In this paper, we give expressions for the generalized Drazin inverse of a (2,2,0) block matrix over a Banach algebra under certain circumstances, utilizing which we derive the generalized Drazin inverse of a 2 × 2 block matrix in a Banach algebra under weaker restrictions. Our results generalize and unify several results in the literature.

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In this paper expressions for the Drazin inverse of a modified matrix A − CD d B are presented in terms of the Drazin inverses of A and the generalized Schur complement D − BA d C under weaker restrictions. Our results generalize and unify several results in the literature and the Sherman-Morrison-Woodbury formula.

We give some statements that are equivalent to the existence of group inverses of Peirce corner matrices of a 2 ×2 block matrix and its generalized Schur complements. As applications, several new results for the Drazin inverses of the generalized Schur complements and the 2 × 2 block matrix are obtained and some of them generalize several results in the literature.2000 Mathematics Subject Classification. 15A09; 15A30; 65F20.

In this paper we give expressions for the generalized Drazin inverse of a (2,2,0) operator matrix and a 2 × 2 operator matrix under certain circumstances, which generalizes and unifies several results in the literature.