Approximate point spectrum and commuting compact perturbations

Abstract

Let X be an infinite-dimensional complex Banach space and denote the set of bounded (compact) linear operators on X by B ( X ) ( K ( X )). Let σ( A ) and σ a ( A ) denote, respectively, the spectrum and approximate point spectrum of an element A of B ( X ). Set σ em ( A )and σ eb ( A ) are respectively Schechter's and Browder's essential spectrum of A ([ 16 ], [ 9 ]). σ ea ( A ) is a non-empty compact subset of the set of complex numbers ℂ and it is called the essential approximate point spectrum of A ([ 13 ], [ 14 ]). In this note we characterize σ ab ( A ) and show that if f is a function analytic in a neighborhood of σ( A ), then σ ab ( f ( A )) = f (σ ab ( A )). The relation between σ a ( A ) and σ eb ( A ), that is exhibited in this paper, resembles the relation between the σ( A ) and the σ eb ( A ), and it is reasonable to call σ ab ( A ) Browder's essential approximate point spectrum of A .