Generalized Jacobson's lemma in a Banach algebra

Published 11 Jun 2020 in math.RA | (2006.06736v1)

Abstract: Let A be a Banach algebra, and let a; b; c 2 A satisfying a(ba)² = abaca = acaba = (ac)^2a: We prove that 1 - ba\in A^d if and only if 1 - ac \in A^d. In this case, (1-ac)^d =1-a(1-ba)^{{\pi}(1-\alpha(1+ba))^{-1}bac} (1+ac)+a((1-ba)^d)bac. This extends the main result on g-Drazin inverse of Corach (Comm. Algebra, 41(2013), 520{531).

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