Single valued extension property and generalized Weyl’s theorem

Abstract

Let $T$ be an operator acting on a Banach space $X$, let $\sigma (T)$ and $ \sigma _{BW}(T) $ be respectively the spectrum and the B-Weyl spectrum of $T$. We say that $T$ satisfies the generalized Weyl’s theorem if $ \sigma _{BW}(T)= \sigma (T) \setminus E(T)$, where $E(T)$ is the set of all isolated eigenvalues of $T$. The first goal of this paper is to show that if $T$ is an operator of topological uniform descent and $0$ is an accumulation point of the point spectrum of $T,$ then $T$ does not have the single valued extension property at $0$, extending an earlier result of J. K. Finch and a recent result of Aiena and Monsalve. Our second goal is to give necessary and sufficient conditions under which an operator having the single valued extension property satisfies the generalized Weyl’s theorem.