Left and right generalized Drazin invertible operators and local spectral theory

Keywords: Left and right generalized Drazin invertible operators, Generalized Drazin invertible operators, SVEP, 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 (C), and property (β), to its generalized Drazin inverse.

Author Biographies

Mohammed Benharrat, Ecole Nationale Polytechnique d’Oran-Maurice Audin.
Département de Mathématiques et Informatique, LMFAO.
Kouider Miloud Hocine, Université des Sciences et de la Technologie d'Oran Mohamed-Boudiaf.
Département de Mathématiques, LMFO.
Bekkai Messirdi, Ecole Supérieure en Génie électrique et energétique d’Oran.
LMFAO.

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Published
2019-12-15
How to Cite
[1]
M. Benharrat, K. M. Hocine, and B. Messirdi, “Left and right generalized Drazin invertible operators and local spectral theory”, Proyecciones (Antofagasta, On line), vol. 38, no. 5, pp. 897-919, Dec. 2019.
Section
Artículos