REPRESENTATIONS FOR THE PSEUDO DRAZIN INVERSE OF ELEMENTS IN A BANACH ALGEBRA

Abstract

In this paper, we investigate the pseudo Drazin invertibility of the sum and the product of elements in a Banach algebra $\mathscr{A}$. Given pseudo Drazin invertible elements $a$ and $b$ such that $a^2b=aba$ and $b^2a=bab$, it is shown that $ab$ is pseudo Drazin invertible and $a+b$ is pseudo Drazin invertible if and only if so is $1+a^{\ddagger} b$, and the related formulae are provided.