problem,  Proc.  Indian  Acad.  Sci.  (Math.Sci.)  110  (2000) 

213-223 

[K3]  Kazmi  K  R,  Existence  of  solutions  for  vector  saddle-point  problems,  in:  Vector  varia- 
tional inequalities  and  vector  equilibria,  Mathematical  Theories  (ed)  F  Giannessi  (Kluwer 
Academic  Publishers,  Dordrecht,  Netherlands)  (2000)  267-275 

[K-A]  Kazmi  K  R  and  Ahmad  K,  Nonconvex  mappings  and  vector  variational-like  inequalities, 
in:  Industrial  and  Applied  Mathematics  (eds)  A  H  Siddiqi  and  K  Ahmad  (New  Delhi, 
London:  Narosa  Publishing  House)  (1998)  103-115 

[Y]  Yang  X-Q,  Vector  complementarity  and  minimal  element  problems,  /.  Optim.  Theory  Appl. 
77(1993)483-495 


©  Printed  in  India 


On  a  generalized  Hankel  type  convolution  of  generalized  functions 


S  P  MALGONDE  and  G  S  GAIKAWAD* 

Department  of  Mathematics,  College  of  Engineering  ,  Kopargaon  423  603,  India 
*Department  of  Mathematics,  S.S.G.M.  College,  Kopargaon  423  601,  India 
E-mail:  sescolk@giaspnol.vsnl.net.  in 

MS  received  9  August  2000;  revised  14  February  2001 

Abstract.  The  classical  generalized  Hankel  type  convolution  are  defined  and  ex- 
tended to  a  class  of  generalized  functions.  Algebraic  properties  of  the  convolution 
are  explained  and  the  existence  and  significance  of  an  identity  element  are  discussed. 

Keywords.  Generalized  Hankel  type  transformation;  Parserval  relation;  generalized 
functions  (distributions);  convolution. 

1.  Introduction 

The  fact  that  there  is  no  simple  expression  for  the  product  J]i[xv]JljL[yv]  in  the  sense  that 
there  is  a  simple  expression  e^*"1"^  for  the  product  e1  v*e/:v  means  that  there  is  no  simple 
Faltung  or  convolution  theorem  for  the  Hankel  transform  corresponding  to  well  known 
transforms  like  Laplace,  Fourier  transform  and  so  on. 

Hankel-convolution  operation  has  been  defined  in  the  classical  sense  by  [4]  and  [2].  We 
consider  here  the  generalized  Hankel  type  convolution  and  an  extension  of  that  definition 
to  a  class  of  generalized  functions  analogous  to  that  introduced  by  [1  1,  5]  and  [6].  This 
extension  has  useful  applications,  when  dealing  with  continuous  linear  systems  which  can 
be  characterized  by  a  Hankel  convolutional  representation;  such  systems,  which  we  may 
call  'generalized  Hankel  translation  invariant  continuous  linear  systems',  may  thereafter 
be  considered  when  developing  sampling  expansions  for  inverse  generalized  Hankel  type 
transforms  of  distributions  of  compact  support  on  the  positive  half  line  of  which  the  work 
of  [8]  is  a  particular  case. 

2.  Notation  and  preliminary  results 

We  use  the  following  definition  for  the  classical  generalized  Hankel  type  transform  of  order 


CO 

(Vu/)(r)  =  F(r)  =  vr-'  j  (Xr)v  Jli[(XT)v]f(X)AX, 


OO 

=  h-]v[F](x)  =  v*-1  /(*r)%[(;tr)'']F(r)dr. 


(1) 


(2) 


471 


472  S  P  Malgonde  and  G  S  Gaikawad 

The  transform  pair  (  1  )  and  (2)  has  been  extended  to  certain  spaces  of  generalized  functions 
in  [6]  by  kernel  method  and  in  [5]  by  mixed  Parseval  equation  (a  new  adjoint  method). 

We  begin  with  a  brief  review  of  the  essential  results  obtained  by  [5]  for  the  generalized 
Hankel  type  transform  of  generalized  functions. 

Lemma  2.1.  If  f(x)  is  of  bounded  variation  and  xv/2f(x]  e  Z/(0,  oo)  then  the  direct 
transform  is  well  defined  by  (1),  and  the  inversion  formula  (2)  holds  almost  everywhere  in 
a  neighbourhood  of  every  point  y  =  x  >  0. 

Lemma  2.2.  For  f  (x)  and  G  (x)  satisfying  the  conditions  of  Lemma  2.  1  we  have  the  Parseval 
relation 

00  00 

f  xf(x)g(x)dx  =  f  rF(r)G(r)dr.  (3) 

o  o 

Finally  we  shall  need  results  involving  the  linear  differential  operator  N]L,  y,  ^  >  —1/2, 
defined  by 


-fW  (4) 

and  the  Bessel  type  differential  operator  of  order  /x,  A,  defined  by 

1  =  x-v~vl^Dx2vil+{Dx-v'L~v+l,  (5) 


where  A!  =  x-vn-iv+\Dx2Vn+[Dx-vn  =  AliA.  and  D  stands  for  the  usual  differential 
operator. 

PROPOSITION  2.3 

Ifxf(x)  —  >  0  as  x  ->  oo  where  f(x)  is  sufficiently  smooth  /z/A  v  -transformable  function, 
then  integration  by  parts  shows  that 


(6) 
/n^  g  =  /i^  v[/]  and  changing  r  into  x, 

.  (7) 


PROPOSITION  2.4 

!n  general,  for  sufficiently  well  behaved  (p(x)  and  non-negative  i,  7  we  can  obtain  from  (6) 
<md  (7) 


(8) 
or  M/^  the  defining  formula  (4)  z>zto  consideration, 

}.    (9) 


Convolution  of  generalized  functions  473 

PROPOSITION  2.5 
For  any  sufficiently  smooth  function  f(x)  on  (0,  oo)  it  can  be  shown  that 

VvfA/(*)](T)  =  -V2T2lVv[/](r)  (10) 

provided  that  f  is  h,^  y -transformable  and  that xf(x)  and  xN,LtVf(x)  both  tend  to  zero 
as  x  ->  oo. 

3.  Spaces  of  fundamental  and  generalized  functions 

3 . 1  Testing  function  spaces 

A  complex  valued  function  0,  defined  and  infinitely  differentiable  on  (0,  oo),,  is  said  to 
belong  to  the  space  H^  „(/)  if  and  only  if  the  numbers  y^WO  defined  by 


sup 

0<jt<oo 


(11) 


are  finite  for  every  pair  m,  k  of  non-negative  integers  where  v  is  real  number  and  ^  > 
—  1/2.  HH^  V(I)  is  a  testing  function  space  with  the  topology  generated  by  the  multinorm 

tyn,'*  WOJm,  *=0  and  We  have 

D(0,  oo)  c  #,,,  »(0,  oo)  C  £"(0,  oo),  (12) 

where  D(0,  oo)  and  E(Q,  oo)  denote  respectively  the  restrictions  of  D(R)  and  E(R)  to  the 
positive  real  axis.  Using  (9)  and  following  the  same  lines  of  [5],  it  can  be  readily  shown 
that  the  h^  v  transformation  is  a  topological  isomorphism  of  #M,  v(0,  oo)  onto  itself. 

3.2  The  space  M  (0,  oo)  of  multipliers 

Denote  by  M(0,  oo)  the  linear  space  of  all  infinitely  smooth  functions  0(x),  0  <  x  <  oo 
such  that  for  each  non-negative  integer  /  there  exists  non-negative  integer  /=/(/)  for  which 

a  WrV~2vwoo  o3) 

is  bounded  on  (0,  oo).  By  using  the  generalized  Leibnitz  formula  it  can  be  shown  that  the 
map  6  ->  6<j>  is  an  isomorphism  of  H,^  y(0,  oo)  for  each  0  €  M(0,  oo);  Af  (0,  oo)  is  the 
space  of  multipliers  on  HIJL^(0,  oo). 

3.3  Duals  of  testing  function  spaces 

We  denote  H*t  v(0,  oo)  the  space  of  all  complex  valued  functions  V>  defined  and  infinitely 
smooth  on  (0,  oo)  which  are  of  the  form 

.  -(14) 


H*  v(0,  oo)  is  again  a  (complete)  testing  function  space,  with  the  topology  generated  by 
the  sequence  of  multinorms 


As  usual  we  denote  the  dual  of  H*  v(0,  oo)  by  H*'<v(0,  oo). 


474  S  P  Malgonde  and  G  S  Gaikawad 

For  any  VOO  =  x<t>(x)  €  #*  y(0,  oo),  and  any  non-negative  integer  r,  set 


Then,  for  each  /  e  #*  v(0,  oo)  there  will  exist  constants  c  and  r  such  that 

0  e  #M,v(0,oo)  =»  |{/(x),  *0(jc)>l  <  Ccrr(0).  (16) 


In  particular,  let  /(*)  be  any  locally  integrable  function  on  (0,  oo)  which  is  such  that 
xf(x)  e  L[  (0,  oo)  and  /(jc)  does  not  grow  more  rapidly  than  a  polynomial  when  x  —  >  oo. 
Then  f(x)  generates  a  regular  generalized  function  in  H*  v(0,  oo)  by  the  formula 


(17) 


Any  generalized  function  in  H*^V(Q,  oo)  not  generated  by  the  formula  of  the  type  (17)  will 
be  described  as  singular. 

In  general,  the  derivative  of  a  generalized  function  in  H*  V(Q,  oo)  (defined  in  the  usual 
sense  of  Schwartz),  is  not  in  H*  v  (0,  oo).  However,  in  certain  cases  the  result  of  applying 
a  differential  operator  to  a  generalized  function  in  H*  v(0,  oo)  does  yield  a  generalized 
function  in  H*tV(Qt  oo).  In  particular,  using  for  differential  operators  in  a  generalized 
sense  the  same  notation  as  the  one  used  for  the  corresponding  operators  which  applied  in 
a  classical  sense,  we  have  the  following  results: 

(i)  /  e  E'(0,  oo)  c  H%v(0,  oo)  =»  Df  e  #*%(0,  oo); 

(ii)  /  e  ff*;v(0,  oo)  =*  (xl'2vD)kf  e  H*%(0,  oo); 

(iii)  /  e  H*[V(Q,  oo)  ==>  A*/  e  H*'  y(0,  oo);  for  any  non-negative  integer  k. 

3.4  Distributional  generalized  h,L^  ^-transform 

We  can  now  define  the  generalized  /z/z,  v-transform  of  any  /  G  JY*'  y  (0,  oo)  by  the  analogue 
of  the  Parseval  relation: 

</«,*0M)  =(Vv[/](r),r<D(r))  (18) 

and  clearly  we  have  that  /  E  #*'  y(0,  oo)  =»  /ZM,  y  [/]  e  //"*'  y(0,  oo).  Moreover,  we  can 
establish  that 

Vu[A*/(*)](r)  =  (-v2T2v)*h^  V[/](T)  (19) 

for  any  non-negative  integer  k. 

The  generalized  /i^  y  -transform  of  any  distribution  a  e  £"(0,  oo),  in  the  sense  of  (18),  is 
a  regular  generalized  function  in  H*[  v  (0,  oo)  generated  by  a  smooth  function  f(x)  defined 
on  (0,  oo)  by 

/(*)  =  (cr(r),  T%[(*T)V]}  =  (d(r),  r^A(r)  JM[(^r)y]}  ,  (20) 


where  A  €  D(0,  oo)  is  such  that  A  (r)  =  i  on  the  support  of  a.  The  function  extend  into 
the  finite  complex-plane  as  an  entire  function  of  exponential  type  which  grows  no  faster 
than  a  polynomial  on  the  positive  real  axis;  it  is  easy  to  show  that  f(x)  e  M(Q,  oo). 


Convolution  of  generalized  functions  475 

lassicai  generalized  Hankel  type  convolution 
INITION4.1 

is  define  Zji,  y[0,  co),  1  <  p  <  oo,  the  space  of  Lebesgue  measurable  functions  on 
o)  such  that 


<  oo. 


_0 


;onsider  the  kernel  DM,  v(x,  >-,  z),  0  <  jt,  v,  z  <  oo  defined  by 

CO 

M  M  M 

0 

Properties  of  the  kernel  DMi  U(A:,  y,  z) 

swing  Watson  [10],  Hirschmann  [4]  and  Cholewinski  [2]  we  can  establish  the  following 

>erties  for  (21): 

or  0  <  x,  y  <  oo  and  0  <  r  <  oo,  we  have 

oo 

0 


»,v(x,  y,  z)     = 


1-v 


00 

/' 


refore, 


hence  the  result.  In  particular,  taking  T  =  0,  gives 


Z2v-lDlt,v(x,  y,  z)dz=l, 


(23) 


476 


S  P  Malgonde  and  G  S  Gaikawad 


that  is,  for  which  x,  y  >  0,  D,^  U(JT,  v,  z)  belongs  to  LQ  v(0,  oo). 

(iii)  0  <  jc,  y,  z  <  oo,  Z)M,  u(jc,  >',  z)  >  0,  and 

(iv)D^v(x,  >',  z)  =  £>/AV(z,  *,  >')  =  £V,v(}',  z,  *)  = 

DEFINITION  4. 3 

We  define  the  classical  /i/A  v -convolution,  for  any  two  function  /(jc)  and  g(jc),  0  <  jc  <  oo 


as 


oo  oo 

f  *  g(x)  =  v  f  J(yzYf(y)g(z)D^  v(x,  y, 

0    0 


(24) 


whenever  the  integral  exists.  We  observe  the  following  properties: 

(i)  Commutativity:  For  any  x  e  /,  (/  *  g)(x)  =  (g  *  /)(*).  Proof  is  obvious  from  the 

relation  (24). 

(ii)  Associativity:  For  any  t  e  /,  (/  *  g)  *  h(t)  =  /  *  (g  *  i 

CO    CO 


00 


•// 

LOO 

oo  oc 

// 

LO    0 


Since  the  integral  exists  due  to  equation  (24),  by  changing  the  order  of  integration  and 
DjltV(x9  >',  z)D,L,v(x,s,  t)  =  DM,  v(z,5,  Jc)D/z<1J(jc,  v,  r)  we  have  the  result.    While  / 
and  g  are  such  that  both  /i^, „(/)  and  h,^v(g)  exists,  we  have  the  convolution  product 
properties, 
(iii) 

v(£).  (25) 


LHS     =     vr 


0 


CO    OC 

// 

L.O          0 


Changing  the  order  of  integration  we  get 

00    CO 


Using  (22)  we  get 


0     0 


^  v(jc,  y, 


LO 


dvdz. 


CO    OO 

=  v^~l+v  f  f 

0    0 


Convolution  of  generalized  functions 


477 


..-l-v 


^  J(yr)vJ,d(yry]f(yWy 
o 


I \z.T)vJ/i 


4.4  hfi,  v- translation 

If  the  /7M>  y -convolution  /  *  g  exists,  then  using  Fubinis  theorem  we  can  write  it  in  the  form 


o 


dy  =  v  f  yvf(y)g(x 
o 


,(26) 


where  we  write 


o;y)  = 


(27) 


with  x  o  3;  denoting  the  /zMi  y-translation  on  the  positive  real  line.   (The  analogue  of  the 
translation  consider  for  the  definition  of  the  usual  convolution  *.) 

The  function  g(jc  o  y)  will  be  called  the  hfit  v  translate  of  g(x);  provided  g(x)  is  locally 
bounded  on  0  <  x  <  oo,  g(x  o  y)  is  well-defined  and  continuous  on  (0,  oo)  x  (0,  oo), 
(Nussbaum  [7]).  The  h,^  ^-translation  is  a  particular  case  of  the  translations  of  Delsarte 
[3],  subsequently  studied  by  Braaksma  [1]. 

Theorem  4.5.  If  g  e  L1Q  u(0,  oo)  fl  L°°(0,  oo)  and  a  e  [0,  oo),  then  a  simple  calculation 
using  Fubinis  theorem  shows  that 


(28) 


Proof. 


LHS     = 


oo 

f(xr)vJfi[(xT)v]x-vg(xoa)dx 
o 

00  00 

,A(xrY]  f  zvg(z)D,i,v(x,  a,  z)dzdx. 


Using  Fubinis  theorem, 


1  j  zvg(z)dz  f  J,A(xT)v]DllrV(x,  a, 

0  0 


Using  (22)  we  get 


478  S  P  Malgonde  and  G  S  Gaikawad 


5.  Generalized  Hankel  type  convolution  of  generalized  functions 

5.1 

For  fixed  x,  y  e  (0,  oo)  then  the  function  DfltV(xt  >',  z),  0  <  z  <  oo  defines  a  regular 
generalized  function  in  H*^  v(0,  oo)  which  we  denote  by  DM>  y(jc  o  y,  z).  In  fact  for  fixed 
xty  e  (0,  oo)  and  0  e  #/z,  u(0,  CXD)  we  have  that 


and  since 


t.v(x  °  y,  z),  z  (/>(z))  =  (^/x,  vU,  >',  ^;, 

00 

zu0(z)D/^,  v(x,  y,  z)dz  =  <p(x  o  j)  (29) 

o 


CO 

|0(jcoy)|  = 
o 


X0°'o  ( 


),     (by  (23))  (30) 


then  DIL,v(x  o  y,  z),  0  <  z   <  oo  truly  generates  a  continuous  linear  functional  on 
H*  V(Q,  oo)  through  (29).  Moreover,  since 


then  we  can  write 

Vv[^.«(*o^z)]  =  ^[(*T)'']7M[(yT)1']>  ()<*,><  oo  (31) 

in  the  sense  of  #*'  ,,(0,  oo)  and  even  in  the  classical  sense. 


Convolution  of  generalized  functions  479 

We  now  show  that,  for  every  fixed  y  >  0,  the  following  implication 

#,,,y(0,  oo)  =>  jr1+1>(*  o  y)  E  #yil  v(0,  oo)  (32) 


holds. 

In  fact,  since  0(jc)  e  #MiV(0,  oo),  then,  <I>  =  /z/Aill[0]  6  #/(t>v(0,  oo).  On  the  other 
hand, 


but  //1[(yr)u]  €  AfT(0,  oo)  and  so  JM[(yT)u]<I>(T)  €  ff;i,  „((),  oo). 

Hence,  since  /i/Zi  v  transformation  is  an  automorphism  on  Ht^  v(0,  oo),  the  function  of 
x  given  by  h~^}[J}l[(yr)v]^(T)}  =  ;c~1+y0U  o  y  also  belongs  to  H^  y(0,  oo). 

5.2  Delsarte  translation 

For  any  0(;c)  €  ///IM  v(0,  oo)  and  0  <  ;c,  y  <  oo,  following  from  well-known  property  of 
the  Delsarte  translation,  we  also  have  that 


;y)  (33) 

for  any  non-negative  integer  m,  by 


PROPOSITION  5.  3 


1,  02  €  ///^  v(0,  oo)  andm  is  a  non-negative  integer,  then 
(i)  0i  *  02  exists  for  allQ  <  x  <  oo; 
(ii)*-1-^*  02  effMlV(0,oo); 
(iii)  Af[0!  *  02]  =  (A?0i)  *  02  -  [01  *  (Af  02)  (34) 


/  In  fact  (34)(i)  follows  since  0i,  02  €  LQ  v(0,  oo)  for  any  p  such  that  1  <  p  <  oo; 
(34)(ii)  is  justified  by  the  fact  that  the  function 


belongs  to  fiT^  v(0,  oo)  and  similarly  for  its  h~  ^-transform;  (34)(iii)  follows  from  (33)  and 
differentiation  under  integral  sign.  Note  finally  that  for  any  0i,  02  €  ////,  y(0,  oo), 

00 

01  *02<»  =  v(0i(;y),  }>V02(*  o  y))  =  v  / 

0 

oo  oo 

=  v  y  /0i(>0  /  zvfa(z)D,lt  v(x,  y,  tfdzdy 

0  0 

Since  0i  ,  0o  €  LQ  u(0,  oc)  we  can  make  use  of  Fubini's  theorem  to  get 
\ 

00  00 

I  y  >i  (y)^.  „(*,  y, 


480  S  P  Malgonde  and  G  S  Gaikawad 

00 

=  VM    ZV<t>2(y)<t>\(x°Z)fc  =  v(02>Zl>lO  O  ~)) 
0 

This  proves  (34)(iv). 

5.4 

If  X  €  #M,  y(0,  oo),  then  for  each  fixed  ;c  e  (0,  oo);  we  have 


it  follows  that  for  any  a  e  Ef(Q,  oo)  the  convolution  a  *  X(x)  is  well-defined  by 

jc"1+V  *  A,(*)  =  H>0,  v/jr1+vA.(;c  o  y)).  (35) 

Further, 


(36) 
where  A(T)  =  /i/Xi  V[X]. 


6  7/MiU(0,oo),andthereforez-+ucr*X(jc)  €  HM%U(0,  oo). 

Hence  jc""  1+wa  *  A(JC)  generates  a  regular  generalized  function  in  //*'  y  (0,  oo),  and  for  any 
0  €  #M,V(0,  oo)  we  get 

-  (V  v^"1+Vflr  *  ^ 


-  (or(jc)f  ^^A  *  <D  W](T)}  =  (OT(JC),  ^UA  *  <D(jc)}.  (37) 

This  could  be  taken  as  the  definition  of  the  generalized  Hankel  type  convolution  of  gen- 
eralized function  (or  generalized  h^v  -convolution),  and  this  in  turn  allows  another  form 
analogous  to  the  direct  product  definition  of  the  generalized  Hankel  type  ordinary  convo- 
lution: 


=  (CT(JC),  xvX  *  0(z)) 

v<l>(x  o  v))) 
=  (cr(x),  vxv(^(y),  y^(x  o  y)))  =  (v(x)  ®  X(y),  v(^)VU  o  }'))•  (38) 


5.5 

For  /  €  H*f  V(Q,  oo)  and  X  e  HjJLtV(Q,  oo).  The  convolution  is  again  well-defined  as  a 
generalized  function  in  #*'y(0,  oo).  By 

<l>(x))  =  (/(Jc),  ;cyA  *  000} 


Convolution  of  generalized  functions  481 

e  #,,,y(0,  oo)  by  (34)(ii).  Using  (18),  we  get 

(x~l+v  f  *  A.(*),  j 


000}  -  (/(*),  *yi  *  0  W) 
,  r[r  l~u  V  V[X](T)0(T)}  -  (T  ^V  ,[/](r)  A(r), 

hat,  in  the  sense  of  #*'  y(0,  oo), 

=  r  ^V  v[/]  V  v  W.  (39) 


illy,  let  /  e  #*'u(0,  oo)  and  a  G  £'(0,  oo).  Since,  for  any  0  e  H^  y(0,  oo)  we 
e  cr  *  0(x)  6  //M,  y(0,  oo),  it  follows  that  x~l+v  f  *  cr  is  well-defined  as  a  generalized 
:tioninff*%(0,oo)by 


(jc-1+u/  *  or  00,  x4>(x))  =  (/(Jc),  zV  *  0(x)).  (40) 

before,  this  may  also  be  expressed  in  the  form 

(x~]+vf  *  o-oo,  x000)  =  (/U)  ®  ^  W.  v(xyY4>(x  o  >0)  (41) 

L,  using  (18)  again,  we  can  derive  the  analogue  of  (39) 

hfJL,vlx-l+vf*a]  =  Tl-vh^v[f]hll,v[cr}.  (42) 

te  that  hllt  y[a]  6  M(0,  oo),  so  that  the  product  in  (42)  makes  sense  in  H*[V(Q,  oo). 

Algebraic  properties  of  the  generalized  h^v-  convolution 

already  remarked,  the  classical  hfli  v-convolution  defined  in  L^v(0,  oo)  is  commutative 
I  associative;  however,  it  possesses  no  identity  element  We  consider  in  turn  these 
perties  with  respect  to  generalized  h^  v  -convolution. 

Commutativity 
cr  e  £;(0,  oo),  X  €  Hjl%  v(0,  oo).  We  have 


^~          *  A,(JC),  JC000   =  CTJC,  x      * 
=  (A/4,  v[cr](r),  rAMf  v[jc 


ere  the  last  manipulation  make  sense  since  h^  v  [cr]  e  M(0,  oo)  and  see  (37)  and  Pinto's 
?er  for  other  type  proof  . 


482  S  P  Malgonde  and  G  S  Gaikawad 

(ii)  If  /  €  #*'y(0,  oo),  A  e  HIJL,  y(0,  oo)  then 

(;t~1+v/  *  AC*),  *0M)  -  (/(*).  *v*  *  000} 


__  lx-l+v^  ^  f(x 

This  is  justified  because  every  function  in  H,lt  y  (0,  oo)  is  also  a  multiplier  in  #*'  y(0,  oo) 

whenever  JC000  €  #M,y(0,  oo). 

(iii)  If  /  e  H*1  y(0,  oo),  a  e  £"(0,  oo)  then  the  same  kind  of  argument  gives 


But  since  h,^  v[f]  does  not  belong  to  M(0,  oo),  no  general  commutativity  property  can 
be  deduced.  If,  in  addition,  we  have  /  e  £7(0,  oo),  then  h^v[f]  e  M(0,  oo),  and  the 
argument  to  establish  commutativity  proceed  as  before. 

6.2  Associativity 

(i)  a  e  £7(0,  oo),  AI  ,  A2  e  #/x,  y (0,  oo).  We  can  establish  the  result 

^-1+vU-1+vcr*A1]*A2  =  x"HV*U-1+vA1*A2]  (43) 

in  the  following  sense,  for  any  0  e  Hllf  y(0,  oo). 

(jc-1+v(^-l+ucr*A)*A2,^0(jc)) 


-  (or (x),  *[*-1+1%  *  (JC-1+yA2  *  0 (x)])} 

=  {^~1+V  *  [J1"^!  *  A,200],  JC0  W). 

The  equality  xvX}  *  (x~I+uA2  *  0)  =  ((JCI;AI  *;c-1+v;A2)  *  0)  is  justified  by  the  fact  that 
A  i,  A.2  and  0  belong  to  L^v  (0,  oo).  (ii)/  e  H*[  v  (0,  oo) ,  or  e  £7(0,cx)),A,  €  HMiV(0,  CXD). 
We  have  that 

jc~  1+y[jr1+ v/  *  a]  *  A(JC)  =  jc~1+v/  *  [^-1+V  *  A.] W.  (44) 


Convolution  of  generalized  functions 


Proof. 


=  (x~l+vf  *<r  (x), 


(iii)  If  /  €  H£  w(0,  oo),  cr,,  a2  6  E'(0,  oo).  We  show,  finally,  that 


Proof. 

(*-l+1'[*-1+w 

=  x~1+v/  *  a\(x),xvaz 


* 


6.3  Identity  element 

For  a,  6  strictly  positive  we  know  that  DM.  „  (a  .  *.  z)  defines  a  regular  generated  function 
DIL,  ,\a.  b,  z)  in  H-  v(0,  oo)  If  either  of  a,  i  takes  the  value  zero  then  D,,  v(a,b,  z) 
longer  defined  as  an  ordinary  function  since 


is  only  a  formal  identity  because  the  integral  fails  to  converge  for  any  z. 
Instead,  for  any  fixed  a  >  0,  we  consider  the  integral 

/• 

0 
which  for  each  R  >  0  is  uniformly  convergent  on  0  <  z  <  oo. 

DEFINITION  6.4 

Define  the  generalized  function  Dfl,  v(a,z)  in  H*'  v(0,  oo)  by 


(46) 


DM.v(a.z)=  lim    f  v 

/?— >OO  J 


484  S  P  Malgonde  and  G  S  Gaikawad 

in  the  sense  that  for  any  <j>  e  HIL,  v(0,  oo), 


=   lim 
R-*oo 


(47) 


For  each  finite  R  >  0  the  integral  (44)  defines  a  function  which  generates  a  regular  gener- 
alized function  in  #*'  v(0,  oo)  (Sneddon  [9]),  Therefore, 


or  by  Fubini's  theorem 
/  R 

(!^, 

\  r\ 


oo 

f 


A2"-1. 


Thus 


R-+OO 


=  a 


l-v 


=   lim  a1^ 


/c 


and  so 


A-v 


(48) 


Convolution  of  generalized  functions  485 

h  shows  that  DMi  v  (a,  z)  e  H*{  u(0,  oo).  Moreover,  since 


(/V  v(a,  z),  z 
btain 

-  JM[(flr)v].  (49) 


let  (fln)J£-i  be  a  monotone  decreasing  sequence  of  positive  real  numbers,  tending  to 
as  n  ~*  oo,  and  consider  the  sequence  of  generalized  functions  (D/x,  „  (an  ,  z))%L  j  in 
,(0,  oo).  Since  H*'  v(0,  oo)  is  complete,  this  limit  is  again  a  generalized  function  in 
;(0,  oo).  For  each  n  and  any  <j>  e  /f/jt,  v(0,  oo), 


therefore  we  define  the  generalized  function  DM,  v(z) 


=      lim  a^V^/i)  =  v<£(0+)  (50) 

«->-00 

ipendently  of  the  particular  sequence  (fl/i)^  chosen).  Moreover,  since 
(OM.v(2),zv*(z))  =  v0(0+)  =  (1,  vrv<D(r))  =  v(l,  vrucD(r)), 
tave 

=  v.l=v  (51) 


equality  being  understood  in  the  sense  of  H*[V(Q,  oo).  The  generalized  function 
u.sOO  €  H*'v(0,  oo)  is  the  required  identity  element  with  respect  to  the  general- 
/i^,  y  -convolution.  In  fact,  it  is  easy  to  show  that  D^  v,z  W  €  £'(0,  oo)  and  therefore 
my  /  €  H*[V(Q,  oo)  and  every  0  €  H,^  v(0,  oo),  by  using  the  results  in  (40),  (41)  and 
we  obtain 


=   /(*> 


:h  shows  that 

/  *  ^^  =  /(*)  (52) 

ic  sense  of  H*'V(Q,  oo),  as  asserted. 

)ifferentiability  properties  of  the  /i^-convolution 

conclude  with  a  brief  remark  on  the  differentiability  properties  of  the  generalized  AM,  „- 
/olution.  Let  fc  be  any  nonnegative  integer  /  €  H*'  v(0,  oo)  and  X  e  HIJL.V(Q,  oo). 


486  S  P  Malgonde  and  G  S  Gaikawad 

Then,  since  for  any  4>  €  Hllt  v(0,  oo), 
A   =    A*[/ 


oo 

/v~ 

0 

oo 

.  /,* 


I^V 

=   v  ( x2v-l<t>(x)^xl-v 


oo  oo 

(  f(yz)vf(yMz)D,ltt(x,  y,  z)dydz 


Lo    o 


dx. 


Differentiating  under  the  integral  sign  and  using  Fubini's  theorem  we  get 


A   =   v 


Similarly 


(53) 


in  the  sense  of  H*f  v(0,  oo). 

If  now  /  €  #*'  v(0,  oo)  and  a  €  £'(0,  oo),  then  by  the  same  kind  of  argument,  and 
using  (53),  we  derive  the  double  equality. 


A*[/  *  cr]  =  /  *  [A^a]  =  [A*/]  *  cr 


(54) 


in  the  sense  of  #  *'  v(0,  oo). 


v 


References 

[1]  Braaksma  B  L  J  and  De  Snoo  H  S  V,  Generalized  translation  operators  associated  with  a 
singular  differential  operator,  ordinary  and  partial  differential  equations,  Dundee  Conference, 
Lecture  Notes  in  Math.  (Berlin:  Springer-  Verlag)  (1974)  vol.  415,  pp.  62-77 

[2]  Cholewinski  F  M,  Hankel  complex  inversion  theory,  Mem.  Am.  Math.  Soc.  58  (1965) 

[3]  Delsarte  J,  Une  extension  nouvelle  de  la  theorie  des  fonctions  presque-periodiques  de  Bohr, 
Ac/a.  Math.  69  (1938)  259-317 

[4]  Hirschmann  1  1  Jr,  Variation  diminishing  Hankel  transforms,  J.  Anal  Math.  8  (1  960/61)  307- 
336 


Convolution  of  generalized  functions  487 

[5]  Malgonde  S  P  and  Gaikawad  G  S,  A  mixed  Parseval  equation  and  the  generalized  Hankel 
type  transformations,  J.  Indian  Acad.  Math.  22(2)  (2000) 

[6]  Malgonde  S  P  and  Gaikawad  G  S,  On  the  generalized  Hankel  type  transformation  of  gener- 
alized functions  (communicated  for  publication) 

[7]  Nussbaun  A  E,  On  functions  positive  definite  relative  to  the  orthogonal  group  and  the  rep- 
resentation of  functions  on  Hankel-Stieltjes  transforms,  Trans.  Am.  Math.  Soc.  175  (1973) 
389-408 

[8]  Pinto  J  De  Sousa,  A  generalised  Hankel  convolution,  SIAM  J.  Math.  Anal.  16(6)  (1985) 
1335-1346 

[9]  Sneddon  I  N,  The  use  of  integral  transforms  (New  York:  Tata  McGraw-Hill)  (1979) 
[10]  Watson  G  N,  A  Treatise  on  the  Theory  of  Bessel  functions  (Cambridge:  Cambridge  Univ. 

Press)  (1944) 

[11]  Zemanian  A  H,  Generalized  Integral  Transformations,  (Interscience)  (1 966);  republished  by 
Dover,  New  York  (1987) 


>c.  Indian  Acad.  Sci.  (Math.  Sci.),  Vol.  Ill,  No.  4,  November  2001,  pp.  489-508. 
Printed  in  India 


onlinear  elliptic  differential  equations  with  multivalued 
cmlinearities 


ANTONELLA  FIACCA*,  NIKOLAOS  MATZAKOS1",  NIKOLAOS 
S  PAPAGEORGIOU1"  and  RAFFAELLA  SERVADEI* 

*  Department  of  Mathematics,  University  of  Perugia,  Via  Vanvitelli  1,  Perugia  06 123, 

Italy 

'''Department  of  Mathematics,  National  Technical  University,  Zografou  Campus, 

Athens  15780,  Greece 

E-mail:  npapg@math.ntua.gr 

MS  received  01  June  2000 

Abstract  In  this  paper  we  study  nonlinear  elliptic  boundary  value  problems  with 
monotone  and  nonmonotone  multivalued  nonlinearities.  First  we  consider  the  case  of 
monotone  nonlineari  ties.  In  the  first  result  we  assume  that  the  multivalued  nonlinearity 
is  defined  on  all  R.  Assuming  the  existence  of  an  upper  and  of  a  lower  solution,  we 
prove  the  existence  of  a  solution  between  them.  Also  for  a  special  version  of  the 
problem,  we  prove  the  existence  of  extremal  solutions  in  the  order  interval  formed 
by  the  upper  and  lower  solutions.  Then  we  drop  the  requirement  that  the  monotone 
nonlinearity  is  defined  on  all  of  R.  This  case  is  important  because  it  covers  variational 
inequalities.  Using  the  theory  of  operators  of  monotone  type  we  show  that  the  problem 
has  a  solution.  Finally  in  the  last  part  we  consider  an  eigenvalue  problem  with  a 
nonmonotone  multivalued  nonlinearity.  Using  the  critical  point  theory  for  nonsmooth 
locally  Lipschitz  functionals  we  prove  the  existence  of  at  least  two  nontrivial  solutions 
(multiplicity  theorem), 

Keywords.  Upper  solution;  lower  solution;  order  interval;  truncation  function; 
pseudomonotone  operator;  coercive  operator;  extremal  solution;  Yosida  approxima- 
tion; nonsmooth  Palais-Smale  condition;  critical  point;  eigenvalue  problem. 

Introduction 

this  paper  we  employ  the  method  of  upper  and  lower  solutions,  the  theory  of  nonlinear 
jerators  of  monotone  type  and  the  critical  point  theory  for  nonsmooth  functionals  in 
der  to  solve  certain  nonlinear  elliptic  boundary  value  problems,  involving  discontinuous 
mlinearities  of  both  monotone  and  nonmonotone  type. 

Most  of  the  works  so  far  have  treated  semilinear  probems.  Only  Deuel-Hess  [12], 
>al  with  a  fully  nonlinear  equation,  but  their  forcing  term  on  the  right  hand  side  is  a 
aratheodory  function.  Deuel-Hess  use  the  method  of  upper  and  lower  solutions,  in  orderto 
ow  that  problem  has  a  solution  located  in  the  order  interval  formed  by  the  upper  and  lower 
lutions.  More  recently  Dancer-Sweers  [11]  considered  a  semilinear  elliptic  problem, 
ith  a  Caratheodory  forcing  term,  which  is  independent  of  the  gradient  of  the  solution  and 
ey  proved  the  existence  of  extremal  solutions  in  the  order  interval  (i.e  the  existence  of  a 
aximal  and  of  a  minimal  solution  there).  Semilinear  elliptic  problems  with  discontinuities 
ive  been  studied  by  Chang  [8]  and  Costa-Goncalves  [10],  who  used  critical  point  theory 
r  nondifferentiable  functionals,  by  Ambrosetti-Turner  [4]  and  Ambrosetti-Badiale  [5], 

489 


490  Antonella  Fiacca  et  al 

who  used  the  dual  variational  principle  of  Clarke  [9]  and  by  Stuart  [23]  and  Carl-Heikkila 
[7],  who  used  monotonicity  techniques.  In  Carl-Heikkila  [7],  we  encounter  differential 
inclusions  but  they  assume  that  the  monotone  term  ft  (•)  corresponding  to  the  discontinuous 
nonlinearity,  is  defined  everywhere  i.e  (dom/?  =  R),  while  here  we  have  a  result  where 
dom/3  •£  R,  a  case  of  special  importance  since  it  incorporates  variational  inequalities. 
We  also  consider  the  case  where  the  term  /?(•)  is  nonmonotone,  which  corresponds  to 
problems  in  mechanics,  in  which  the  constitutive  laws  are  nonmonotone  and  multivalued 
and  so  are  described  by  the  subdifferential  of  nonsmooth  and  nonconvex  potential  functions 
(hemivariational  inequalities). 

2.  Preliminaries 

Let  X  be  a  reflexive  Banach  and  X*  its  topological  dual.  In  what  follows  by  (•,  •)  we  denote 
the  duality  brackets  of  the  pair  (X,  X*).  A  map  A  :  A  -*  2X*  is  said  to  be  monotone,  if 
for  all  [jci,  **],  [JC2,  JtJ]  €  GrA,  we  have  (xj  -  **,  *2  -  *i)  >  0.  The  set  D  =  {x  e  X  : 
A(x)  7^  0}  is  called  the  'domain  of  A'.  We  say  that  A(-)  is  maximal  monotone,  if  its 
graph  is  maximal  with  respect  to  inclusion  among  the  graphs  of  all  monotone  maps  from 
X  into  X*.  It  follows  from  this  definition  that  A(-)  is  maximal  monotone  if  and  only  if 
(v*  —  jc*,  v  —  x)  >  0  for  all  DC,**]  e  GrA,  implies  [u,  v*]  e  GrA.  For  a  maximal 
monotone  map  A(-),  for  every  x  e  D,  400  is  nonempty,  closed  and  convex.  Moreover, 
GrA  C  X  x  X*  is  demiclosed,  i.e.  if  xn  -»  x  in  X  and  jc*  -^  x*  in  X*  or  if  xn  A  x  in  X  and 
x*  ->  x*  in  X*,  then  [jc,  x*]  6  GrA.  A  single-  valued  A  :  X  -*  X*  with  domain  all  of  X,  is 
said  to  be  hemicontinuous  if  for  all  jc,  y,z  €  X,  the  map  A  —  >  (A  (x  +  A.y  ),  z)  is  continuous 
from  [0,  1]  into  R  (i.e.  for  all  jc,  y  e  X,  the  map  A  -*•  A(;c  -f  A.y)  is  continuous  from 
[0,  1]  into  X*  furnished  with  the  weak  topology).  A  monotone  hemicontinuous  operator 
is  maximal  monotone.  A  map  A  :  X  ->  2X*  is  said  to  be  'pseudomonotone',  if  for  all 
x  e  X,A(x)  is  nonempty,  closed  and  convex,  f  or  every  sequence  {[*„,  **]}„>!  C  GrAsuch 
ihoixn  -^  jc  in  X,  jc*  -*  x*  in  X*  and  lim  sup(**,  xn  —  x)  <  0,  we  have  that  for  each  y  e  X, 
there  corresponds  a  y*(y)  e  A(x)  such  that  (y*(y),x-y)  <  lim  inf(x*,xn  —  y)  and  finally 
A  is  upper  semicontinuous  (as  a  set-valued  map)  from  every  finite  dimensional  subspace 
of  X  into  X*  endowed  with  the  weak  topology.  Note  that  this  requirement  is  automatically 
satisfied  if  A(-)  is  bounded,  i.e.  maps  bounded  sets  into  bounded  sets.  A  map  A  :  X  -> 
2X*  with  nonempty,  closed  and  convex  values,  is  said  to  be  generalized  pseudomonotone 
if  for  any  sequence  {[xn,x*]}n>\  £  GrA  such  that  xn  -*  x  in  X,  x*  -^  jc*  in  X*  and 
limsup(x*,xn  -  Jt)  <  0,  we  have  [jc.jc*]  €  GrA  and  (x*,xn)  -*  (x*,x)  (generalized 
pseudomonotonicity).  The  sum  of  two  pseudomonotone  maps  is  pseudomonotone  and  a 
maximal  monotone  map  with  domain  D  =  X,  is  pseudomonotone.  A  pseudomonotone 
map  which  is  also  coercive  (i.e  mf[(*^**eA(*)]  ^  ^  as  \\x  ||  _^  ooj  js  surjective. 

A  function  q>  :  X  ~>  R  =  R  U  {-hoc}  is  said  to  be  proper,  if  it  is  not  identically  +00, 
i.e  dom<p  =  {x  e  X  :  <p(x)  <  .-f  oo)  (the  effective  domain  of  <p)  is  nonempty.  By  Fo(X) 
we  denote  the  space  of  all  proper,  convex  and  lower  semicontinuous  functions.  Given  a 
proper,  convex  function  <?(•),  its  subdifferential  d<p  :  X  ->•  2**  is  defined  by 

d(p(x)  =  {**  E  X*  :  (jc*,  y  -  x)  <  (^(y)  -  (^(z)  for  all  y  e  dom<p}. 


If  (p  e  Fo(X),  then  fyp(-)  is  maximal  monotone  (in  fact  cyclically  maximal  monotone). 
Finally  recall  that  a  <p  6  T0(X)  is  locally  Lipschitz  in  the  interior  of  its  effective  domain. 


Nonlinear  elliptic  bvp  's  491 

Next  let  (p  :  X  ->  R  be  locally  Lipschitz.    For  such  a  function  we  can  define  the 
generalized  directional  derivative  of  <p  at*  e  X  in  the  direction  /z  e  X,  as  follows 

0,     ,,       r  <p(x'  +  Ui)-<p(x') 

(p^(x\  h)  —  hmsup  -  . 


It  is  easy  to  see  that  <p°(jt;  •)  is  sublinear  and  continuous  and  so  by  the  Hahn-Banach 
theorem  we  can  define  the  nonempty,  weakly  compact  and  convex  set 

d(p(x)  =  {x*  e  X*  :  (**,  h)  <  <A*;  h)  for  all  h  e  X}. 

The  set  d<p(x)  is  called  the  (generalized)  subdifferential  of  <p  at  A:  (see  Clarke  [9]).  If  (p  is 
also  convex,  then  this  subdifferential  coincides  with  the  subdifferential  of  <p  in  the  sense 

of  convex  analysis  defined  earlier.  Moreover,  in  this  case  CP°(JC;  h)  =  lim  V^+W-yM 

>40          A 

=  (p'(x;  h)  (the  directional  derivative  of  (p  at  x  in  the  direction  h).  A  function  (p  for  which 
0>°(jt;  •)  =  <pf(x\  •)  is  said  to  be  regular  at  x.  Finally  recall  that  if  jc  isalocalextremumof<p, 
then  0  €  d<p(x).  More  generally  a  point  x  e  X  for  which  we  have  0  €  dip(jc),  is  said  to  be 
a  critical  point  of  (p.  For  further  details  on  operators  of  monotone  type  and  subdifferentials, 
we  refer  to  Hu-Papageorigiou  [16]  and  Zeidler  [25]. 

3.  Existence  results  with  monotone  nonlinearities 

Let  Z  c  RN  be  a  bounded  domain  with  a  C1  -boundary  P.  In  what  follows  by  Aj(-) 
we  denote  the  nonlinear,  second  order  differential  operator  in  divergence  form  defined  by 
AI  (*)(•)  =  —  Yjc=\  AfcflJkG,*O»  £*(•))•  In  this  section  we  study  the  following  boundary 
value  problem: 


First  using  the  method  of  upper  and  lower  solutions,  we  establish  the  existence  of  (weak) 
solutions  for  problem  (1),  when  dom/?  =  R.  Let  us  start  by  introducing  the  hypotheses  on 
the  coefficient  functions  ak(z,  x,  y},  k  e  {1,  2, . . . ,  TV},  and  on  the  multifunction  /?(r). 

H(ce^):  ak  :  E  x  R  x  RN  ->  R,  k  e  {1,  2, . . . ,  TV},  are  functions  such  that 

(i)  for  all  jc  E  R  and  all  y  e  R^,  z  ->  <z*(z,  *,  )0  is  measurable; 

(ii)  for  almost  all  z  €  Z,  (x,  y)  -»  a*(z,  ;c,  y)  is  continuous; 
(iii)  for  almost  all  z  €  Z,  all  ;c  e  R  and  all  y  6  R^,  we  have 

!<**(*» *,)OI  <Y(z)  +  c(\x\p~l  +  \\y\\p~l) 

with  y  €  L*(Z),  c>  0,  2  <  /?  <  oo  and  |  -f  ^  =  1; 
(iv)  for  almost  all  z  €  Z,  all  x  €  R  and  all  y,  /  €  R^,  y  ^  y',  we  have 

AT 

,  x,  y)  -  ak(z,  x,  y'))(yk  -  y'k)  >  0; 


492  Antonella  Fiacca  et  al 

(v)  for  almost  all  z  €  Z,  all  x  e  R  and  all  y  e  RN ,  we  have 


N 


withcj  >0,  yi  6  L!(Z). 
Remark.  By  virtue  of  these  hypotheses,  we  can  define  the  semilinear  form 

a\  W^P(Z)  x 
by  setting 


H(/l):  £  •  ZxR  ->•  2R  is  a  graph  measurable  multifunction  such  that  for  all  z  €  Z,  £(z,  •) 
is  maximal  monotone,  dom/?(z,  •)  =  R,  0  €  /?(z,0)  anc*  \P(z,x)\  =  max[|t>|  :  v  e 
P(z,  x)]  <  k(z)  +  r\\x\P~l  a.e  on  Z  with  fc  G  L^(Z),  77  >  0. 


Remark.  It  is  well-known  (see  for  example  [16],  example  III.4.28(a),  p.  348  and  theorem 
III.5.6,  p.  362),  that  for  all  z  6  Z,  j8(z,;t)  =  dj(z,  x)  with  y(z»  •*)  a  jointly  measurable 
function  such  that  j  (z,  •)  is  convex  and  continuous  (in  fact  locally  Lipschitz).  Ifp®(z,x)  = 
proj(0;  j8(z,  ;c))  (=  the  unique  element  of  ^(z,  ;c)  with  the  smallest  absolute  value),  then 
x  -*•  jS°(z,jc)  is  nondecreasing  and  for  every  (z,x)  e  Z  x  R,  we  have  f)(z,x)  = 
[j}*(z,x-),PQ(z,x+)].  Moreover,  ;  (z,x)  =  7(2,  0)  +  !Q  PQ(z,s)ds.  Since  j(z,:)  is 
unique  up  to  an  additive  constant,  we  can  always  have  j(z,  0)  =  0.  Since  by  hypothesis 
0  €  0(z,  0),  we  infer  that  for  all  z  e  Z  and  all  x  e  R,  j(z,  x)  >  0.  In  what  follows 
£_(z,  x)  =  pP(z,  x')  and  0+(z,  ^)  -  ^(z,  x+).  So  ^(z,  jc)  =  [£_(z,  jc),  ^(z,  ^)].  Ev- 
idently we  have  for  almost  all  z  €  Zandall*  e  R,  |£_(z,  jc)|,  |j8+(z,^)[  <  A:(z)  +  r?|jc|y;~1. 
To  introduce  the  hypotheses  on  the  rest  of  the  data  of  (1  ),  we  need  the  following  defini- 
tions. 

DEFINITION 

A  function  q>  e  Wl*p(Z)  is  said  to  be  an  'upper  solution'  of  (1)  if  there  exists  x*  e  Lq(Z) 
such  that  jc*(z)  6  0(z,  ^(^))  a.e.  on  Z  and 


a(<p,v)+  I  ao(z,(p,  D(p)v(z)dz+  I  x*{(z)v(z)dz>  I  g((p(z))v(z)dz 
Jz  Jz  Jz 


for  all  v  €  WQ'P  n  V\Z)+  and  ^,r  >  0. 
DEFINITION 


A  function  $  e  Wl*p(Z)  is  said  to  be  a  'lower  solution'  of  (1),  if  there  exists  x%  e  LP(Z) 
such  that  jcj(z)  €  )5(z,  V^fe))  a.e.  on  Z  and 


I  JcJ(z)u(z)dz<  f 
Jz  Jz 

for  all  i;  e  W0Up  h  Ly;(Z)+  and  ^,r  <  0. 


Nonlinear  elliptic  bvp  's  493 

We  can  continue  with  the  hypotheses  on  the  data  of  (1): 

HQ:  There  exist  an  upper  solution  cp  e  Wl-p(Z)  and  a  lower  solution  V"  €  W['P(Z)  such 
that  i/r(z)  <  0  <  <p(z)  a.e.  on  Z  and  for  all  y  e  LP(Z)  such  that  VOO  <  y(z)  <  <p(z)  a.e. 
on  Z  we  have  g(y(-))  e  Lfl(Z).  Moreover,  #(•)  is  nondecreasing. 

H(<*o):  «o  :  Z  x  R  x  RN  ->  R,  is  a  function  such  that 

(i)  for  all  x  €  R  and  all  y  e  R^,  z  ->  tfofe  *>  >')  is  measurable; 

(ii)  for  almost  all  z  e  Z,  (;c,  >»)  ->  «o(z.  x,  y)  is  continuous; 
(iii)  for  almost  all  z  €  Z  and  all  jc  e  [^(z).  ^(z)],  we  have 


DEFINITION 

By  a  *(weak)  solution'  of  (1),  we  mean  a  function  x  e  W$P(Z)  such  that  there  exists 
/  €  L*(Z)  with  /(z)  e  j8(z,  jc(z))  a.e.  on  Z  and 


f 
Jz 


for  all  ue 


Let  ^  =  ty,(p]  =  {y  e  Wl^(Z)  :  ty(z)  <  y(z)  <  <p(z)  a.e.  on  Z}.  Our  approach 
will  involve  truncation  and  penalization  techniques.  So  we  introduce  the  following  two 
functions: 

r  :  Wl*p(Z)  ->  Wl'p(Z)  (the  truncation  function)  defined  by 


if 
x(z)     if 

if    x(z)<if(z) 


and  u  :  Z  x  R  -»•  E  (the  penalty  function)  defined  by 

)Y~l       if 
0  if 

*)"""1    if 


It  is  easy  to  check  that  the  following  are  true  (see  also  Deuel-Hess  [12]). 


Lemma  1.  (a)  The  truncation  function  map  r  :  W!''J(Z)  ->  W!^(Z)  £s  bounded  and 
continuous,  (b)  The  penalty  function  u(z,  x)  is  a  Caratheodory  function  such  that 


/or  a//  j:  6  LP(Z)  and  some  cs,  04  >  0. 


494  Antonella  Fiacca  et  al 

To  solve  (1),  we  first  investigate  the  following  auxiliary  problem,  with  y  e  K: 

4-  p(z,  *(z))  +  p«(z,  *(z))  3  £(>'(*))  on  Z 


Here  A2(*)  is  the  nonlinear,  second  order  differential  operator  in  divergence  form,  defined 
by 

N 

A2(x)(z)  =  —  2_J  Dk<Zk(z,  ?(x),  Dx). 

k=\ 

In  the  next  proposition  we  establish  the  nonemptiness  of  the  solution  set  S(y)  c  W^p  (Z) 
of  (2)  for  all  ye  K. 

PROPOSITION  2 


If  hypotheses  H(^),H(^),  Ho.H(flo)  A0W  0nd  >'  6  K,  then  the  solution  set  S(y)  c 
W0ltp(Z)  0/(2)  «  nonempty  for  p  >  0  large. 

Proof  Let  0  :  WQll/;(Z)  x  wJ'^Z)  -^  R  be  the  semilinear  Dirichlet  form  defined  by 


By  virtue  of  hypotheses  H(ajk),  this  Dirichlet  form  defines  a  nonlinear  operator  A\  : 
Wo^(Z)  ->  W-{*(Z)  by  (AI(AC),  y)  =  ^(jc,  y)  (here  by  {-,  -)  we  denote  the  duality 
brackets  of  the  pair  (W01>P(Z),  W'l^(Z))).  Also  let  OQ  :  W^P(Z)  -+  L«(Z)  be  defined 
by  OQ(X)(Z)  =  ao(z,  r(;c)(z),  DT(JC)(Z)).  This  is  continuous  and  bounded  (see  hypothesis 
H(ao))- 

Claim  1.  The  operator  A2  =  AI  +2"0  :  WQl'/7(Z)  ->  W"!^(Z)  is  pseudomonotone. 


To  this  end,  let  JCM-^  jcin  W0'/;(Z)asn  ~>  oo  and  assume  that  limsup(A2(jc«),xn--Jc}  < 


0.  Then  limsup(Ai(jto)  -f-2b(^/i)»^/i  —  ^)  <  0.  From  the  Sobolev  embedding  theorem, 
we  have*n  -+  x  in  LP(Z)  and  so  (2b(xn),  x«  "  ^)  =  (?bUn).  ^n  -  Jc)/,^  ->  0  (by£,  -)^ 
we  denote  the  duality  brackets  of  (L/7(Z),  Lq(Z)).  Therefore  we  obtain  lim  sup(Ai  (*„), 
xn-x)  <0. 
We  have 


r 

(A\(xn),  xn  -  x)     =      / 

Jz 


z  k=\ 


*=i 

N 

P0*fe.  r(xn),  Dx)(Dkxn  - 


Nonlinear  elliptic  bvp  's  495 

[  x^ 

-      /   2_^^'  T(*")'  ®x)(DkXn  -  Dkx)dz  (hypothesis  H(a*)(iv)). 
Jz  *=i 

Since  ^rt  -^  ^c  in  WQ'y)(Z),  we  have  ^  -»  x  in  L/;(Z)  and  then  directly  from  the  definition 
of  the  truncation  map  r,  we  have  r(xn)  -»  r(;c)  in 
Therefore 


,  Dx)(Dkxn  -  Dkx)dz  -»  Oasn  -*  oo. 


On  the  other  hand  we  already  know  that  lim  sup(A  i  (;cn),  *„  -*)  <  0.  Hence  (Aj  (xn),  xn 
x)  ->  0  as  n  -*  oo.  From  this  it  follows  that 


,  T  (*,,),  Djcn)  -  ak(z,  r(xn),  Dx))(Dkxn  -  Dkx)dz  -+  Oas  n  ->  oo. 


Then  invoking  Lemma  6  of  Landes  [17],  we  infer  that  Dfcxn(z)  -»  DjtJc(z)  a.e  on  Z  for  all 
A  €  {1,  2,  .  .  .  ,  N}.  So  using  Lemma  3.2  of  Leray-Lions[  18],  we  have  that  A\(xn)  -^  AI(^) 
in  Wl*Q(Z).  We  have  already  established  earlier  that  (Ai(jcn),jcn  —  x]  -»  0.  Since 
(Ai(jcn),  j:)  ->  (A  i  (*),*},  we  obtain  that  (A  [(xn),xn)  ~>  (AI(JC),^).  Also  (2b(jcn),^)  = 

(2b(*n)»  ^i)y^/-  But  again  by  Lemma  3.2,  Leray-Lions  [18],  we  have  that  <5bC*/i)  -^  2b(jc) 
in  Z/y(Z).  Since  xn  ->  *  in  L/;(Z)  (by  the  Sobolev  imbedding  theorem,  we  have  that 
(2b(*n),  */?)  =  (2b(^/i)»^n)^  -^  (2bW.  *)/?</  =  (2bW»^>.  Therefore  finally  we  have 
^2(^/1)  -^  ^2(^)  in  W~!'^(Z)  and  JAifoi),  Jf/i)  ->•  (A2(jc),  ^},  which  proves  that  A-2  is  a 
generalized  pseudomonotone.  But  A2  is  everywhere  defined,  single-  valued  and  bounded. 
So  from  Proposition  111.6.  1  1,  p.  366  of  Hu-Papageorgiou  [16],  it  follows  that  AI  is  pseu- 
domonotone. This  proves  the  claim. 

Next  let  17  :  W^P(Z)  -+  L«  (Z)  be  defined  by  U(x)(z)  =  u(z,x(z)).  From  the  compact 
embedding  of  W^P(Z)  in  L/;(Z)  and  Lemma  1,  we  infer  that  17(0  is  completely  continuous 
(i.e.  sequentially  continuous  from  WQ  '/;  (Z)  with  the  weak  topology  into  Z/7  (Z)  with  strong 
topology).  Therefore  A  =  A2  +  pU  :  W^P(Z)  ->  W{^(Z)  is  pseudomonotone. 

From  Lebourg's  subdifferential  mean  value  theorem  (see  Clarke  [9],  theorem  2.3.7, 
p.  41),  we  have  that  for  almost  all  z  6  Z  and  all  x  €  E,  \j(z,x)\  <  k(z)\x\  +  rj\x\p.  Thus 
if  we  define  G  :  LP(Z)  ->  Rby  G(jc)  =  /z  ;(^^W)dz,  we  have  that  G(-)  is  continuous 

(in  fact  locally  Lipschitz)  and  convex.  Let  G  =  G\W\,P(Z).  Then  from  Lemma  2.1  of 

*^o 

Chang  [8],  we  have  that  for  all  x  e  W^P(Z),  3G(x)  =  8G(x)  c  L^(Z). 

Then  the  auxiliary  boundary  value  problem  is  equivalent  to  the  following  abstract  oper- 
ator inclusion 


with  g(>0(.)  =  £(?(•))  €  L*(Z)  (see  hypothesis  H0). 

Ctom  2.  jc  ->  A(^)  4-  3GW  is  coercive  form  W0ll/J(Z)  into  W"^(Z),  for  p  >  0  large. 


496  Antonella  Fiacca  et  al 

To  this  end,  we  have 


(A(x),x)  =  (Ai(x)  +  2bW  +  pU(x),  x). 
From  hypothesis  H(a^)(v),  we  have 

(Ai  (*),*)  >  c,||Z>jc||£  -  llyilli  >  csWIi,/;  -  c6,  with  c5,  c6.  >  0.  (3) 

Also  from  hypothesis  H(flo)  (Hi),  we  have 

>  >  -C7WI/J*!!?'1  ~  C8ll*ll/;  for  some  c7,c8  >  0.  (4) 


From  Young's  inequality  with  6  >  0,  we  obtain 


and  so  using  that  in  (4),  we  have 


(5) 

Finally  from  lemma  1,  we  have 

(pU(x)tx)  >  cgp\\x]\p  -CIQ  for  some  C9,cio  >  0.  (6) 

From  (3),  (5)  and  (6)  it  follows  that 


>     c5  -  cj—    ||jt||      +    c9p  -  ci-          \\x\\p  -  ct\\x\\p  -  c6. 


(7) 


~ 


Choose  €  >  0  so  that  c$  >  cj~.  Then  with  €  >  0  fixed  this  way  choose  p  >  0  so  that 

cip  >  ci-~r-.  From  (7)  it  follows  that  A  is  coercive. 

Moreover,  since  by  hypothesis  H(/2)  we  have  that  0  €  0(zt  0),  it  follows  that  0  e  3G(0) 
and  so  (jc*,  x)  >  0  for  all  Jt*  e  3G(x).  Thus  A  4-  dG  is  coercive  and  this  proves  the  claim. 

Finally  because  3G(-)  is  maximal  monotone  and  domdG  =  X,  we  have  that  dGQ  is 
pseudomonotone.  So  A  -f  3G  is  pseud  omonotone  (Claim  1)  and  coercive  (Claim  2).  Apply 
Corollary  III.6.30,  p.  372,  of  Hu-Papageorgiou  [16]  to  conclude  that  A  -f  8G  is  surjective. 
So  there  exists  jc  €  W0lt7'(Z)  such  that  A(x)  +  6G(x)  3  ?(>') 

Having  this  auxiliary  result,  we  can  now  prove  the  first  existence  theorem  concerning 
our  original  problem  (1). 


Theorem  3.  If  hypotheses  H(ajO,  H(<2o)>  HO  and  H(/?)  /zo/J,  //z^n  problem  (I)  has  a 
nonempty  solution  set. 

Proof.  We  consider  the  solution  multifunction  S  :  K  ->  2wn  (Z)  for  the  auxiliary  problem 
(2),  i.e  for  every  y  e  K,  S(y)  c  W0ll/?(Z)  is  the  solution  set  of  (2).  From  Proposition  2, 
we  know  that  5(0  has  nonempty  values. 

Claim  I.  S(K)  OK. 


Nonlinear  elliptic  bvp  's  497 

Let  y  €  K  and  let  ;c  e  $(>•).  We  have 
}  +  (**,  v) 


for  some**  €  9G(jc)  and  all  v  €  W0lt/;(Z).  Since  $  e  W^P(Z)  is  a  lower  solution,  by 
definition  we  have 

)+  f  floU,  ^  D^)v(z)dz  +  (*J,  u)  <  (?W,  »>  for  all  v  e  W0l'*(Z)  n 
Jz 


and  for  some  jc*  €  L'7(Z)  withz^(z)  €  0(s,  V(z))  a.eonZ. 

Let  u  =  (^  -  ^c)+  €  W0K/?(Z)  n  L/;(Z)+  (see  for  example  Gilbarg-Trudinger  [13], 
Lemma  7.6,  p.  145).  From  the  definition  of  the  convex  subdifferential,  we  have 

<jc*,  (Vr  -  Jc)+  )  <  G(JC  +  (^  -  ^)+) 
and 


Using  these  two  inequalities,  we  obtain 

+  W  ~  ^)+)  +  G  W 

-x)+)  (8) 


and 

J(Vr,  (iA  ~  x)+)  4- 


z 


Note  that  G(x)  +  G(^)  -  G(x  4-  (^  -  x)+)  ~  G(Vr  -  (^  ~*)+)  =  0.  So  adding  (8)  and 
(9),  we  obtain 


&Q(Z, 

First  we  estimate  the  quantity 

-  +        f 

Jz 
We  have 

Jz 

r    r 

-/,? 


Since 

n./,/          ^+,,^_  Q  ^ 


498  Antonella  Fiacca  et  al 

(see  Gilbarg-Trudinger  [13]),  we  have 


(see  hypothesis  H(^))  (iv)).  Also  because 


D<p(z)     if    <p(z)<x(z) 
=      Dx(z)     if 
if 


we  have 

-  flo(z, 


(fl0(Z,  ^r,  />^r)  -  *0(Z,  f,  />^))(^  -  jr)fe)dz  =  0. 

^1 

Therefore  finally  we  can  write  that 

fl«%  (1^  ~  ^)+)  +  /  *o(z,  ^  ^)^)(^  -  ^)+dz  -  (A2,  (^  -  A:)+)  >  0.  (1  1) 

«/  2, 

Because  g(-)  is  nondecreasing  (see  hypothesis  HO)  and  y  €  K,  we  have 

(?(^)  -  ?()>),  (^  -  ^)+)  =  /*  fe(^fe))  -  ^(^(z)))(^  -  x^(z)dz  <  0. 

J2 

(12) 
Using  (11)  and  (12)  in  (10),  we  obtain 


I  -W-x)'>-[(zM-x)+ 
Jz 


Similarly  we  show  that  jc  <  <p,  hence  jc  €  K.  This  proves  the  claim. 

Claim  2.  If  y\  <  x\  €  S(y\)  and  y\  <  y2  €  K,  then  there  exists  xi  e  S(yi)  such  that 

X\  <  ^2- 


Since  x\  e  S(y\)  c  £,  we  have  for  some  /i  €  L^(Z)  with  f[(z)  e  p(z,x\(z))  a.e  on 
Z, 

?(jci,v)+  /  flo(z,Jci,D^i)u(z)dz+  /  /i(z)u(z)dz=  / 
Jz  Jz  Jz 

for  all  u  e  W01|P(Z), 

/i(z)u(z)dz  < 


Nonlinear  elliptic  bvp's  499 

all  n  6  W0'P(Z)  n  L/;(Z)+,  since  g(-)  is  nondecreasing  and  y\  <  yi.   Thus  x\  e 
|P(Z)  is  a  lower  solution  of  the  problem 


argument  similar  to  that  of  Claim  1,  gives  us  a  solution  KI  e  W0'/;(Z)  of  (13)  such  that 
<*2  <<P-  Note  that  <p  e  Wlt/J(Z)  remains  an  upper  solution  of  (13),  since  y2  e  K  and 
<  g(<p(z))  a.e  on  Z.  This  proves  the  claim. 


im  3.  For  every  y  e  K,  S(y)  c  W0';'(Z)  is  weakly  closed. 

b  this  end,  let  xn  €  S(>0,  w  >  1,  and  assume  that;c,z  —  >  x  in  W0'^(Z),  By  definition 
have 

A(xn)+x*  =  ?(y),  n  >  1,  with  x*  e  3G(xn) 

=»  (A(xn),  Jc/i  -  Jc)  =  (g(y),xn  -  x)pq  -  (x*,  xn  -  x). 

rn  the  compact  embedding  of  W0lt/;(Z)  into  L;;(Z),  we  have  that^  ->  x  in  Z/(Z)  and 
?(y)»^«~  *)pq  ->  0.  Also{jc*}n>i  c  Z/7(Z)  is  bounded  (see  the  proof  of  Proposition  2) 
so  (**,  Jt,i  —  *)  =  (^*,  xn  -  *)/j<7  -^  0.  Therefore 

lim  A(jc,,),  xn  -  Jc  =  0  ==»  A(JCW)  -^  A(x)  in  W-KC/(Z) 


ce  A  is  bounded,  pseudomonotone). 

>Jso  we  may  assume  that  x*  -^  ;c*  in  Lq(Z).  Since  [*„,**]  €  Gr9G  =  GrOG  n 
)llp(Z)  x  L^(Z))  (see  the  proof  of  Proposition  2  and  Chang  [8],  Lemma  2.1)andGr9G 
emiclosed,  we  conclude  that^:*  €  9G(x).  Thus  finally  we  have 

A(JC)  +  x*  =  g*(y),  with  ;c*  €  9G(jc), 

-  Jc  e  S(>')>  which  proves  the  claim. 

"laims  1,  2  and  3  and  that  fact  that  W1>/;(Z)  is  separable,  permit  the  application  of 
position  2.4  of  Heikkila-Hu  [15],  which  gives  x  e  S(x)  (fixed  point  of  £(•)).  Evidently 
is  a  weak  solution  of  problem  (1).  D 


In  fact  with  a  little  additional  effort,  we  can  show  that  the  result  is  still  valid, 
nstead  we  assume  that  there  exists  M  >  0  such  that  x  -+  g(x)  +  MX  is  nonde- 
ising.  However,  to  simplify  our  presentation  we  have  decided  to  proceed  with  the 
>nger  hypothesis  that  #(•)  is  nondecreasing.  Moreover,  it  is  clear  from  our  proof,  that  if 
Z  x  R  x  RN  ->  R^  is  defined  by  a(z,  x,  >')  =  (ak(z,  x,  >0)f=1  and  x  e  W^P(Z)  is  a 
ition  of  (1),  then  -divafe,  Jc,  Dx)  e  Lq(Z)  and 


f   -div 
Ulr= 


i  /  €  Z/'CZ),  /(z)  €  )8(z,  jc(z))  a.e  on  Z  (i.e  jc  is  a  strong  solution). 

?or  a  particular  version  of  problem  (1),  we  can  show  the  existence  of  extremal  solutions 

he  order  interval;  K,  i.e  of  solutions  xl  ,  xu  in  K  such  that  for  every  solution  x  e  K,  we 


500  Antonella  Fiacca  et  al 

So  let  A3;t(z)  =  -  £f=i  DkQk(z,  Dx)  (second  order  nonlinear  differential  operator  in 
divergence  form)  and  consider  the  following  boundary  value  problem 


A3  0000  +  aQ(z,  x(z))  +  ]8(z,  *(0)  3  £(*(z))  on  Z 


The  hypotheses  on  the  functions  ak  and  aQ  are  the  following: 

H(cck))f:  ak  :  Z  x  RN  -»  R,  A:  €  {1,  2,  .  .  .  ,  N],  are  functions  such  that 

(i)  for  all  y  €  E^,  z  ->  ak(z,  y)  is  measurable; 

(ii)  for  almost  all  z  €  Z,  y  -»  ^(z,  >')  is  continuous; 
(iii)  for  almost  all  z  €  Z,  and  all  >'  €  RN,  we  have 


with  y  €  Z/'(Z),  c>  0,  2  <  p  <  oo  and  ^  +  5-  =  1; 
(iv)  for  almost  all  z  6  Z  and  all  y,  /  €  R7^,  y  7^  >J/,  we  have 


(v)  for  almost  all  z  €  Z  and  all  y  €  RN,  we  have 


j  >  0,  yj  e  Ll(Z). 

H(oc0)':  ^0  :  Z  x  R  -»•  R,  is  a  function  such  that 
(i)  for  all  x  €  R  z  ->  a0(z,  ^)  is  measurable; 
(ii)  for  almost  all  z  €  Z,  x  -»  <20(z>  Jc)  is  continuous,  nondecreasing; 
(iii)  for  almost  all  z  6  Z  and  all  x  e  [^  (z)  t  (p(z)]9  we  have  ||fl0(z,^)||  <  KO(Z)  with 


Then  we  can  prove  the  following  result. 
PROPOSITION  4 


If  hypotheses  H(aA:)/,  H(00)',  H(^)  an^  H0  hold,  then  problem  (14)  has  extremal  solutions 
in  the  order  interval  K. 

Proof.  Hypotheses  E(ak)f  and  H(floy,  imply  that  the  map  S  :  K  ->  K  is  actually  single- 
valued.  Also  we  claim  that  it  is  increasing  with  respect  to  the  induced  partial  order  on  K. 
Indeed  let  y}  ,  y2  e  K,  y{  <  y2  and  let  xl  =  5^),  A:2  =  5(>'2).'  We  have 


Nonlinear  elliptic  bvp  's  501 


ithjcf  edG(xi)J  =  1,2. 

Using  (jcj  -  Jt2)+  6  WQP(Z)  H  L^(Z)+  as  our  test  function,  we  have 


*i-*2)}.          05) 

/  virtue  of  hypotheses  H(a^)x  and  H(a0)'  (ii),  we  have 

(A(xi)  -  A(*2),  (x{  -  *2)+)  >  0  (strictly  if  x\  ^  x2).  (16) 

Iso  from  the  monotonicity  of  the  subdifferential,  we  have 

(*?  -  *2,  (jci  -  jc2)+  )  =  (*J  -  *J,  (xi  -  x2)+  )pq  >  0.  (17) 

nally  since  by  hypothesis  HO,  #(•)  is  nondecreasing  it  follows  that 

X2)+)pq  <  0.          (18) 


sing  (16),  (17)  and  (18)  in  (15),  we  infer  that  Ui  —JC2)+  =  0,hencexi  <  x^.  Thisproves 
e  claim.  Using  Corollary  1.5  of  Amann  [2],  we  infer  that  S(-)  has  extremal  fixed  points 
K.  Clearly  these  are  the  extremal  solutions  of  (14)  in  K.  D 

Now  we  will  consider  a  multivalued  nonlinear  elliptic  problem,  with  a  /3(«)  such  that 
)m/5  T>£  R.  This  case  is  important  because  it  covers  variational  inequalities. 
So  now  we  examine  the  following  boundary  value  problem: 

|  Ai(jc)(z)+a0(zfJc(z))+^U(z))s^(z)onZ  1  Q 

Mr=Q  I' 

ur  hypotheses  on  a§  and  /?  are  the  following: 

(a0)//:  a0  :  Z  x  E  -»  R,  is  a  function  such  that 

T)  for  all  x  €  R,  z  -*  «0(z,  x)  is  measurable; 

ii)  for  almost  all  z  €  Z,  ^  -^  fl0(z,  z)  is  continuous,  nondecreasing; 

li)  for  almost  all  z  e  Z  and  all  x  e  R,  we  have  |a0(z,  Jc)|  <  72(2)  4-  c2\x\  with  /2  € 


(j8)i:  jS  :  R  ~>  2R,  is  a  maximal  monotone  map  with  0  €  /J(0). 

heorern  5.  If  hypotheses  H(ak),  H(a0)/;,  H(y3)i  hold  and  g  e  LP(Z),  then  the  solution 
•t  of  problem  (19)  is  nonempty. 


roof.  Recall  $  =  8;  with  ;  e  r0(R).  Let  ft  =  i(l  -  (1  4-  ^)~1),  £  >  0,  be  the  Yosida 
>proximation  of  fi(-)  and  consider  the  following  approximation  of  problem  (19): 

J  A(xi)  -  fl0(z,  jc(z))  +  ft  (*(z))  =  ^(z)  on  Z  1  (2Q) 

{  o:|r  =0  J  ' 


502  Antonella  Fiacca  et  al 

As  before  let  2"  :  WQl'y)(Z)  x  WQ'/;(Z)  be  the  semilinear  form  defined  by 

N 


and  let  AI  :  W0K/;(Z)  ->  W^(Z)  be  defined  by 

(A!(JC),  >'}  =  a(x,  v)  for  all  *,  >>  e  W^P(Z). 

Also  let  SQ  -  LP(Z)  ->  L^(Z)  be  the  Nemitsky  operator  corresponding  to  aQ  i.e.  2*0  ( 
=  a0(-,*(-))  (in  fact  note  that  by  H(a)"  (iii)  ofr)  6  L';(Z)  c  Z/'(Z)  since/?  >  2  >  g). 

From  Theorem  3,1  of  Gossez-Mustonen  [  14]  we  know  that  A  i  is  pseudomonotone,  while 
exploiting  the  compact  embedding  of  W0ll/;(Z)  in  L/;(Z),  we  can  easily  see  that  fl^l^i,,,  is 

completely  continuous.  Therefore  A  2  =  A\  -\-  2b  is  pseudomonotone. 

Let  G£  :  WQ'/;(Z)  -»  R  be  the  integral  functional  defined  by  Gs(x)  =  /z  je(x(z))dz 
with  jK(r)  being  the  Moreau-Yosida  regularization  of  j(-)  (see  for  example  Hu-Papageor- 
giou  [16],  Definition  IIL4.30,  p.  349).  We  know  that  Gfi(-)  is  Gateaux  differentiable 
and  3GK(x)  =  a;e(x(-))  (see  Hu-Papageorgiou  [16],  Proposition  III.4.32,  p.  350).  Then 
problem  (20)  is  equivalent  to  the  following  operator  equation 

A2W  +  8Ge(x)  =  g.  (21) 


Note  that  3  GE  is  maximal  monotone,  with  dom3Gfi  =  WQ'/;(Z).  Therefore3G^  is  pseudo- 
monotone  and  hence  so  is  A?  +  3GS.  We  will  show  that  A2  +  3G£  is  coercive.  Since 
0=  Gfi(0)  and  (3Gfi(jc),  x)  >  0,  to  establish  the  desired  coercivity  of  AI  +  3  G^,  it  suffices 
to  show  that  AI  is  coercive.  To  this  end  we  have 


Since  aQ(z,  •)  is  nondecreasing  (hypothesis  H(a);/  (ii))  (fl0(z,^(z))  —  fl0(z,  0))^(z)  >  0 
a.e  on  R  and  so 


f 

Jz 


Therefore  is  follows  that 

>     ci\\Dx\\pp  -  \\yi\\i 

z 

O,  0)\\q\\x\\P, 

from  which  we  infer  the  coercivity  of  x  -*  (A2  +  3GR)(x).  Thus  Corollary  III.6.30, 
p.  372,  of  Hu-Papageorgiou  [16],  implies  that  there  exists  xs  e  W^P(Z)  which  solves 
(21).  Now  let  sn  |  0  and  set  xn  =  xKnn  >  1.  We  will  derive  some  uniform  bounds  for  the 
sequence  {xn}n>\  c  W^P(Z).  To  this  end,  we  have 


(  f 

I  aQ(z,xn)xn(z)Az  +  /  pen 

Jz  JZ 


(xn)xn(z)dz 


=  I  g(z)xn(z)dz 
Jz 


Nonlinear  elliptic  bvp's  503 

xn\\Pp  ~  ||  XI  111  ™ 


(since  ps(xn(z))xn(z)  >  0  a.e  on  Z). 

From  this  inequality  we  deduce  that  {xn}n>\  c  W0K;;(Z)  is  bounded.  Also  note  that 
lln(r)  =  !&„(>)  |y;~2&/?0")  is  locally  Lipschitz  on  R  and  qn(0)  =  0.  So  from  Marcus- 
Mizel  [20],  we  know  that  rjn(xn(-))  e  W^p(Z),n  >  1.  Using  this  as  our  test  function,  we 
have 


/  Y^    ak(z,xn,Dxn)Dkrin(xn)dz     +      /  a0(z,^)^(^)dz  -f  /  \pEn(xn)\pdz 
Jz       /c=l  Jz  Jz 

-     f  g(z)^(jfn(z))dz.  (22) 

Jz 


^^ 

Mizel  [20],  and  recall  that  fiKn  (•)  being  Lipschitz  is  differentiable  almost  everywhere). 
Since  &„(•)  is  nondecreasing,  (p  -  l)ift,,te(z))|/;~2^7(jc/2))  >  0  a.e  on  Z.  Thus  using 
hypothesis  H(a&)(v),  we  have 


Moreover,  from  hypothesis  HCao)'^"!)  floG. •*«(•))  €  L/;(Z).  In  addition  since  ^,,(0 
is  j— Lipschitz  and  0  =  Ar;?(0),  we  have  |j6fi/I(r)|  <  j-|r|,  from  which  it  follows  that 
!&/(*(•)) I  €  LCI(Z).  So  by  Holder's  inequality,  we  have" 


/  ao(ztxn 
Jz 


But  since  [xn}n>\  c  WQ'/;(Z)  is  bounded,  we  have  sup,2>j  ||2b(^/i)ll/;  £  ^i  (see  hypoth- 


esis H(flo)"(iiO)-  So  we  obtain 

f  flo(z^/, 
Jz 

Returning  to  (22),  we  can  write 


>  - 


is  bounded,  hence  is  bounded  also  in  L2(Z).  Hence  by  passing  to  a  subsequence  if  neces- 
sary, we  may  assume  that  xn  -^  x  in  W01/;(Z)  and^e/l(^)  -^  u*  in  L2(Z)  as  n  -»•  oo. 
Also  we  have 


Exploiting  the  compact  embedding  of  W0I>;;(Z)  into  L/?(Z),  we  obtain 


504  Antonella  Fiacca  et  al 

(recall  that  A  2  is  pseudomonotone  and  bounded).  Hence  in  the  limit  as  n  —  >  oo  we  have 
A2(;c)  +  u*  =  g  in  W~{4(Z).  Let  J8  :  L2(Z)  ->  2L2(Z)  be  defined  by 


{w  €  L(Z)  :  a(z)  €  0(*(z))  a.e.  on  Z}. 

We  know  that  /f  is  maximal  monotone  (see  Hu-Papageorgiou  [  1  6],  p.  328).  Using  Proposi- 
tion III.2.29,  p.  325,  of  Hu-Papageorgiou  [16],  we  have  that  v*  e  /T(jt)  and  so  u*(z)  € 
).  Sox  e  WQ'^Z)  is  a  solution  of  [19].  D 


4.  Existence  results  with  nonmonotone  nonlinearities 

In  this  section  we  examine  a  quasilinear  elliptic  problem  with  a  multivalued  nonmonotone 
nonlinearity.  The  problem  that  we  study  is  a  hemivariational  inequality.  Hemivariational 
inequalities  are  a  new  type  of  variational  inequalities,  where  the  convex  subdifferential  is 
replaced  by  the  subdifferential  in  the  sense  of  Clarke  [9],  of  a  locally  Lipschitz  function. 
Such  inequalities  are  motivated  by  problems  in  mechanics,  where  the  lack  of  convexity 
does  not  permit  the  use  of  the  convex  superpotential  of  Moreau  [21].  Concrete  applications 
to  problems  of  mechanics  and  engineering  can  be  found  in  the  book  of  Panagiotopoulos 
[22].  Also  our  formulation  incorporates  the  case  of  elliptic  boundary  value  problems  with 
discontinuous  nonlinearities.  Such  problems  have  been  studied  (primarily  for  semilinear 
systems)  by  Ambrosetti-Badiale  [5],  Ambrosetti-Turner  [3],  [4],  Badiale  [6],  Chang  [8] 
and  Stuart  [23]. 

Let  Z  c  R^  be  a  bounded  domain  with  a  C  l  -boundary  F.  We  start  with  a  few  remarks 
concerning  the  first  eigenvalue  of  the  negative  p-Laplacian  —  A/;;t  =  —  div(||Z)jtp~~2Z)je), 
2  <  p  <  oo,  with  Dirichlet  boundary  conditions.  We  consider  the  following  nonlinear 
eigenvalue  problem: 

-div(\\Dx(z)\\p~2Dx(z»  =  *.\x(zW-2x(z)  a.e.  on  Z 


The  least  A  6  R  for  which  (20)  has  a  nontrivial  solution  is  called  the  first  eigenvalue  of 
—  (Ap,  W01>/;(Z)).  From  Lindqvist  [19]  we  know  that  A.I  >  0,  is  isolated  and  simple. 
Moreover,  A.]  >  0  is  characterized  via  the  Rayleigh  quotient,  namely 


AI  =  mm    jj-  :  x  e 

This  minimum  is  realized  at  the  normalized  first  eigenfunction  u  \ ,  which  we  know  that 
it  is  positive,  i.e  u  \  (z)  >  0  a.e  on  Z  (note  that  by  nonlinear  elliptic  regularity  theory 
ui  €  C/jf  (Z),  0  <  p  <  1;  seeTolksdorf  [24]). 
We  consider  the  following  nonlinear  eigenvalue  problem: 

€  A9/(z,  x(z))  a.e  on  Z 


*lr=0,2</?  <oo,A.>0  '  (    } 

Our  approach  to  problem  (24)  will  be  variational,  based  on  the  critical  point  theory  for 
nonsmooth  locally  Lipschitz  functionals,  due  to  Chang  [8].  In  this  case  the  classical  Palais- 
Smale  condition  (PS-condition  for  short)  takes  the  following  form.  Let  X  be  a  Banach  space 
and  /  :  X  ->  R  a  locally  Lipschitz  function.  We  say  that  /(•)  satisfies  the  nonsmooth 


Nonlinear  elliptic  bvp  's  .  505 

PS-condition,  if  any  sequence  [xn}n>\  c  X  for  which  {/(*„)}„>!  is  bounded  and  m(xn)  = 
nindl**  ||  :  x*  e  3f(xn)}  ->  0  as  n  -»  oo,  has  a  strongly  convergent  subsequence.  When 
f  6  C1  (X),  we  know  that  8f(xn)  =  {/'(*„)}  and  so  we  see  that  the  above  definition  of 
:he  PS-condition  coincides  with  the  classical  one. 
Our  hypotheses  on  the  function  7(2,  r)  in  problem  (24),  are  the  following: 

H(j):  j  :  Z  x  R  ->  E  is  a  function  such  that 
(i)  for  all  *  e  R  ->  ;  (z,  *)  is  measurable; 
(ii)  for  almost  all  z  €  Z,  x  -*  /  (z,  jc)  is  locally  Lipschitz; 
Jiii)  for  almost  all  z  e  Z,  all  x  e  R  and  all  u  e  3y  (z,  jc),  we  have 


with  ci  >  0,  1  <  r  <  p; 

(iv)  ;(•,  0)  €  L°°(Z),  /z  jf(z,  0)dz  =  0  and  there  exists  JCQ  E  R  such  that  for  almost  all 
z  e  Z,  j(z,xQ)  >  0; 

(v)  limjc-^o  sup  ^4%^  <  0  uniformly  for  almost  all  z  e  Z. 

We  will  need  the  following  nonsmooth  variant  of  the  classical  'Mountain  Pass  theorem'. 
The  result  is  due  to  Chang  [8]. 

Theorem  6.  IfX  is  a  reflexive  Banach  space,  V  :  X  ->  R  is  a  locally  Lipschitz  functional 
which  satisfies  the  (PS)  -condition  and  for  some  r  >  0  and  y  e  X  with  \\y\\  >  r  we  have 

max[V(0),  V001  <  inf[V(*)  :  ||*||  -  r]. 

Then  there  exists  a  nontrivial  critical  point  x  e  X  of  V  (Le  0  €  9  V(jc))  such  that  the 
critical  value  c  =  V(x)  is  characterized  by  the  following  minimax  principle 

c  =  inf  max    V(y(r)), 
yero<r<i 

where  r  =  {y  €  C([0,  1],  X)  :  y(0)  =  0,  y(l)  =  v}. 
We  have  the  following  multiplicity  result  for  problem  (1). 

Theorem  7.  If  hypotheses  H(y')  /ioW,  f/z^/i  f/z^re  exists  A,Q  >  0  such  that  for  all  X  >  XQ 
problem  (24)  /za^  af  /^^.y?  fwo  nontrivial  solutions. 


Proof.  For  A.  >  0,  let  ^  :  wJ'^CZ)  ->  R  be  defined  by 

Vx(x)  =  -\\Dx\\t-),  [  j(z, 
P  Jz 


We  know  that  VA,  is  locally  Lipschitz  (see  Clarke  [9]). 

Vx  satisfies  the  nonsmooth  (PS)-condition.  L 

<  M\  for  all  n  >  1  and  m(^/7)  ->  0  as  n  ->  CXD.  Let  jc*  e  8V^(^n)  such  that 


Claim  1.  Vx  satisfies  the  nonsmooth  (PS)-condition.  Let  {xn}n>\  C  W0'y;(Z)  be  such  that 


506  Antonella  Fiacca  et  al 

m(jc/7)  =  ||jt*||  for  all  n  >  1.  Its  existence  follows  the  fact  that  a  Vx(xn)  is  w  -compact  and 
the  norm  functional  is  weakly  lower  semicontinuous.  We  have 


Here  A  :  W^P(Z)  -+  W~l^(Z)  is  defined  by 


=   I 
J  z 


for  all  x,  y  €  W0lf/7(Z)  and  u*  €  di!/(xn)  where  VOO  =  /2  j(z,  x(z))dz.  It  is  easy  to  see 
that  A  is  monotone,  demicontinuous,  thus  maximal  monotone. 

From  theLebourg  mean  value  theorem  (see  Clarke  [9],  Therorem  2.3.7,  p.  41),  we  know 
that  there  exists  v*  e  dj(z,  17*  ).  0  <  77  <  1  such  that  j(z,  x)  -  j  (z,  0)  =  u**.  Using  this, 
together  with  hypothesis  H(j)(iii)  and  the  fact  that  j  (-,0)  e  L°°(Z),  we  can  write  that  for 
almost  all  z  e  Z,  and  all  x  e  R,  we  have  \j(z,  x)\  <  0[  4-  fa\x\r  with  0{,  #>  >  0.  Hence 
we  have  that 


j(z,xn(z»dz 

P 


>     -\\Dxnfp  -W\\Z\-\fo\\xn\\rp  for  some  ft  >  0. 

Here|Z|  denotes  the  Lebesgue  measure  of  the  domain  Z  c  R^.  Using  Young's  inequality 
with  s  >  0,  we  have 


for  some  Mg  >  0.  Let  s  <  ^-.  We  have 


Mi       > 


1        ^    .  .,  «     •• "  _  x0i  |Z|  -  Affi  (Rayleigh  quotient).         (25) 


Since  -  —  —-  >  0  (recall  the  choice  of  s  >  0),  from  the  above  inequality  it  follows 

P 


i        £_ 

i 

that  {xn}n>]  c  WQP(Z)  is  bounded.  So  we  may  assume  that  xn  -^  x  in  W^P(Z)  and  so 
xn  ->  x  in  LP(Z)  as  n  ->•  oo.  We  have 


From  Theorem  2.2  of  Chang  [8],  we  have  that  {v*}n>\  c  L^(Z)  and  is  bounded.  So  we 
have 

lim  (A(xn),  xn  ~  x)  =  lim  A,(u*,  *n  -  x);^. 

Since  A  is  maximal  monotone,  we  have  (A (xn),xn}  ->  (A(^),^)  =^  ||D;cw.||y,  — >•  ||Z)^||y7. 

Since  DJCW  A  D*  in  L^(Z,  R^)  and  L';(Z,  R^)  is  uniformly  convex,  from  the  Kadec- 
Klee  property  (see  Hu-Papageorgiou  [16],  Definition  1.1.72  and  Lemma  1.1.74,  p.  28)  it 
follows  that  Dxn  ->  Dx  in  LP(Z>RN ),  hence  xn  ->  x  in  T^'^Z).  This  proves  the  claim. 


Nonlinear  elliptic  bvp  's  507 

From  (25),  we  have  that  VA.(-)  is  coercive.  This  combined  with  Claim  1,  allow  the  use 
Df  Theorem  3.5  of  Chang  [8]  which  gives  us  yi  G  W^P(Z)  such  that  0  e  dV^(y\)  and 

c\  =     inf      V\  - 


rrom  hypothesis  H(;)  (iv),  for?  =  XQ,  we  have  ^(jc)  >  0  where  ^  :  Lr(Z)  ->  R  is 
iefined  by  V(v)  =  fz  j(z,  y(z))dz.  Evidently  ^  is  locally  Lipschitz  and  ^\wi.Pf7.  -  \[r. 

W  £ 


Since  W0ll/;(Z)  is  embedded  continuously  and  densely  in  U  (Z),  from  the  continuity  of 


it  follows  that  we  can  find  x  €  WQiy;(Z)  such  that  \fs(x)  =  ^r(jc)  >  0.  Then  there  exists 
\,0  >  0  such  that  for  A,  >  1Q  we  have  Vx(yi)  =  ±\\Dy\\$  ~  W(yi)  <  0  =  Vx(0).  So 
Vi  7^  0. 

Claim  2.  There  exists  r  >  0  such  that  inf[Vx(;c)  :  ||jc||  =  r]  >  0. 

By  virtue  of  hypotheses  H(y')(v),  we  can  find  8  >  0  such  that  for  almost  all  z  €  Z  and 
all  |;c|  <  5,  we  have  for  some  y  <  0, 


Also  recall  that  j(z,x)  <  fi\  +  p2\x\r  .  Thus  we  can  find  $4  >  0  large  enough  such  that 
for  almost  all  z  €  Z  and  all  x  e  R,  we  have 


j(z>  x)  < h  P4\x\x  with  p  <  s  <  p*  = 


Therefore,  we  can  write  that 

VA(*)  >  -  f  1  «  ^)  ||Z)jc||^  -^5II^II»  for  some  ft  >  0. 
P  \        Ai  / 

Note  that  (1-^-)  >  0  (since  y  <  OandO  <  A.0  <  A.,  A}  >  0).  Thus  for  every  A.  >  X0  >  0 
we  can  find  H^l^p  >  0  (depending  in  general  on  X)  such  that  inf[V^(x)  :  \\x\\  =  p]  >  0. 
Then  Vx(^i)  <  V'A.(O)  <  inf[Vx(jt)  :  \\x\\  =  p]  and  so  we  can  apply  Theorem  6  and  obtain 
y2  ^  0,  y2  ^  y\  such  that  0  €  3  VA.^)- 

Now  let  v  =  y\  or  3;  =  V2.  From  0  e  3^(v)  we  have 

A(y)  =  Xu* 


for  some  u*  € 

From  Clarke  [9]  we  know  that  u*  e  L^(Z)  and  u*(z)  e  dj(z,  >'(z))  a.e  on  Z.  From 
the  representation  therorem  for  the  elements  in  W~l^(Z)  (see  Adams  [1],  Theorem  3.10, 
p.  50)  we  have  that  divdlZtyp-1  Dy)  e  W~l^(Z),  So  we  have  for  all  u  6  W0lf/'(Z), 

,  u)  =  (  -  div(\\Dy\\P-2Dy)t  u}  =  X(u*,  w)w 
-div(||Dy(z)||^-2Dy(z)  =  Xv*(z)  e  A8/fe,  >'(z))  a.e  on 
Ji  ,  >'2  are  distinct,  nontrivial  solutions  of  (24). 


508  Antonella  Fiacca  et  al 

Remark.  Our  theorem  extends  Theorem  3. 5  of  Chang  [8],  who  studies  a  semilinear  problem 
and  proves  the  existence  of  one  solution  for  some  A  €  R.  Moreover,  in  Chang  ;  (z,  ;c)  = 
JQ  h(z,  s)ds.  In  addition  our  result  extends  Theorem  5.35  of  Ambrosetti-Rabinowitz  [2] 
to  nonlinear  problems  with  multivalued  terms. 

References 

[1]  Adams  R,  Sobolev  Spaces  (New  York:  Academic  Press)  (1975) 

[2]  Ambrosetti  A  and  Rabinowitz  P,  Dual  variational  methods  in  critical  point  theory  and  appli- 
cations, J.  Funct.  Anal  14  (1973)  349-381 

[3]  Ambrosetti  A  and  Turner  R,  Dual  variational  methods  in  critical  point  theory  and  applications, 
J.  Funct.  Anal.  14  (1973)  349-381 

[4]  Ambrosetti  A  and  Turner  R,  Some  discontinuous  variational  problems,  Diff.  Integral  Eqns  I 
(1988)341-350 

[5]  Ambrosetti  A  and  Badiale  M,  The  dual  variational  principle  and  elliptic  problems  with  dis- 
continuous nonlinearities,  7.  Math.  Anal.  Appl.  140  (1989)  363-373 
[6]  Badiale  M,  Semilinear  elliptic  problems  in  E^  with  discontinuous  nonlinearities,  An.  Sem. 

Mat.  Fis.  Univ  Modena  43  (1995)  293-305 

[7]  Carl  S  and  Heikkika  S,  An  existence  result  for  elliptic  differential  inclusions  with  discontin- 
uous nonlinearity,  Nonlin.  Anal.  18  (1992)  471-472 
[8]  Chang  K  C,  Variational  methods  for  nondifferentiabla  functional  and  its  applications  to  partial 

differential  equations,  J.  Math.  Anal.  Appl.  80  (1981)  102-129 
[9]  Clarke  F  H,  Optimization  and  Nonsmooth  Analysis  (New  York:  Wiley)  (1983) 

[10]  Costa  D  and  Goncalves  J,  Critical  point  theory  for  nondifferentiable  functionals  and  applica- 
tions, J.  Math.  Anal.  Appl.  153  (1990)  470-^85 

[11]  Dancer  E  and  Sweers  G,  On  the  existence  of  a  maximal  weak  solution  for  a  semilinear  elliptic 
equation,  Diff  Integral  Eqns  2  (1989)  533-540 

[12]  Deuel  J  and  Hess  P,  A  criterion  for  the  existence  of  solutions  of  nonlinear  elliptic  boundary 
value  problems,  Proc.  R.  Soc.  Edinburg  74  (1974,  75)  49-54 

[13]  Gilbarg  D  and  Trudinger  N,  Elliptic  Partial  Differential  Equations  of  Second  Order  (Berlin: 
Springer- Verlag)  (1983) 

[14]  Gossez  J-P  and  Mustonen  V,  Pseudomonotonicity  and  the  Leray-Lions  condition,  Diff.  Inte- 
gral Eqns  6  (1 993)  37-46 

[15]  Heikkila  S  and  Hu  S,  On  fixed  points  of  multifunvtions  in  ordered  spaces  Appl.  Anal.  54 
(1993)  115-127 

[16]  Hu  S  and  Papageorgiou  N  S,  Handbook  of  Multivalued  Analysis.  Volume  I:  Theory  (The 
Netherlands:  Kluwer,  Dordrecht)  (1997) 

[17]  Landes  R,  On  Galerkin's  method  in  the  existence  theory  of  quasilinear  elliptic  equations,  J. 
Fund  Anal  39  (1980)  123-148 

[1 8]  Leray  J  and  Lions  J-L,  Quelques  resultants  de  Visik  sur  les  problems  elliptiques  nonlinearairies 
par  methods  de  Minty-Browder  Bull.  Soc  Math.  France  93  (1 965)  97-1 07 

[19]  Lindqvist  P,  On  the  equation  div(\Dx\p~2Dx)  +  h.\x\p~2x  =  0,  Proc.  Am.  Math.  Soc.  109 
(1990)  157-164 

[20]  Marcus  M  and  Mizel  V,  Absolute  continuity  on  tracks  and  mappings  of  Sobolev  spaces,  Arch. 
Rational  Mech.  Anal.  45  (1972)  294-320 

[21]  Moreau  J-J,  La  notion  de  sur-potentiel  et  les  liaisons  unilaterales  en  elastostatique,  Compt.es 
RendusAcad.  Sci.  Paris  267  (1968)  954-957 

[22]  Panagiotopouios  P  D,  Hemivariational  Inequalities.  Applications  in  Mechanics  and  Engineer- 
ing (Berlin:  Springer- Verlag)  (1993) 

[23]  Stuart  C,  Maximal  and  minimal  solutions  of  elliptic  differential  equations  with  discontinuous 
nonlinearities,  Math.  163  (1978)  239-249 

[24]  Tolksdorf  P,  Regularity  for  a  more  general  class  of  quasilinear  elliptic  equations,  J.  Diff.  Eqns 
51(1894)  126-150 

[25]  Zeidler  E,  Nonlinear  Functional  Analysis  and  its  Applications  II  (New  York:  Springer- Verlag) 
(1990) 


Proceedings  (Mathematical  Sciences) 


Volume  111,2001 


SUBJECT  INDEX 


[n,  k,  d]  cyclic  code 

Cyclic  codes  of  length  T  371 

2  categories 

Algebraic  stacks  1 

4x4  matrices 

Unitary  tridiagonalization  in  M(4,  C)     38 1 


Abel's  theorem 

On  a  Tauberian  theorem  of  Hardy  and 
Littlewood  221 

Absolute  Euler  summability 
Multipliers     for     the     absolute     Euler 
summability  of  Fourier  series  203 

Absolute  summability 
Multipliers     for     the     absolute     Euler 
summability  of  Fourier  series  203 

Absolutely  continuous  spectrum 
Spectra  of  Anderson  type  models  with  de- 
caying randomness  179 

Ahlfors-Bers  variational  formulae 
Variational  formulae  for  Fuchsian  groups 
over  families  of  algebraic  curves  33 

Algebraic  curve 
Unitary  tridiagonalization  in  M(4,  C)     381 

Algebraic  stacks 
Algebraic  stacks  1 

Anderson  model 

Spectra  of  Anderson  type  models  with  de- 
caying randomness  179 

Arzela-Ascoli  theorem 
Periodic  and  boundary  value  problems  for 
second  order  differential  equations         1 07 

Asymptotic  behaviour 

On  oscillation  and  asymptotic  behaviour  of 
'     solutions  of  forced  first  order  neutral  differ- 
ential equations  337 


Banach  lattice 

On  property  (j3)  in  Banach  lattices,  Calderdn- 
LozanowskiTand  Orlicz-Lorentz  spaces 

319 

Binary  forms 
On  totally  reducible  binary  forms:  I       249 

Boundary  controllability 
Boundary     controllability     of     integro- 
differential  systems  in  Banach  spaces     127 


C*-enveloping  algebra 
Topological  *-algebras  with  C*-enveloping 
algebras  II  65 


Calder6n-Lozanowskii  space 
On  property  (/3)  in  Banach  lattices,  Calder6n- 
Lozanowskii  and  Orlicz-Lorentz  spaces 

319 

Caratheodory  function 
Periodic  and  boundary  value  problems  for 
second  order  differential  equations"        107 

Character 

Obstructions  to  Clifford  system  extensions 
of  algebras  151 

Clifford  system 

Obstructions  to  Clifford  system  extensions 
of  algebras  151 

Coercive  operator 

Nonlinear  elliptic  differential  equations  with 
multivalued  nonlinearities  489 

Cohomology  groups 

Obstructions  to  Clifford  system  extensions 
of  algebras  151 

Compact  embedding 

Periodic  and  boundary  value  problems  for 
second  order  differential  equations  107 

Contraction  mapping  theorem 
Monotone  iterative  technique  for  impulsive 
delay  differential  equations  351 

Convolution 

On  a  generalized  Hankel  type  convolution 
of  generalized  functions  47 1 

Critical  point 

Nonlinear  elliptic  differential  equations  with 
multivalued  nonlinearities  489 

C-totally  real  submanifold 
On    Ricci    curvature    of    C-totally    real 
submanifolds  in  Sasakian  space  forms   399 

Cuspidal  curve 

The  multiplication  map  for  global  sections 
of  line  bundles  and  rank  1  torsion  free  sheaves 
on  curves  163 

Cyclotomic  cosets 
Cyclic  codes  of  length  T  37 1 


Decaying  randomness 
Spectra  of  Anderson  type  models  with  de- 
caying randomness  179 

Degree 
Unitary  tridiagonalization  in  M(4,  C)     381 

Dilation  matrix 

Multiwavelet  packets  and  frame  packets  of 
L\R(I)  439 

Divergent  series 

On  a  Tauberian  theorem  of  Hardy  and 
Littlewood  221 


509 


510 


Subject  index 


Dunford-Pettis  theorem 
Periodic  and  boundary  value  problems  for 
second  order  differential  equations         1 07 

Eigenvalue  problem 

Nonlinear  elliptic  differential  equations  with 
multivalued  nonlineadties  489 

Embeddable  measures 
Limits  of  commutative  triangular  systems  on 
locally  compact  groups  49 

Equisummability 

On  the  equisummability  of  Hermite  and 
Fourier  expansions  95 

Euler-Bernoulii  beam  equation 

On  initial  conditions  for  a  boundary  stabi- 
lized hybrid  Euler-Bernoulli  beam  365 

Exponential  energy  decay 

On  initial  conditions  for  a  boundary  stabi- 
lized hybrid  Euler-Bernoulli  beam  365 

Extremal  solution 

Periodic  and  boundary  value  problems  for 
second  order  differential  equations  107 
Monotone  iterative  technique  for  impulsive 
delay  differential  equations  35 1 

Nonlinear  elliptic  differential  equations  with 
multivalued  nonlinearities  489 

Fixed  point  theorem 

Boundary  controllability  of  integro- 
difTerential  systems  in  Banach  spaces  127 

Fixed  points 

Common  fixed  points  for  weakly  compatible 
maps  24 1 

Frame  packets 

Multiwavelet  packets  and  frame  packets  of 
L2(tf')  439 

Frechet  *-algebra 

Topological  ^-algebras  with  C*-enveloping 
algebras  II  65 

Fuchsian  groups 

Variational  formulae  for  Fuchsian  groups 
over  families  of  algebraic  curves  33 

Galois  group 

Descent  principle  in  modular  Galois  theory 

139 

Generalized  functions  (distributions) 
On  a  generalized  Hankel  type  convolution 
of  generalized  functions  471 

Generalized  Hankel  type  transformation 
On  a  generalized  Hankel  type  convolution 
of  generalized  functions  47 1 

Generator  polynomial 
Cyclic  codes  of  length  2"'  37 1 

Groupoid  C*-algebra 

Topological  *-algebras  with  C*-enveloping 
algebras  II  65 


Hardy-Littlewood  Tauberian  theorem 
On  a  Tauberian  theorem  of  Hardy  and 
Littlewood  221 

Hardy's  theorem 

Uncertainty  principles  on  two  step  nilpotent 
Lie  groups  293 

Hecke  lines 

Stability  of  Picard  bundle  over  moduli  space 
of  stable  vector  bundles  of  rank  two  over  a 
curve  263 

Heisenberg's  inequality 
Uncertainty  principles  on  two  step  nilpotent 
Lie  groups  293 

Hermite  functions 

On  the  equisummability  of  Hermite  and 
Fourier  expansions  95 

Homogeneous  operators 
Homogeneous  operators  and  projective  rep- 
resentations of  the  Mobius  group:  A  survey 

415 

Hybrid  system 

On  initial  conditions  for  a  boundary  stabi- 
lized hybrid  Euler-Bernoulli  beam  365 

Idempotent  generator 

Cyclic  codes  of  length  21"  37 1 

Impulsive  delay  differential  equations 
Monotone  iterative  technique  for  impulsive 
delay  differential  equations  351 

Infmitesimally  divisible  measures 
Limits  of  commutative  triangular  systems  on 
locally  compact  groups  49 

Initial  conditions 

On  initial  conditions  for  a  boundary  stabi- 
lized hybrid  Euler-Bernoulli  beam  365 

Integrodifferential  system 
Boundary     controllability     of     integro- 
differential  systems  in  Banach  spaces     127 

Iteration 

Descent  principle  in  modular  Galois  theory 

139 

Line  bundle 

The  multiplication  map  for  global  sections 
of  line  bundles  and  rank  1  torsion  free  sheaves 
on  curves  163 

Unitary  tridiagonalization  in  A/(4,  C)     381 

Loop  groups 
Principal  C-bundles  on  nodal  curves      271 

Lower  solution 

Periodic  and  boundary  value  problems  for 
second  order  differential  equations  1 07 
Nonlinear  elliptic  differential  equations  with 
multivalued  nonlinearities  489 

Mobility  edge 

Spectra  of  Anderson  type  models  with  de- 
caying randomness  1 79 


Subject  index 


511 


oduli  spaces 

Algebraic  stacks 

ultipliers 

Multipliers     for     the     absolute 

summability  of  Fourier  series 


Euler 
203 


*utral  equations 

On  oscillation  and  asymptotic  behaviour  of 
solutions  of  forced  first  order  neutral  differ- 
ential equations  337 
Ddal  curve 

The  multiplication  map  for  global  sections 
of  line  bundles  and  rank  1  torsion  free  sheaves 
on  curves  '  163 

Dnoscillation 

On  oscillation  and  asymptotic  behaviour  of 
solutions  of  forced  first  order  neutral  differ- 
ential equations  337 
ansmooth  Palais— Smale  condition 
Nonlinear  elliptic  differential  equations  with 
multivalued  nonlinearities  489 

^-algebra 

Topological  *-algebras  with  C*-enveloping 
algebras  II  65 

bstructions 

Obstructions  to  Clifford  system  extensions 
of  algebras  151 

rder  interval 

Periodic  and  boundary  value  problems  for 
second  order  differential  equations  107 
Nonlinear  elliptic  differential  equations  with 
multivalued  nonlinearities  489 

rlicz-Lorentz  space 

On  property  (/3)  in  Banach  lattices,  Calderon- 
Lozanowskii  and  Orlicz-Lorentz  spaces 

319 

scillation 

On  oscillation  and  asymptotic  behaviour  of 
solutions  of  forced  first  order  neutral  differ- 
ential equations  337 

irabolic  bundles 

Principal  G-bundies  on  nodal  curves      271 

irserval  relation 

On  a  generalized  Hankel  type  convolution 

of  generalized  functions  47 1 

-convexity 

A  variational  principle  for  vector  equilibrium 

problems  465 

sdersen  ideal  of  a  C*-algebra 

Topological  *~algebras  with  C*-enveloping 

algebras  II  65 

malty  function 

Periodic  and  boundary  value  problems  for 

second  order  differential  equations         1 07 


Periodic  points 

Boundedness  results  for  periodic  points  on 
algebraic  varieties  173 

Periodic  problem 

Periodic  and  boundary  value  problems  for 
second  order  differential  equations  107 

Picard  bundle 

Stability  of  Picard  bundle  over  moduli  space 
of  stable  vector  bundles  of  rank  two  over  a 
curve  263 

P-y/-monotonicity 

A  variational  principle  for  vector  equilibrium 
problems  465 

Prescribed  Gaussian  curvature 
A  variational  proof  for  the  existence  of  a 
conformal  metric  with  preassigned  negative 
Gaussian  curvature  for  compact  Riemann 
surfaces  of  genus  >  1  407 

Principal  bundles 
Principal  G-bundles  on  nodal  curves      271 

Projective  representations 
Homogeneous  operators  and  projective  rep- 
resentations of  the  Mobius  group:  A  survey 

415 

Property  (j3) 

On  property  (j3)  in  Banach  lattices,  Calderon- 
Lozanowskii  and  Orlicz— Lorentz  spaces 

319 

Proximinality  and  strong  proximinality 
Proximinal  subspaces  of  finite  codimension 
in  direct  sum  spaces  229 

Pseudomonotone  operator 
Nonlinear  elliptic  differential  equations  with 
multivalued  nonlinearities  489 


Rank  1  torsion  free  sheaf 
The  multiplication  map  for  global  sections 
of  line  bundles  and  rank  1  torsion  free  sheaves 
on  curves  163 

Real  algebraic  groups 

Limits  of  co mm utative  triangular  systems  on 
locally  compact  groups  49 

Reproducing  kernels 

Homogeneous  operators  and  projective  rep- 
resentations of  the  Mobius  group:  A  survey 

415 

Ricci  curvature 

On  Ricci  curvature  of  C-totally  real  sub- 
manifolds  in  Sasaki  an  space  forms  399 

Riemann  surfaces 

Variational  formulae  for  Fuchsian  groups 
over  families  of  algebraic  curves  33 

A  variational  proof  for  the  existence  of  a 
conformal  metric  with  preassigned  negative 
Gaussian  curvature  for  compact  Riemann 
surfaces  of  genus  >  1  407 


512 


Subject  index 


Sasakian  space  form 

On  Ricci  curvature  of  C-totally  real  sub- 
manifolds  in  Sasakian  space  forms        399 
Semigroup  theory 

Boundary  controllability  of  integro- 
differential  systems  in  Banach  spaces  127 
Singular  projective  curve 
The  multiplication  map  for  global  sections 
of  line  bundles  and  rank  1  torsion  free  sheaves 
on  curves  163 

Small  deflection 

On  initial  conditions  for  a  boundary  stabi- 
lized hybrid  Euler-Bernoulli  beam        365 
Smooth  Frechet  algebra  crossed  product 
Topological  ^-algebras  with  C*-enveloping 
algebras  II  65 

Sobolev  space 

Periodic  and  boundary  value  problems  for 
second  order  differential  equations         1 07 
Special  divisor 

The  multiplication  map  for  global  sections 
of  line  bundles  and  rank  1  torsion  free  sheaves 
on  curves  163 

Special  Hermite  expansions 
On  the  equisummability  of  Hermite  and 
Fourier  expansions  95 

Sturm-Liouville  boundary  conditions 
Periodic  and  boundary  value  problems  for 
second  order  differential  equations         107 
Summ ability  of  factored  Fourier  series 
Multipliers     for     the     absolute     Euler 
summability  of  Fourier  series  203 

Sz-Nagy-Foias  characteristic  functions 
Homogeneous  operators  and  projective  rep- 
resentations of  the  Mobius  group:  A  survey 

415 

Tauberian  theorem 

On  a  Tauberian  theorem  of  Hardy  and 
Littlewood  221 

Topological  *-algebra 
Topological  *-algebras  with  C*-enveloping 
algebras  II  65 

Totally  disconnected  groups 
Limits  of  commutative  triangular  systems  on 
locally  compact  groups  49 

Transitivity 

Descent  principle  in  modular  Galois  theory 

139 

Triangular  systems  of  measures 
Limits  of  commutative  triangular  systems  on 
locally  compact  groups  49 

Truncation  function 

Nonlinear  elliptic  differential  equations  with 
multivalued  nonlinearities  489 


Truncation  map 

Periodic  and  boundary  value  problems  for 
second  order  differential  equations         107 

Two  step  nilpotent  Lie  groups 
Uncertainty  principles  on  two  step  nilpotent 
Lie  groups  293 


Unbounded  operator  representation 
Topological  *-algebras  with  C*-enveloping 
algebras  II  65 

Uncertainty  principles 
Uncertainty  principles  on  two  step  nilpotent 
Lie  groups  293 

Uniformization  theorem 
A  variational  proof  for  the  existence  of  a 
conformal  metric  with  preassigned  negative 
Gaussian  curvature  for  compact  Riemann 
surfaces  of  genus  >  1  407 

Unitary  tridiagonalization 
Unitary  tridiagonalization  in  M(4,  C)     381 

Universal  algebra  on  generators  with  relations 
Topological  *-algebras  with  C*-enveloping 
algebras  II  65 

Upper  solution 

Periodic  and  boundary  value  problems  for 
second  order  differential  equations  107 
Nonlinear  elliptic  differential  equations  with 
multivalued  nonlinearities  489 


Variational  principle 

A  variational  principle  for  vector  equilibrium 
problems  465 

Vector  bundles 
Algebraic  stacks  1 

Vector  equilibrium  problem 
A  variational  principle  for  vector  equilibrium 
problems  465 


Wavelet 

Multiwavelet  packets  and  frame  packets  of 
L\R(I)  439 

Wavelet  packets 

Multiwavelet  packets  and  frame  packets  of 
L\Rd)  439 

Weakly  compatible  maps 
Common  fixed  points  for  weakly  compatible 
maps  241 


Yosida  approximation 

Nonlinear  elliptic  differential  equations  with 
multivalued  nonlinearities  489 


AUTHOR  INDEX 


Abhyankar  Shreeram  S 

Descent  principle  in  modular  Galois  theory 

139 
Anandhi  E  R 

see  Balachandran  K  127 

Bagchi  Bhaskar 

A  survey  of  homogeneous  operators  and  pro- 
jective  representations  of  the  Mobius  group 

415 

Balachandran  K 

Boundary  controllability  of  integro- 
differential  systems  in  Banach  spaces  127 

Ballico  E 

The  multiplication  map  for  global  sections 
of  line  bundles  and  rank  1  torsion  free  sheaves 
on  curves  163 

Behera  Biswaranjan 

Multiwavelet  packets  and  frame  packets  of 
L2(R'<)  439 

Bhatt  S  J 

Topological  ^'-algebras  with  C*-enveloping 
algebras  II  65 

Bhattacharyya  Dakshini 
Variational  formulae  for  Fuchsian  groups 
over  families  of  algebraic  curves  33 

BhosleUshaN 

Principal  G-bundles  on  nodal  curves      271 

Biswas  Indranil 

Stability  of  Picard  bundle  over  moduli  space 
of  stable  vector  bundles  of  rank  two  over  a 
curve  263 

Bose  Sujit  K 

On  initial  conditions  for  a  boundary  stabi- 
lized hybrid  Euler-Bernoulli  beam  365 

Cegarra  Antonio  M 

Obstructions  to  Clifford  system  extensions 
of  algebras  151 

Chandra  Prem 

Multipliers  for  the  absolute  Euler 
summability  of  Fourier  series  203 

Chugh  Renu 

Common  fixed  points  for  weakly  compatible 
maps  241 

Deswal  Sunita 
see  Ku  mar  Raj  neesh  1 37 

Dey  Rukmini 

A  variational  proof  for  the  existence  of  a 
conformal  metric  with  preassigned  negative 
Gaussian  curvature  for  compact  Riemann 
surfaces  of  genus  >  1  407 


Fakhruddin  Najmuddin 
Boundedness  results  for  periodic  points  on 
algebraic  varieties  173 

FiaccaAntonella 

Nonlinear  elliptic  differential  equations  with 
multivalued  nonlinearities  489 

Fu  Xilin 
see  Yan  Baoqiang  35 1 

Gaikawad  G  S 

see  Malgonde  S  P  471 

Garzon  Antonio  R 

see  Cegarra  Antonio  M  151 

G6mezTomdsL 

Algebraic  stacks  1 

see  Biswas  Indranil  263 

Hooley  C 
On  totally  reducible  binary  forms:  I       249 

Indumathi  V 

Proximinal  subspaces  of  finite  codimension 
in  direct  sum  spaces  229 

Kazmir  K  R 

A  variational  principle  for  vector  equilibrium 
problems  465 

Keskar  Pradipkumar  H 
see  Abhyankar  Shreeram  S  1 39 

Kolwicz  Pawei 

On  property  (/3)  in  Banach  lattices,  Calder6n~ 
Lozanowskii  and  Orlicz-Lorentz  spaces 

319 

Krishna  M 

Spectra  of  Anderson  type  models  with  de- 
caying randomness  179 

Kumar  Raj  neesh 

Steady-state  response  of  a  micropolar  gen- 
eralized thermoelastic  half-space  to  the  mov- 
ing mechanical/thermal  loads  137 

KumarSanjay 
see  Chugh  Renu  241 

Malgonde  S  P 

On  a  generalized  Hankel  type  convolution 

of  generalized  functions  47 1 

Matzakos  Nikolaos 

see  Fiacca  Antonella  489 

Misra  Gadadhar 

see  Bagchi  Bhaskar  415 


513 


514 


Author  index 


Narayanan  E  K 

On  the  equisummability  of  Hermite  and 
Fourier  expansions  95 

Papageorgiou  Nikolaos  S 
Periodic  and  boundary  value  problems  for 
second  order  differential  equations         1 07 
s^FiaccaAntonella  489 

Papalini  Francesca 
see  Papageorgiou  Nikolaos  S  107 

Parhi  N 

On  oscillation  and  asymptotic  behaviour  of 
solutions  of  forced  first  order  neutral  differ- 
ential equations  337 

PatiT 

On  a  Tauberian  theorem  of  Hardy  and 
Littlewood  221 

PatiVishwambhar 
Unitary  tridiagonalization  in  M(4,  C)     381 

Pruthi  Manju 
Minimal  cyclic  codes  of  length  2"'         37 1 


Rath  R  N 
see  Parhi  N 


RaySK 

Uncertainty  principles  on  two  step  nilpotent 
Lie  groups  293 

Servadei  Raffaella 

seeFiaccaAntonella  489 

Shah  Riddhi 

Limits  of  commutative  triangular  systems  on 

locally  compact  groups  49 

Sinha  K  B 

see  Krishna  M  179 


Thangavelu  S 
see  Narayanan  E  K 


65 


337 


XiminLiu 

On    Ricci    curvature    of    C-totally    real 
submanifolds  in  Sasakian  space  forms   399 

Yan  Baoqiang 

Monotone  iterative  technique  for  impulsive 
delay  differential  equations  351 


End  of  one  hundred  and  eleventh  volume