Generalized Jacobson's lemma for Drazin inverses and its applications

ABSTRACT

Let R be any associative ring with unit 1 and suppose that $a,b,c,d\in R$ satisfy $\begin{array}{rl}acd& =dbd,\\ dba& =aca;\end{array}$ we prove that 1−ac is Drazin (respectively, generalized Drazin, pseudo-Drazin) invertible if and only if 1−bd is Drazin (respectively, generalized Drazin, pseudo-Drazin) invertible. In other words, we give an affirmative answer to the conjecture of D. Mosić. Moreover, some applications to Banach algebra elements and Banach space operators are also given.

KEYWORDS:

• Jacobson's lemma
• Drazin inverse
• generalized Drazin inverse
• pseudo-Drazin inverse
• ring

Mathematics Subject Classifications:

• 15A09
• 16U99
• 47A05

Acknowledgements

The authors are highly grateful to the referees for constructive comments and careful reading of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the Office of the Fujian Provincial Education Fund (Grant No. JAT170098), the Natural Science Foundation of Fujian Province, China (Grant Nos. 2016J01014 and 2018J05004), and the Scientific Research Start-up Fund of Fuzhou University (Grant No. XRC-1674).