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

Abstract

In an associative ring \(\mathcal {R}\), 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 \(\mathcal {R}\) satisfying $$\begin{aligned} \left\{ \begin{array}{c}acd=dbd,\\ dba=aca,\end{array}\right. \end{aligned}$$ 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.