COMPLETING THE FUNCTIONAL CALCULUS

15

= f(R) and g is the inverse of the restriction of f to G, then R = multiplication

by g on

H2(3A)

.

(b) C(A) e f(5(E)) if and only if there is an open set G with cl G

included in E such that f is one-to-one on G and f(G) = A . Moreover, if C(A)

= f(R) and g is the inverse of the restriction of f to G, then R is

multiplication by g(z) on

H2(3A

) .

Proof, (a) If there is an open set G with cl G included in E such that f is

one-to-one on G , g is the inverse of the restriction of f to G, and R is

multiplication by g , then R e 5(E) and f(R) = A(A) . Conversely, if A(A) =

f(R) for some R in 5(E) , then R must commute with A(A) . Hence (see, for

example, [31] or page 147 of [11]) there is a g in H°°(A) such that R =

multiplication by g on H

2

$A) . It is routine to check that f(g(Q) = £ for all £

in A . If G = g(A) , then f is one-to-one on G . The Spectral Mapping

Theorem implies cl G = a(R) c E .

(b) This follows from part (a) and the definition of C(A) . •

Before stating the next result, two additional pieces of notation are

needed. For any operator T , let min ind T = min{ nul T , nul T } . Also, if o

is a closed and relatively open subset of c(T) , let 9{[T; o) denote the range of

the Riesz idempotent, E(T; a) , associated with a . If o consists of a single

isolated point X , let E(T; X) = E(T; {X} ) and #(T; X) = 9{T\ [X] ) .

The next result is a special case of the Similarity Orbit Theorem from

[5]. Also see [3], page 5.

1.8 Theorem (Special case of the Similarity Orbit Theorem) Assume the

operator X has the property that if X is an isolated point of ae(X) , k^ x(z) is

defined to be X - z on a neighborhood of X and 0 on a neighborhood of ae(X) \

{X} , and X is the image of X in the Calkin algebra, then [k^ X(X) ]

m

* 0 for all

m 1 . If the operator Y satisfies the conditions:

(a) OQ(Y) C G0(X) and each component of crlre(Y) meets oe(X) ;