THE P-DRAZIN INVERSE FOR OPERATOR MATRIX OVER BANACH ALGEBRAS

Abstract

An element $\mathcal{A}$ in a Banach algebra $\mathcal{A}$ has p-Drazin inverse provided that there exists $b\in comm(a)$ such that $b=b^2a, a^k-a^{k+1}b
\in J(\mathcal{A})$ for some $k\in {\Bbb N}$. In this paper, we present new conditions for a block operator matrix has p-Drazin inverse. As applications, we prove the p-Drazin invertibility of the block operator matrix under certain spectral conditions.