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.2010 Mathematics Subject Classification. 47A10.
In this paper we investigate the quasi-nilpotent part and the analytical core for the upper triangular operator matrix M C in terms of those of A and B. We give some necessary and sufficient conditions for M C to be left or right generalized Drazin invertible operator for some C ∈ B(K, H). As an application, we study the existence and uniqueness of the solution for abstract boundary value problems described by upper triangular operator matrices with right generalized Drazin invertible component.
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