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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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