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

Jianlong Chen, Zhengqian Zhu & Guiqi Shi

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.

  • AMS Subject Headings

15A09, 16U90, 46H05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-37-484, author = {Chen , Jianlong and Zhu , 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 = {

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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