Property $(aBw)$ and perturbations

Abstract

A bounded linear operator T acting on a Banach space X satisfies property (aBw), a strong version of a-Weyl's theorem, if the complement in the approximate point spectrum σ a (T ) of the upper semi-B-Weyl spectrum σ U SBW (T ) is the set of all isolated points of approximate point spectrum which are eigenvalues of finite multiplicity. In this paper we investigate the property (aBw) in connection with Weyl type theorems. In particular, we show that T satisfies property (aBw) if and only if T satisfies a-Weyl's theorem and σ U SBW (T ) = σ U SW (T ), where σ U SW (T ) is the upper semi-Weyl spectrum of T . The preservation of property (aBw) is also studied under commuting nilpotent, quasi-nilpotent, power finite rank or Riesz perturbations. The theoretical results are illustrated by some concrete examples.