The Sherman-Morrison-Woodbury formula for the generalized inverses

Abstract

In this paper, we investigate the Sherman-Morrison-Woodbury formula for the {1}-inverses and the {2}-inverses of bounded linear operators on a Hilbert space. Some conditions are established to guarantee that (A+YGZ*)? = A? ?A?Y(G? +Z*A?Y)?Z*A? holds, where A? stands for any kind of standard inverse, {1}-inverse, {2}-inverse, Moore-Penrose inverse, Drazin inverse, group inverse, core inverse and dual core inverse of A.