Generalized invertibility in two semigroups of Banach algebras

Authors : Wende LI, Jianlong CHEN, Yuanyuan KE

Pages : 956-964

Doi:10.15672/hujms.1099257

View : 135 | Download : 272

Publication Date : 2023-08-15

Article Type : Research Paper

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 insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(resp., generalized Drazin invertibility); between the operators in the semigroup $AA^{-}\\mathscr{B}insert ignore into journalissuearticles values(Y);AA^{-}+I_Y-AA^{-}$ and the semigroup $A^{=}A\\mathscr{B}insert ignore into journalissuearticles values(X);A^{=}A+I_X-A^{=}A$, where $A^{-}, A^{=}\\in \\mathscr{B}insert ignore into journalissuearticles values(Y,X);$ are inner inverses of $A\\in \\mathscr{B}insert ignore into journalissuearticles values(X,Y);$.
Keywords : generalized Drazin invertibility, pseudo Drazin invertibility, Banach algebra, bounded linear operator