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 $\sigma_{a}(T)$ of the upper semi-B-Weyl spectrum $\sigma_{USBW}(T)$ 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 $\sigma_{USBW}(T)=\sigma_{USW}(T)$, where $\sigma_{USW}(T)$ 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.