A Note on Property (g^b) and Perturbations

An operator T∈ℬ(X) defined on a Banach space X satisfies property (gb) if the complement in the approximate point spectrum σa(T) of the upper semi-B-Weyl spectrum σSBF+-(T) coincides with the set Π(T) of all poles of the resolvent of T. In this paper, we continue to study property (gb) and the stability of it, for a bounded linear operator T acting on a Banach space, under perturbations by nilpotent operators, by finite rank operators, and by quasinilpotent operators commuting with T. Two counterexamples show that property (gb) in general is not preserved under commuting quasi-nilpotent perturbations or commuting finite rank perturbations.