11 THE SPECTRAL THEORY OF HUBERT 397

for all functions /e ®c(Sp(N)). Also it follows from (15.11.1.3) that
\(f(W-x\y)\ ^ \\f\\ • \\x\\ - \\y\\ for all continuous functions/on Sp(N), and
therefore

(15.11.2.1) \\mx,y\\^ \\x\\-\\y\\

for all x9 y in E.

If fe ^c(Sp(N)) is the restriction to Sp(N) of a complex-valued function
g defined on a subset of C containing Sp(JV), we shall write g(N) in place
of/(N).

(15.11.3) We shall say that N is a simple normal operator if the representa-
tion /W/(AO of #c(Sp(JV)) on E is topologically cyclic. By virtue of (15.5.6),
there exists a decomposition of E as a Hilbert sum of closed subspaces En
which are stable under stf (and therefore under N and JV* (cf. Problem 3)
such that, for each «, the restriction Nn ofN to En is a simple normal operator.
The space En may then be identified with Lc(Sp(NB), juw), where nn is a positive
measure, and Nn is then identified with the operator MMn(lc) of multiplication
by the class of the function lc in L£(Sp(Nn), JUB). If an is a totalizing vector for
the representation/^/(NJ, then (15.10.3.3) we may take

(15.11.3.1) ^ = mflM>fln.

With this notation:

(15.11.4) The support of fin is the whole of Sp(Nn).

If a ^ Supp(jun), then the function £i—>(a— C)""1 is continuous and
bounded on Supp(/zB); it extends to a bounded continuous function g on
SpW (4.5.1), and since 0(Q(a -0 = 1 for all C e Supp(jun), it follows that the
continuous operator on Lc(Sp(JVn), /(„), defined by multiplication by the class
of the function Ch-»a — £, is invertible. Hence (as this operator is identified
with alEn — Nn) we have a £ Sp(JVn).

(15.11.5) SpC/V) is the closure in Cof(J Sp(JVn).

n

Clearly Sp(Nn) c Sp(AT); since Sp(N) is closed, it therefore contains the
closure of (J Sp(JVrt). If a does not lie in this closure, then there exists/ > 0

such that "a-Cl^r for M C e Sp(Nn) and all n. Hence (15:11.1.3) we
have IKal^-^)"1!! £l/r, from which we conclude (15.10.8.1 y that the
(alEn — N^)""1 are the restrictions of a continuous operator on E, the inverse
of an!E - N.operators on LcO"*). With this notation, the restriction of