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

TL;DR

This paper introduces pseudo B-Weyl operators, explores their spectra, and investigates generalized Drazin invertibility of operator matrices, providing new spectral characterizations and conditions for invertibility.

Contribution

It generalizes B-Weyl operators to pseudo B-Weyl operators and establishes spectral equalities involving the generalized Drazin spectrum.

Findings

Pseudo B-Weyl spectrum satisfies a specific spectral equality.

Conditions are provided for the generalized Drazin invertibility of operator matrices.

Spectral properties of upper triangular operator matrices are characterized.

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 $\sigma_{pBW}(T)\cup[{\mathcal S}(T)\cap{\mathcal S}(T^*)]=\sigma_{gD}(T);$ where $\sigma_{gD}(T)$ is the generalized Drazin spectrum of $T\in L(X)$ and ${\mathcal S}(T)$ (resp., ${\mathcal S} (T^*)$) is the set where $T$ (resp., $T^*$) 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.