A Generalization of the SMW Formula of Operator A + Y G Z * to the { 2 } -Inverse Case

Abstract

The classical Sherman-Morrison-Woodbury (for short SMW) formula ${\left(A+YG{Z}^{*}\right)}^{-1}=A^{-1}-A^{-1}Y{\left(G^{-1}+Z^{*}{A}^{-1}Y\right)}^{-1}{Z}^{*}{A}^{-1}$ is generalized to the $\{2\}$-inverse case. Some sufficient conditions under which the SMW formula can be represented as ${\left(A+YG{Z}^{*}\right)}^{-}=A^{-}-A^{-}Y{\left(G^{-}+Z^{*}{A}^{-}Y\right)}^{-}{Z}^{*}{A}^{-}$ are obtained.