Glasnik Matematicki, Vol. 57, No. 1 (2022), 1-15. \( \)

JACOBSON'S LEMMA FOR THE GENERALIZED \(n\)-STRONG DRAZIN INVERSES IN RINGS AND IN OPERATOR ALGEBRAS

Yanxun Ren and Lining Jiang

School of Mathematics and Statistics, Beijing Institute of Technology, 100081 Beijing, China
e-mail:renyanxun110@126.com

School of Mathematics and Statistics, Beijing Institute of Technology, 100081 Beijing, China
e-mail:jianglining@bit.edu.cn

Abstract.   In this paper, we extend Jacobson's lemma for Drazin inverses to the generalized \(n\)-strong Drazin inverses in a ring, and prove that \(1-ac\) is generalized \(n\)-strong Drazin invertible if and only if \(1-ba\) is generalized \(n\)-strong Drazin invertible, provided that \(a(ba)^{2}=abaca=acaba=(ac)^{2}a\). In addition, Jacobson's lemma for the left and right Fredholm operators, and furthermore, for consistent in invertibility spectral property and consistent in Fredholm and index spectral property are investigated.

2020 Mathematics Subject Classification.   15A09, 47A53

Key words and phrases.   Jacobson's lemma, generalized \(n\)-strong Drazin inverse, Fredholm operator, consistent in invertibility

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https://doi.org/10.3336/gm.57.1.01

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