Property (gb) through local spectral theory

Property (gb) for a bounded linear operator T ∈ L(X), on a Banach space X, means that the points λ of the approximate point spectrum for which λI − T is upper semi B-Weyl are exactly the poles of the resolvent. In this paper we shall give several characterizations, of operators T for which property (gb) holds. These characterizations are obtained by using typical tools from local spectral theory. We also show that property (gb) holds for large classes of operators and prove the stability of property (gb) under some commuting perturbations.