Generalized Drazin Invertibility of Operator Matrices

Abstract

In this paper, we investigate the generalized Drazin invertibility of upper triangular operator matrices acting on Banach spaces. Among other things, we explicit the defect set with respect to the local spectral theory. Moreover, we exhibit some sufficient conditions which assure that the generalized Drazin spectrum of a 3 × 3 upper triangular block operator matrix is the union of its diagonal entries spectra.