Abstract
For bounded linear operators A, B, C and D on a Banach space X, we show that if BAC = BDB and CDB = CAC then I — AC is generalized Drazin—Riesz invertible if and only if I — BD is generalized Drazin—Riesz invertible, which gives a positive answer to Question 4.9 in Yan, Zeng and Zhu [Complex Anal. Oper. Theory 14, Paper No. 12 (2020)]. In particular, we show that Jacobson’s lemma holds for generalized Drazin—Riesz inverses.
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The authors are grateful to the referee for several helpful remarks and suggestions concerning this paper.
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Hadji, S., Zguitti, H. Jacobson’s Lemma for Generalized Drazin–Riesz Inverses. Acta. Math. Sin.-English Ser. 39, 481–496 (2023). https://doi.org/10.1007/s10114-023-1376-7
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DOI: https://doi.org/10.1007/s10114-023-1376-7