Generalized invertibility in two semigroups of Banach algebras

Year 2023, Volume: 52 Issue: 4, 956 - 964, 15.08.2023

Wende Lı , Jianlong Chen , Yuanyuan Ke

https://doi.org/10.15672/hujms.1099257

Abstract

Motivated by the results involving Drazin inverses of Patrício and Puystjens, we investigate the relevant results for pseudo Drazin invertibility and generalized Drazin invertibility in two semigroups of Banach algebras. Given a Banach algebra $\mathcal{A}$ and $e^2=e\in \mathcal{A}$, we firstly establish the relation between pseudo Drazin invertibility (resp., generalized Drazin invertibility) of elements in $e\mathcal{A}e$ and $e\mathcal{A}e+1-e$. Then this result leads to a remarkable behavior of pseudo Drazin invertibility (resp., generalized Drazin invertibility) between the operators in the semigroup $AA^{-}\mathscr{B}(Y)AA^{-}+I_Y-AA^{-}$ and the semigroup $A^{=}A\mathscr{B}(X)A^{=}A+I_X-A^{=}A$, where $A^{-}, A^{=}\in \mathscr{B}(Y,X)$ are inner inverses of $A\in \mathscr{B}(X,Y)$.

Keywords

generalized Drazin invertibility, pseudo Drazin invertibility, Banach algebra, bounded linear operator

Supporting Institution

National Natural Science Foundation of China and Qing Lan Project of Jiangsu Province

Project Number

12171083, 11871145, 12071070, 11901245

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