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,
is lower triangular and the diagonal entries must be {X
and (X -
. Conversely, if X - X and X - Y are both invertible, the lower
triangular matrix with {X -
and {X -
on the diagonal and - (X -
(X -
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
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)
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