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)
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