New extensions of Jacobson’s lemma and Cline’s formula

Abstract

In an associative ring R, if elements a, b and c satisfy aba= aca then Corach et al. (Comm Algebra 41:520–531, 2013) proved that 1 - ac is (left/right) invertible if and only if 1 - ba is left/right invertible; which is an extension of the Jacobson’s lemma. Also, Lian and Zeng (Turk Math J 40:166–165, 2016) and Zeng and Zhong (J Math Anal Appl 427:830–840, 2015) proved that if the product ac is (generalized/pseudo) Drazin invertible, then so is ba extending the Cline’s formula to the case of the (generalized/pseudo) Drazin invertibility. In this paper, for elements a, b, c, d in an associative ring R satisfying (Formula presented.),we study common spectral properties for 1 - ac (resp. ac) and 1 - bd (resp. bd). So, we extend Jacobson’s lemma for (left/right) invertibility and we generalize Cline’s formula to the case of the (generalized/pseudo) Drazin invertibility. In particular, as application, for bounded linear operators A, B, C, D satisfying ACD= DBD and DBA= ACA, we show that AC is B-Weyl operator if and only if BD is B-Weyl operator.