14 CONWAY, HERRERO, AND MORREL

to int [ f(5(E)) ] , then X and Y e int f(5(E)) .

Proof, (a) The stated condition on X and Y implies that the commutant of T

consists of lower triangular operator- valued matrices. Thus if X - T is in-

vertible, (A,

-T}"1

is lower triangular and the diagonal entries must be {X

-X)"1

and (X -

Y)"1

. Conversely, if X - X and X - Y are both invertible, the lower

triangular matrix with {X -

X}"1

and {X -

Y)"1

on the diagonal and - (X -

X)"1

Q

(X -

Y)"1

in the lower left corner is the inverse of X - T .

(b) Let e 0 such that I I X 0 Y - T I I e implies T e f(5(E)) . By a

result of [14], there is a c 0 such that I I XS - SY I I c I ISI I for all S in

B(M) . Let 5 e be sufficiently small that I I XXS - SY I I (c/2) I ISI I when-

ever I IX - Xl I I 5 . Then I I X e Y - X ^ Y I I e whenever I IX - X

:

I I 5

and so Xx 0 Y e f(5(E)) . But XXT = TY implies T = 0 . Hence XY e f(5(E)) .

That is, I I X - Xl I I 6 implies Xx e f(5(E)) . Thus X e int[ f(5(E)) ] .

Similarly, Y e int f(5(E)) . •

For an analytic Cauchy domain A define the following operators.

A(A) = Mz on H

2

0A) C(A) = A(A*)*.

1.6 Proposition Let A1 and A2 be analytic Cauchy domains. If X is a bounded

operator, n and m are extended positive integers, and AfA^ X = XC(A2)

then X = 0 .

Proof. In fact, A(AX) is a subnormal operator and C(A2) is a cosubnormal

operator. The result now follows by a standard result of subnormal operator

theory [25] (also see [11], p. 199). •

1.7 Proposition Let A be an open subset of C .

(a) A(A) G 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 A(A)