The generalized Drazin inverse of operator matrices

Abstract

Representations for the generalized Drazin inverse of an operator matrix $\begin{pmatrix} A & B\\C& D \end{pmatrix}$ are presented in terms of $A,B,C,D$ and the generalized Drazin inverses of $A,D,$ under the condition that $BD^d = 0$, and $BD_iC = 0$, for any nonnegative integer i. Using the representation, we give a new additive result of the generalized Drazin inverse for two bounded linear operators $P,Q \in B(X)$ with $PQ^d = 0$ and $PQ_iP = 0$, for any integer $ i \geqslant 1$. As corollaries, several well-known results are generalized