The Pseudo Drazin Inverses in Banach Algebras (Communications in Mathematical Research, 2022)
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 onlyif $a+J$ is generalized Drazin invertible in $\mathscr{A}$/$J$. Then we prove that $a$ is pseudoDrazin invertible in $\mathscr{A}$ if and only if $a+J$ is Drazin invertible in $\mathscr{A}$/$J$. As itsapplication, the pseudo Drazin invertibility of elements in a Banach algebrais explored. (2) The pseudo Drazin order is introduced in $\mathscr{A}$. We give thenecessary and sufficient conditions under which elements in $\mathscr{A}$ have pseudoDrazin order, then we prove that the pseudo Drazin order is a pre-order.
Keywords: Drazin inverse; pseudo Drazin inverse; generalized Drazin inverse.