Additive property of pseudo Drazin inverse of elements in a Banach algebra

Abstract

We study properties of pseudo Drazin inverse in a Banach algebra with unity 1. If $ab=ba$ and $a,b$ are pseudo Drazin invertible, we prove that $a+b$ is pseudo Drazin invertible if and only if $1+a^{‡}b$ is pseudo Drazin invertible. Moreover, the formula of $(a+b{)}^{‡}$ is presented . When the commutative condition is weaken to $ab=\lambda ba(\lambda \ne 0)$, we also show that $a-b$ is pseudo Drazin invertible if and only if $a{a}^{‡}(a-b)b{b}^{‡}$ is pseudo Drazin invertible.