COMPLETING THE FUNCTIONAL CALCULUS 13

then N e f(5(E)) if and only if there is a bounded set E

}

c E such that a(N) c

f(Ex) . If f is analytic and completely non-constant on the open set G , then a

normal operator N belongs to f(5(G)) if and only if a(N) e f(G) .

Proof. If N = f(A) for some A in f(5(E)) , the n the Spectral Mapping

Theorem implies that the set E

x

may be taken to be a (A) . Conversely, if

such a bounded set E

x

exists, then there is a Borel function g: a(N) - Ex

such that g{X) e E

1

n f

1

!!) for all X in a(N) . (This last statement follows by

any of many measurable selection theorems, or, using the analyticity of f, the

reader can give a direct proof.) So g is bounded and if A = g(N) , then N =

f(A) .

In the second statement of the proposition, one implication is, again,

an immediate consequence of the Spectral Mapping Theorem. If a(N) c

f(G) , then the assumptions about f imply that there is a compact set K

contained in G such that f(K) = a(N) . To see this note that G = k J

n

1 ^ ,

where each Kn is compact and Kn c int K

n + 1

. Since f is completely non-

constant on G , f(int K J is open for each n . Thus { f(int Kn) } is an open

cover of a(N) and there is an integer n such that a(N) c fCK^) . The result

now follows by the first part of the proposition. •

The stated condition on f and E in the preceding proposition is not

always satisfied. For example, let E = {0} u {

n"1

+ 2?ini : n e M } and let f(z)

=

ez

. So f(E) = { 1, e

1 / 2

, e

1 / 3

, . . . } . If N is the diagonal operator with

entries 1, e

1 / 2

, e

1 / 3

, . . . , then N e f(5(E)) . It is not too difficult to see that

N E cl[ f(5(E)) ] . See Corollary 2.7 for a characterization of the normal

operators belonging to cl[ f(5(E)) ] .

1.5 Proposition (a) If X and Y are operators such that there is no non-zero

[ V Q ]

operator S with XS = SY , then for any operator Q the spectrum of T =

is G(X) U a(Y) . Moreover, if T e f(5(E)) , then X and Y belong to f(5(E)) .

(b) If X and Y are operators with Gj(X) n ar(Y) = 0 and X 0 Y belongs