16 CONWAY, HERRERO, AND MORREL

(b) P(Y) c P(X) and ind {X - Y) = ind {X - X) and min ind {X - Y)k

min ind {X - X)k for all X in P(Y) and all k 1 ;

(c) dim itf(Y; X.) = dim #(Y; X) for all X in o0(Y) ; then Y is in the

closure of the similarity orbit of X ,

5(X) = { WXW"1 : W is invertible in CB(fhQ} .

The next proposition is a variation on a result of [6] (also see page 136

of [20]). Essentially it is a special case of that result that will be needed in

this paper.

1.9 Proposition If T e B(9{) and e 0 , then there is an operator S such

that I I S - Tl I e and S is similar to a direct sum Sx © • • • © S

n

© F , where

these direct summands satisfy the following properties:

(a) F is a finite rank operator with a(F) c J0(T) and for X in a(F),

F | ^(F; X) similar to T | #|T; X) ;

(b) for each j ,

CJ(SJ)

is connected;

(c)

G(F)

n

G(SJ)

= 0 = a(Sj) n afS^ for i * j and 1 i, j n ;

(d) for each j , alre(Sj) is the closure of an analytic Cauchy domain and

a(Sj) \ cilre(Sj) is an analytic Cauchy domain;

(e) P(S) c P(T) , P+(S) has only a finite number of components, and

for each X in P(S) , nul (X - S) = nul {X - T) and nul (X - S)* = nul (X - T)* .

(f) oe(T) c oe(S) , alre(T) c cJlre(S) , and each component of c

e

(S)

contains at least one component of tfe(T) .

Proof. Let 8 0 and let A be an analytic Cauchy domain with alre(T) e A c

[alre(T) ]g and 3A n a0(T) = 0 . Let N be a normal operator with o(N) = cl A .

By Proposition 1.4 of [6] (also see Chapter 3 in [20]) there is an operator S

with I I T - SI I 25 such that S is similar to T © N . (This could be

obtained as a consequence of the Similarity Orbit Theorem, but this would

be putting the cart before the horse.) Consequently