Further Results on the Generalized Drazin Inverse of Block Matrices in Banach Algebras

Abstract

The objective of this paper is to derive formulae for the generalized Drazin inverse of a block matrix in a Banach algebra \(\mathcal{A}\) under different conditions. Let \(x=\left[ \begin{array}{c@{\quad }c} a&{}b\\ c&{}d\end{array}\right] \in \mathcal{A}\) relative to the idempotent \(p\in \mathcal{A}\) and \(a\in p\mathcal{A}p\) be generalized Drazin invertible. The formulae for the generalized Drazin inverse are obtained under the more general case that the generalized Schur complement \(s=d-ca^db\) is generalized Drazin invertible, which covers the cases that \(s\) is Drazin invertible, \(s\) is group invertible, or \(s\) is equal to zero. Thus, recent results on the Drazin inverse of block matrices and block-operator matrices are extended to a more general setting.