@article{article_1099257, title={Generalized invertibility in two semigroups of Banach algebras}, journal={Hacettepe Journal of Mathematics and Statistics}, volume={52}, pages={956–964}, year={2023}, DOI={10.15672/hujms.1099257}, author={Lı, Wende and Chen, Jianlong and Ke, Yuanyuan}, keywords={generalized Drazin invertibility, pseudo Drazin invertibility, Banach algebra, bounded linear operator}, 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 $A\in \mathscr{B}(X,Y)$.}, number={4}, publisher={Hacettepe University}, organization={National Natural Science Foundation of China and Qing Lan Project of Jiangsu Province}