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
G 5(E) and
. 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
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