Abstract
Motivated by the results involving Drazin inverses of Patr\'{i}cio 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 first 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 $AA^{-}\mathscr{B}(X)AA^{-}+I_X-AA^{-}$ and the semigroup $A^{=}A\mathscr{B}(Y)A^{=}A+I_Y-A^{=}A$, where $A^{-}, A^{=}\in \mathscr{B}(Y,X)$ are inner inverses of $A\in \mathscr{B}(X,Y)$.
Subject
Geometry and Topology,Statistics and Probability,Algebra and Number Theory,Analysis
Reference21 articles.
1. E. H. Benabdi and M. Barraa, The Drazin and generalized Drazin invertibility of linear combinations of idempotents, J. Math. Anal. Appl., 478, 1163--1171, 2019.
2. J. Ben\'{i}tez, X. J. Liu and Y. H. Qin, Representations for the generalized Drazin inverse in a Banach algebra, Bull. Math. Anal. Appl., 5, 53--64, 2013.
3. N. Castro-Gonz\'{a}lez and J. J. Koliha, New additive results for the g-Drazin inverse, Proc. Roy. Soc. Edinburgh Sect. A, 134 , 1085--1097, 2004.
4. H. Y. Chen and M. Sheibani, The g-Drazin inverse of the sum in Banach algebras, Linear Multilinear Algebra, 70, 53--65, 2022.
5. D. S. Cvetkovi\'{c}-Ili\'{c}, D. S. Djordjevi\'{c} and Y. M. Wei, Additive results for the generalized Drazin inverse in a Banach algebra, Linear Algebra Appl., 418 (1), 53--61, 2006.