LEFT AND RIGHT GENERALIZED DRAZIN INVERTIBILITY OF AN UPPER TRIANGULAR OPERATOR MATRICES WITH APPLICATION TO BOUNDARY VALUE PROBLEMS

When A is an element of B(H) and B is an element of B(K) are given, we denote by M-C the operator on the Hilbert space H circle plus K of the form M-C = (A C 0 B) . 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 is an element of 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.