On pseudo B-Weyl operators and generalized Drazin invertibility for operator matrices

Abstract

We introduce a new class which generalizes the class of B-Weyl operators. We say that $T\in L(X)$ is pseudo B-Weyl if $T=T_1\oplus T_2$ where $T_1$ is a Weyl operator and $T_2$ is a quasi-nilpotent operator. We show that the corresponding pseudo B-Weyl spectrum ${\sigma }_{pBW}(T)$ satisfies the equality ${\sigma }_{pBW}(T)\cup [\mathcal{S}(T)\cap \mathcal{S}(T^{\ast })]={\sigma }_{gD}(T);$ where ${\sigma }_{gD}(T)$ is the generalized Drazin spectrum of $T\in L(X)$ and $\mathcal{S}(T)$ (resp., $\mathcal{S}(T^{\ast })$) is the set where $T$ (resp., $T^{\ast }$) fails to have SVEP. We also investigate the generalized Drazin invertibility of upper triangular operator matrices by giving sufficient conditions which assure that the generalized Drazin spectrum or the pseudo B-Weyl spectrum of an upper triangular operator matrices is the union of its diagonal entries spectra.