The Drazin inverse for perturbed block-operator matrices

Abstract

We present new formulas of the Drazin inverse for anti-triangular block-operator matrices. If $B^{\pi}A^DB=0, B^{\pi}AB^D=0$ and $B^{\pi}ABA^{\pi}=0$, the explicit representation of the Drazin inverse of a block-operator triangular matrix $\left(
\begin{array}{cc}
A&I\\
B&0
\end{array}
\right)$ is given. Thus a generalization of [A note on the Drazin inverse for an anti-triangular matrix, Linear Algebra Appl., {\bf 431}(2009), 1910--1922] is obtained. Some applications to full block-operator matrices are thereby considered.