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

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.