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

Kouider Miloud Hocine ,Bekkai Messirdi ,Mohammed Benharrat

doi:10.28924/2291-8639-17-2019-105

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.

Highlights

Introduction and preliminariesLet B(H) be the Banach algebra of all bounded linear operators acting on an infinite-dimensional complex Hilbert space H
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 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

Summary

Introduction and preliminaries

Let B(H) be the Banach algebra of all bounded linear operators acting on an infinite-dimensional complex Hilbert space H. An operator T ∈ B(H) is said to be right generalized Drazin invertible if K(T ) is closed and complemented with a subspace N in H such that T (N ) ⊂ N ⊆ H0(T ). The basic existence results of generalized Drazin inverses and its relation to the quasi-nilpotent part and the analytical core are summarized in the following theorems. The following assertions are equivalent: (i) T is generalized Drazin invertible, (ii) 0 is an isolated point in the spectrum σ(T ) of T ;. The following assertions are equivalent: (i) T is left generalized Drazin invertible; (ii) 0 is an isolated point in σap(T );. The following assertions are equivalent: (i) T is right generalized Drazin invertible; (ii) 0 is an isolated point in σsu(T );. Right) generalized Drazin invertibility of MC using the isolated point in the approximate spectrum

The quasi-nilpotent part and the analytical core of the operator MC

Left and right generalized Drazin invertibility of MC

Application to a spectral boundary value matrix problem