The Pseudo Drazin Inverses in Banach Algebras

Abstract

Let $\mathscr{A}$ be a complex Banach algebra and $J$ be the Jacobson radical of $\mathscr{A}$. (1) We firstly show that $a$ is generalized Drazin invertible in $\mathscr{A}$ if and only if $a+J$ is generalized Drazin invertible in $\mathscr{A}$/$J$. Then we prove that $a$ is pseudo Drazin invertible in $\mathscr{A}$ if and only if $a+J$ is Drazin invertible in $\mathscr{A}$/$J$. As its application, the pseudo Drazin invertibility of elements in a Banach algebra is explored. (2) The pseudo Drazin order is introduced in $\mathscr{A}$. We give the necessary and sufficient conditions under which elements in $\mathscr{A}$ have pseudo Drazin order, then we prove that the pseudo Drazin order is a pre-order.