2 CONWAY, HERRERO, AND MORREL

open) non-empty subset E of the complex plane C , let 5(E) denote the

collection of all operators A defined on some separable Hilbert space with

a (A) c E . (Note that in the definition of 5(E) neither the Hilbert space nor

the dimension of the Hilbert space is specified or restricted, save the

restriction that the space is separable. This is done to ensure that such

statements as " A 0 B e 5(E) if and only if A and B e 5(E) " are valid without

modification or qualification.) If f is analytic in a neighborhood of E , what is

the characterization of the operators that belong to f(5(E)) = { f(A) : A e

5(E) } ? As discussed in [13] , which can be considered as the predecessor

of this paper, such a question seems beyond the present capabilities of

operator theory even for such nice functions as zP and the exponential. In

particular, such a description in terms of spectral properties alone is

impossible as two operators can be found with the same spectral picture,

only one of which has a square root.

Instead, the characterization of cl[ f(5(E)) ] , the closure of f(5(E)) , is

obtained. The methods used are those of non-abelian approximation as

presented in [20] and [3], Some background material will be presented to

ease the reader's burden.

These results provide an important illustration of the Closure

Theorem [22] that states that a closed set of operators on Hilbert space that

is similarity invariant and has "sufficient structure" (in a certain technical

sense described in [22]) can be characterized in terms of spectral

properties alone. Indeed, if A e 5(E) and R is an invertible operator, then

RAR"1

G 5(E) and

ffRAR"1)

=

RffAJR"1

. Thus cl[ f(5(E)) ] is a closed

similarity invariant set and membership in this set is characterized solely in

terms of spectral properties (Theorem 2.1).

Some ruminations and reflections seem appropriate here as a caution

and encouragement for the reader. The results of this paper may strike our

audience as extremely complicated. This is, undoubtedly, a correct

perception. In order to achieve total generality, something must be