Generalized invertibility in two semigroups of Banach algebras

Year 2023, , 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

References

• [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. Benítez, 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-González 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. Cvetkovic-Ilic, D.S. Djordjevic and Y.M. Wei, Additive results for the generalized Drazin inverse in a Banach algebra, Linear Algebra Appl. 418 (1), 53–61, 2006.
• [6] D.S. Cvetkovic-Ilic, X.J. Liu and Y.M. Wei, Some additive results for the generalized Drazin inverse in a Banach algebra, Electron. J. Linear Algebra 22, 1049–1058, 2011.
• [7] C.Y. Deng and Y.M. Wei, New additive results for the generalized Drazin inverse, J. Math. Anal. Appl. 370, 313–321, 2010.
• [8] M.P. Drazin, Pseudo-inverse in associative rings and semigroups, Amer. Math. Monthly 65, 506–514, 1958.
• [9] Q.L. Huang, L.P. Zhu, X.R. Chen and C. Zhang, On stable perturbations of the generalized Drazin inverses of closed linear operators in Banach spaces, Abstr. Appl. Anal. 131040, 2012.
• [10] J.J. Koliha, A generalized Drazin inverse, Glasgow Math. J. 38, 367–381, 1996.
• [11] J.J. Koliha, D.S. Cvetkovic-Ilic and C.Y. Deng, Generalized Drazin invertibility of combinations of idempotents, Linear Algebra Appl. 437, 2317–2324, 2012.
• [12] M.Z. Kolundžija, D. Mosic and D.S. Djordjevic, Further results on the generalized Drazin inverse of block matrices in Banach algebras, Bull. Malays. Math. Sci. Soc. 38, 483–498, 2015.
• [13] X.J. Liu and Y.H. Qin, Perturbation bound for the generalized Drazin inverse of an operator in Banach space, Filomat 31 (16), 5177–5191, 2017.
• [14] D. Mosic, Reverse order laws for the generalized Drazin inverse in Banach algebras, J. Math. Anal. Appl. 429, 461–477, 2015.
• [15] D. Mosic, Additive results for the generalized Drazin inverse in a Banach algebra, Bull. Malays. Math. Sci. Soc. 40 (4), 1465–1478, 2017.
• [16] P. Patrício and R. Puystjens, Generalized invertibility in two semigroups of a ring, Linear Algebra Appl. 377, 125–139, 2004.
• [17] Z. Wang and J.L. Chen, Pseudo Drazin inverses in associative rings and Banach algebras, Linear Algebra Appl. 437 (6), 1332–1345, 2012.
• [18] Z.L. Ying and J.L. Chen, On quasipolar rings, Algebra Colloq. 19 (4), 683–692, 2012.
• [19] H.H. Zhu and J.L. Chen, Additive property of pseudo Drazin inverse of elements in a Banach algebra, Filomat 28 (9), 1773–1781, 2014.
• [20] H.L. Zou and J.L. Chen, On the pseudo Drazin inverse of the sum of two elements in a Banach algebra, Filomat 31 (7), 2011–2022, 2017.
• [21] H.L. Zou, D. Mosic and J.L. Chen, Generalized Drazin invertibility of product and sum of two elements in a Banach algebra and its applications, Turkish J. Math. 41, 548–563, 2017.