Left and right generalized Drazin invertible operators and Local spectral theory

Abstract

In this paper, we give some characterizations of the left and right generalized Drazin invertible bounded operators in Banach spaces by means of the single-valued extension property (SVEP). In particular, we show that a bounded operator is left (resp. right) generalized Drazin invertible if and only if admits a generalized Kato decomposition and has the SVEP at 0 (resp. it admits a generalized Kato decomposition and its adjoint has the SVEP at 0. In addition, we prove that both of the left and the right generalized Drazin operators are invariant under additive commuting finite rank perturbations. Furthermore, we investigate the transmission of some local spectral properties from a bounded linear operator, as the SVEP, Dunford property , and property , to its generalized Drazin inverse. Subjects: Spectral Theory (math. SP) MSC classes: 47A10 Cite as: arXiv: 1512.02623 [math. SP] (or arXiv: 1512.02623 v2 [math. SP] for this version) Submission history From: Mohammed Benharrat Dr.[view email] [v1] Tue, 8 Dec 2015 20: 42: 56 UTC (326 KB) [v2] Fri, 15 Apr 2016 08: 26: 08 UTC (14 KB)