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A note on formulae for the generalized Drazin inverse of anti-triangular block operator matrices in Banach spaces

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Abstract

In this paper, we separately obtain new explicit expressions for the generalized Drazin inverse of anti-triangular block operator matrices under certain restrictions. As applications, we utilize the relationship between the anti-triangular block operator matrix and a \(2\times 2\) block operator matrix to establish several formulae for the generalized Drazin inverse of a \(2\times 2\) block operator matrix. Our results generalize and unify a series of results about the Drazin inverse and the generalized Drazin inverse in the literature. We give certain numerical examples to illustrate our results.

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Acknowledgements

D. Zhang: This work was completed with the support of the National Natural Science Foundation of China (NSFC) (No. 11901079), China Postdoctoral Science Foundation (No. 2021M700751), and the Scientific and Technological Research Program Foundation of Jilin Province, China (No. JJKH20190690KJ; No. 20200401085GX; No. JJKH20220091KJ). D. Mosić: This work was completed with the support of the Ministry of Education, Science and Technological Development, Republic of Serbia (No. 174007/451-03-9/2021-14/200124) and the bilateral project between Serbia and Slovenia (Generalized inverses, operator equations and applications, No. 337-00-21/2020-09/32). The authors are thankful to the editor and the referees for their very useful comments and suggestions.

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Authors and Affiliations

  1. College of Sciences, Northeast Electric Power University, Jilin, People’s Republic of China

    Daochang Zhang & Yu Jin

  2. Faculty of Sciences and Mathematics, University of Niš, P.O. Box 224, 18000, Niš, Serbia

    Dijana Mosić

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  1. Daochang Zhang
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  2. Yu Jin
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Correspondence to Daochang Zhang.

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Communicated by Fuzhen Zhang.

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Zhang, D., Jin, Y. & Mosić, D. A note on formulae for the generalized Drazin inverse of anti-triangular block operator matrices in Banach spaces. Banach J. Math. Anal. 16, 28 (2022). https://doi.org/10.1007/s43037-022-00176-8

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