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New extensions of Jacobson’s lemma and Cline’s formula. (English) Zbl 1390.15014
Summary: In an associative ring $$\mathcal {R}$$, if elements $$a$$, $$b$$ and $$c$$ satisfy $$aba=aca$$ then G. Corach et al. [Commun. Algebra 41, No. 2, 520–531 (2013; Zbl 1269.47002)] 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, H. Lian and Q. Zeng [“An extension of Cline’s formula for a generalized Drazin inverse”, Turk. J. Math. 40, 161–165 (2016; doi:10.3906/mat-1505-4)] and Q. Zeng and H. Zhong [J. Math. Anal. Appl. 427, No. 2, 830–840 (2015; Zbl 1327.47001)] 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 $\begin{cases} acd=dbd,\\ dba=aca,\end{cases}$ 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 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 a B-Weyl operator if and only if $$BD$$ is a B-Weyl operator.
##### MSC:
 15A09 Theory of matrix inversion and generalized inverses 16S10 Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) 47A10 Spectrum, resolvent 47A53 (Semi-) Fredholm operators; index theories
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