§1 SPECTRAL PRELIMINARIES

In this section several results that will be used in the remainder of the

paper are presented. Some of these are new or at least do not seem to be

stated in the literature; some are here only for the reader's convenience.

Proofs are given where it is appropriate.

The convention will be adopted in this paper that elements and

subsets of the domain of a function will be denoted by Roman letters, while

elements and subsets of the range will be Greek. This convention will be

abandoned if the circumstances warrant it.

The reader is assumed to be familiar with spectral theory, including

the properties of the Fredholm index as contained in the last chapter of

[12]. A few concepts are recalled here.

For an operator T , c(T) , ae(T) , ale(T) , and a

r e

(T) denote the

spectrum, essential spectrum, left essential spectrum, and right essential

spectrum, respectively, of T . Let alre(T) = Jle(T) n are(T) . For any operator

T , nul T = dim [ ker T ] and for X e alre(T) , ind ( X - T) = nul (X - T) -

nul (X - T)* .

For an operator T , a0(T) denotes the isolated eigenvalues of T such

that the corresponding Riesz idempotent ha s finite rank. If n is an

extended integer, Pn{T) = { X e C : X - T is semi-Fredholm and ind (X - T)

= n } ; let P±(T) = U { Pn(T) : n * 0 } and P

±

JT) = P.JT) u P

+

JT) .

For any subset F of C let F

g

= { z e C : dist (z, F) 8 } .

The proof of the next theorem, a refinement of the Spectral Mapping

Theorem, is left to the reader.

1.1 Theorem Let A e (B(9{) and let f be an analytic function defined in a

6