Generalization of the Sherman-Morrison-Woodbury formula involving the Schur complement

Let $X\in {\mathbb{C}}^{m\times m}$ and $Y\in {\mathbb{C}}^{n\times n}$ be nonsingular matrices, and let $N\in {\mathbb{C}}^{m\times n}$. Explicit expressions for the Moore-Penrose inverses of $M=XNY$ and a two-by-two block matrix, under appropriate conditions, have been established by Castro-Gonz'{a}lez et al. [Linear Algebra Appl. 471 (2015) 353-368]. Based on these results, we derive a novel expression for the Moore-Penrose inverse of $A+U{V}^{\ast }$ under suitable conditions, where $A\in {\mathbb{C}}^{m\times n}$, $U\in {\mathbb{C}}^{m\times r}$, and $V\in {\mathbb{C}}^{n\times r}$. In particular, if both $A$ and $I+V^{\ast }{A}^{-1}U$ are nonsingular matrices, our expression reduces to the celebrated Sherman-Morrison-Woodbury formula. Moreover, we extend our results to the bounded linear operators case.