4

CONWAY, HERRERO, AND MORREL

characterize cl 5(E) (as done in [6]) as well as int 5(E) , cl[ int 5(E) ] , and

so on. We also separate out the results for biquasitriangular and compact

operators. The finite dimensional results, while interesting, are mostly left

to the reader.

In Section 1, some preliminary material is presented, including most

of results from non-abelian approximation theory that will be used here. A

finer version of the Spectral Mapping Theorem is also given which analyzes

the various parts of the spectrum of f(A) and relates it to the parts of the

spectrum of A .

Section 2 presents the main theorem characterizing cl[ f(5(E)) ], to-

gether with its corollaries. In particular, a necessary and sufficient

condition on f is given for cl[ f(5(E)) ] to be all of *B(J{) . It is interesting to

see tha t cl[ f(5(E)) ] = B(tt) if and only if f(5(E)) = $(?{). The

characterization of the analytic functions f for which f(5(E)) = ^Bffl) has

already been obtained by A Brown [9]. In this section it will also be shown

how to obtain the results of [13] characterizing cl[ f(5(E)) ] for the special

case in which E = C and f(z) = TP or

ez

.

Section 3 contains a numbe r of examples illustrating the main

theorem and some of the results that come later in the paper.

Section 4 considers when cl[ f(5(E)) ] is invariant under compact

perturbations. This happens if E = C and f(z) = z

p

or

ez

, but is not true in

general. One clearly necessary condition for cl[ f(5(E)) ] to be invariant

under compact perturbations is that f(E) must be dense. Even if E = C and f

is a polynomial, however, it is not necessarily true that T + K e cl[ f(5(E)) ]

when T e cl[ f(5(E)) ] and K is compact. (See Example 3.8.) Theorem 4.4

provides a necessary and sufficient condition for cl[ f(5(E)) ] to be invariant

under compact perturbations. Moreover, using the results of Chapter 12 of

[3] it is possible to determine a formula for the distance from an operator to

the set cl[ f(5(E)) ] +

{B0(M)

(Theorem 4.2).

Section 5 characterizes the interior of the set f(5(E)) (Theorem 5.1)