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Some converse problems on the g-Drazin invertibility in Banach algebras

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Abstract

The main purpose of this paper is to investigate the converse problems of some well-known results related to the generalized Drazin (g-Drazin for short) inverse in Banach algebras. Let \({\mathcal {A}}\) be a Banach algebra and \(a,b\in {\mathcal {A}}\). First, we give the relationship between the Drazin (g-Drazin, group) invertibility of a, b and that of the sum \(a+b\) under certain conditions. Then, for a given polynomial \(f(x)\in {\mathbb {C}}[x]\), the g-Drazin invertibility of f(a), \(f(a^{d})\), f(ab), \(f(1-ab)\) and \(f(a+b)\) are investigated.

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Acknowledgements

The author would like to thank the referees for their helpful suggestions for the improvement of this paper. This research was supported by talent introduction project of Zhejiang Shuren University (No. 2023R025), Key Laboratory of Applied Mathematics of Fujian Province University (Putian University) (No. SX202202), China Postdoctoral Science Foundation (No. 2020M671281), the National Natural Science Foundation of China (No.11871145), NSF of Jiangsu Province (No. BK20200944).

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  1. College of Basic Science, Zhejiang Shuren University, Hangzhou, 310015, China

    Honglin Zou

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  1. Honglin Zou
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Correspondence to Honglin Zou.

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Communicated by Qing-Wen Wang.

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Zou, H. Some converse problems on the g-Drazin invertibility in Banach algebras. Ann. Funct. Anal. 15, 41 (2024). https://doi.org/10.1007/s43034-024-00344-x

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