The Drazin Inverse of the Sum of Two Bounded Linear Operators and it’s Applications

Abstract

Let $P$ and $Q$ be bounded linear operators on a Banach space. The existence of the Drazin inverse of
$P+Q$ is proved under some assumptions, and the representations of $(P+Q)^D$ are also given. The results recover the cases $P^2 Q=0, QPQ=0$ studied by Yang and Liu in \cite{YL} for matrices, $Q^2
P=0, PQP=0$ studied by Cvetkovi$\acute{c}$ and Milovanovi\'c in\cite{CM2} for operators and $P^2 Q+QPQ=0$, $P^3 Q=0$ studied by Shakoor, Yang and Ali in \cite{Abdul} for matrices. As an
application, we give representations for the Drazin inverse of the operator matrix $ M=\left(\begin{smallmatrix} A & B
\\
C & D
\end{smallmatrix}\right)
$.