Property (gb) and perturbations

Abstract

An operator T acting on a Banach space X possesses property (gb) if σ a ( T ) ∖ σ SBF + − ( T ) = π ( T ) , where σ a ( T ) is the approximate point spectrum of T, σ SBF + − ( T ) is the essential semi- B-Fredholm spectrum of T and π ( T ) is the set of all poles of the resolvent of T. In this paper we study property (gb) in connection with Weyl type theorems, which is analogous to generalized Browderʼs theorem. Several sufficient and necessary conditions for which property (gb) holds are given. We also study the stability of property (gb) for a polaroid operator T acting on a Banach space, under perturbations by finite rank operators, by nilpotent operators and, more generally, by algebraic and Riesz operators commuting with T.