@Article{CMR-37-484,
author = {Chen , JianlongZhu , Zhengqian and Shi , Guiqi},
title = {The Pseudo Drazin Inverses in Banach Algebras},
journal = {Communications in Mathematical Research },
year = {2021},
volume = {37},
number = {4},
pages = {484--495},
abstract = {<p style="text-align: justify;">Let $\mathscr{A}$ be a complex Banach algebra and $J$ be the Jacobson radical of&nbsp;<span style="text-align: justify;">$\mathscr{A}$</span>. (1) We firstly show that $a$ is generalized Drazin invertible in <span style="text-align: justify;">$\mathscr{A}$</span> if and only
if $a+J$ is generalized Drazin invertible in&nbsp;<span style="text-align: justify;">$\mathscr{A}$</span>/$J$. Then we prove that $a$ is pseudo
Drazin invertible in <span style="text-align: justify;">$\mathscr{A}$</span> if and only if $a+J$ is Drazin invertible in&nbsp;<span style="text-align: justify;">$\mathscr{A}$</span>/$J$. As its
application, the pseudo Drazin invertibility of elements in a Banach algebra
is explored. (2) The pseudo Drazin order is introduced in <span style="text-align: justify;">$\mathscr{A}$</span>. We give the
necessary and sufficient conditions under which elements in&nbsp;<span style="text-align: justify;">$\mathscr{A}$</span> have pseudo
Drazin order, then we prove that the pseudo Drazin order is a pre-order.</p>},
issn = {2707-8523},
doi = {https://doi.org/10.4208/cmr.2021-0013},
url = {http://global-sci.org/intro/article_detail/cmr/19440.html}
}