TY - JOUR T1 - The Pseudo Drazin Inverses in Banach Algebras AU - Chen , Jianlong AU - Zhu , Zhengqian AU - Shi , Guiqi JO - Communications in Mathematical Research VL - 4 SP - 484 EP - 495 PY - 2021 DA - 2021/08 SN - 37 DO - http://doi.org/10.4208/cmr.2021-0013 UR - https://global-sci.org/intro/article_detail/cmr/19440.html KW - Drazin inverse, pseudo Drazin inverse, generalized Drazin inverse. AB -

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.