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arXiv · 2104.09846

Extensions of Jacobson's lemma for generalized inverses in a ring

Yanxun Ren·Lining Jiang
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Abstract

Let $R$ be an associative ring with unit $1$, and $a, b, c\in R$ satisfy $a(ba)^{2}=abaca=acaba=(ac)^{2}a$, this paper proves that $1-ac$ has generalized Drazin inverse (Drazin inverse, pseudo Drazin inverse, respectively) if and only if $1-ba$ has generalized Drazin inverse (Drazin inverse, pseudo Drazin inverse, respectively). In particular, we obtain new common spectral properties for $ac$ and $ba$ in Banach algebras. As applications, new extension of Jacobson's lemma for B-Fredholm elements and generalized Fredholm elements in rings is established.

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Yanxun Ren, Lining Jiang. 2021-04-20. Extensions of Jacobson's lemma for generalized inverses in a ring. https://arxiv.org/abs/2104.09846

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