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 $(A+YGZ^{\ast})^{\odot}=A^{\odot}-A^{\odot}Y(G^{\odot}+Z^{\ast}A^{\odot}Y)^{\odot}Z^{\ast}A^{\odot}$ holds,
where $A^{\odot}$ 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 $A$.