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 - <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>