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 is pseudo B-Weyl if , where is a Weyl operator and is a quasi-nilpotent operator. We show that the corresponding pseudo B-Weyl spectrum satisfies the equality where is the generalized Drazin spectrum of and (resp., ) is the set where (resp., ) 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.