Volume 37, Issue 4
The Pseudo Drazin Inverses in Banach Algebras

Commun. Math. Res., 37 (2021), pp. 484-495.

Published online: 2021-08

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• 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.

• Keywords

Drazin inverse, pseudo Drazin inverse, generalized Drazin inverse.

15A09, 16U90, 46H05

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@Article{CMR-37-484, author = {Jianlong and Chen and and 18040 and and Jianlong Chen and Zhengqian and Zhu and and 18041 and and Zhengqian Zhu and Guiqi and Shi and and 18042 and and Guiqi Shi}, title = {The Pseudo Drazin Inverses in Banach Algebras}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {37}, number = {4}, pages = {484--495}, 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.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2021-0013}, url = {http://global-sci.org/intro/article_detail/cmr/19440.html} }
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.

Jianlong Chen, Zhengqian Zhu & GuiqiShi. (2021). The Pseudo Drazin Inverses in Banach Algebras. Communications in Mathematical Research . 37 (4). 484-495. doi:10.4208/cmr.2021-0013
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