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Volume 111 Number 1
February 2001
Proceedings of the of
Mathematical
Editor
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Indian Statistical Institute, Bangalore
Associate Editors
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The Institute of Mathematical Sciences, Chennai
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Tata Institute of Fundamental Research, Mumbai
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Purdue University, West Lafayette, USA
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Iowa State University, Iowa, USA
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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
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University of California, Los Angeles, USA
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Courant Institute of Mathematical Sciences, USA
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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
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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/-
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Indian Academy of Sciences, C V Raman Avenue,
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©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.
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[Ar2] Artin M, Versal deformations and algebraic stacks, Invent. Math. 27 (1974) 165-189
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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)
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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)
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Grenzgebiete. 3. Folge, 39 (Springer Verlag) (2000)
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1992
[S] Simpson C, Moduli of representations of the fundamental group of a smooth projective
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(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.
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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
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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.
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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.
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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.
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195-197
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(Liverpool) (1983)
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EC. Norm. Sup. 20 (1987) 65-87
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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.
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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).
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Austral. Math. Soc. A65 (1998) 289-302
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[5] Cowling M G, Sitaram A and Sundari M, Hardy's uncertainty principle on semisimple Lie
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Part 1 - Basic theory and examples (NY: Cambridge Univ. Press, Cambridge) (1990)
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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)
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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
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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)
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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
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(1999)
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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.
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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
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[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?
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>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.
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726-739
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712-738
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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