On deriving the generalized Drazin inverse of block matrices in a Banach algebra

Abstract

This paper is devoted to the generalized Drazin inverse of a block matrix x = [a b c d] in a Banach algebra A, under specic conditions. We focus on deriving formulae for the generalized Drazin inverse of x in terms of the generalized Drazin inverses of the elements a, a^πbc, a²a^d + aa^dbca^d and the generalized Schur complement s = d - ca^db.

Keywords: Generalized Drazin inverse, Schur complement, block matrix