Further results on common properties of the products ac and bd

Zeng, Qingping; Yan, Kai; Wu, Zhenying

doi:10.3336/gm.55.2.07

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2022

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“…More recently, in [26], the author generalized Jacobson's lemma to generalized strong Drazin inverses. In [31], Zeng and Yan considered Jacobson's lemma for the generalized Hirano inverses. In [9], Chen and Sheibani investigated Jacobson's lemma for the generalized n-strongly Drazin inverses.…”

Section: Introductionmentioning

confidence: 99%

Jacobson's lemma for the generalized \(n\)-strong Drazin inverses in rings and in operator algebras

Ren¹,

Jiang²

2022

Glas. Mat. Ser. III

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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.

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Section: Introductionmentioning

confidence: 99%