Left and Right Generalized Drazin Invertibility of an Upper Triangular Operator Matrices with Application to Boundary Value Problems

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Kouider Miloud Hocine, Bekkai Messirdi, Mohammed Benharrat, Left and Right Generalized Drazin Invertibility of an Upper Triangular Operator Matrices with Application to Boundary Value Problems, Int. J. Anal. Appl., 17 (1) (2019), 105-121.

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Kouider Miloud Hocine, Bekkai Messirdi, Mohammed Benharrat

Abstract

When A ∈ B(H) and B ∈ B(K) are given, we denote by MC the operator on the Hilbert space H ⊕ K of the form MC = (A C 0 B). In this paper we investigate the quasi-nilpotent part and the analytical core for the upper triangular operator matrix MC in terms of those of A and B. We give some necessary and sufficient conditions for MC 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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