Properties of the Local Functional Calculus

Abstract

ABSTRACT: We develop the local functional calculus for continuous operators with the SVEP, obtain-ing a spectral mapping theorem for the local spectrum and a stability theorem for the SVEP. As anapplication of this calculus, we obtain local resolvent equations. 1 Introduction The holomorphic functional calculus was developed by Dunford and Taylor in the forties (see[?], [?]). Given a continuous operator T acting on a complex Banach space X , this calculusassociates an operator f ( T ) 2 L ( X ) to each holomorphic function f defined on a neighborhood ofthe spectrum of T . Gindler [?] extended it by associating a closed operator to each meromorphicfunction of a specific class. This meromorphic functional calculus was also studied in [?].Many efforts have been devoted to extend the holomorphic functional calculus in other direc-tions. For example, assume that T 2 L ( X ) has the Single-Valued Extension Property (SVEP inshort) and let f be a holomorphic function on a neighborhood of a fixed compact subset