Let f be any analytic function defined in a neighborhood of the non-
empty set E and let 5(E) denote the set of all operator having spectrum
included in E . In this paper the closure and interior of the set f(5(E)) =
{ f(A) : A e 5(E) } a r e characterized. In addition the sets cl[ int{ f(5(E)) } ] ,
int{ cl[ f(5(E)) ] ) , cl[ int{ cl[ f(5(E)) ] } ] , and int{ cl[ int{ f(5(E)) } ] } are
characterized. Several examples and applications are given.
Key words and phrases: Functional calculus, analytic function, spectrum,
Fredholm index, non-abelian approximation.
John B. Conway and Domingo A. Herrero were partially supported during the
preparation of this paper by grants from the National Science Foundation.
Received by the editors Jun e 7, 1988.
Received in revised form December 15, 1988.
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