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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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##### References:
 [1] Barnes, BA, Common operator properties of the linear operators $$RS$$ and $$SR$$, Proc. Am. Math. Soc., 126, 1055-1061, (1998) · Zbl 0890.47004 [2] Benhida, C; Zerouali, EH, Local spectral theory of linear operators $$RS$$ and $$SR$$, Integral Equ. Oper. Theory, 54, 1-8, (2006) · Zbl 1104.47006 [3] Berkani, M, Index of B-Fredholm operators and generalization of a Weyl theorem, Proc. Am. Math. Soc., 130, 1717-1723, (2002) · Zbl 0996.47015 [4] Berkani, M; Sarih, M, An atkinson-type theorem for B-Fredholm operators, Stud. Math., 148, 251-257, (2001) · Zbl 1005.47012 [5] Cline, R.E.: An application of representation for the generalized inverse of a matrix, MRC Technical Report, vol. 592 (1965) · Zbl 1305.47004 [6] Corach, G; Duggal, B; Harte, R, Extensions of jacobsons lemma, Commun. Algebra, 41, 520-531, (2013) · Zbl 1269.47002 [7] Drazin, MP, Pseudo-inverse in associative rings and semigroups, Am. Math. Mon., 65, 506-514, (1958) · Zbl 0083.02901 [8] Harte, R, On quasinilpotents in rings, Panam. Math. J., 1, 10-16, (1991) · Zbl 0761.16009 [9] Koliha, JJ, A note on generalized Drazin inverse, Glasg. Math. J., 38, 367-381, (1996) · Zbl 0897.47002 [10] Koliha, JJ; Patrício, P, Elements of rings with equal spectral idempotents, J. Aust. Math. Soc., 72, 137-152, (2002) · Zbl 0999.16025 [11] Liao, YH; Chen, JL; Cui, J, Cline’s formula for the generalized Drazin inverse, Bull. Malays. Math. Sci. Soc., 37, 37-42, (2014) · Zbl 1298.15009 [12] Lian, H; Zeng, Q, An extension of cline’s formula for generalized Drazin inverse, Turk. Math. J., 40, 161-165, (2016) · Zbl 1438.16069 [13] Lin, C; Yan, Z; Ruan, Y, Common properties of operators $$RS$$ and $$SR$$ and $$p$$-hyponormal operators, Integral Equ. Oper. Theory, 43, 313-325, (2002) · Zbl 1015.47018 [14] Patrício, P; Hartwig, RE, Some additive results on Drazin inverses, Appl. Math. Comput., 215, 530-538, (2009) · Zbl 1182.65067 [15] Wang, Z; Chen, JL, Pseudo Drazin inverses in associative rings and Banach algebras, Linear Algebra Appl., 437, 1332-1345, (2012) · Zbl 1262.47002 [16] Yan, K; Fang, X, Common properties of the operator products in spectral theory, Ann. Funct. Anal., 6, 60-69, (2015) · Zbl 1334.47003 [17] Yan, K; Fang, X, Common properties of the operator products in local spectral theory, Acta Math. Sin. Engl. Ser., 31, 1715-1724, (2015) · Zbl 1327.47004 [18] Zeng, QP; Zhong, HJ, New results on common properties of the bounded linear operators $$RS$$ and $$SR$$, Acta Math. Sin. (Engl. Ser.), 29, 1871-1884, (2013) · Zbl 1305.47004 [19] Zeng, QP; Zhong, HJ, Common properties of bounded linear operators $$AC$$ and $$BA$$: local spectral theory, J. Math. Anal. Appl., 414, 553-560, (2014) · Zbl 1308.47002 [20] Zeng, QP; Zhong, HJ, New results on common properties of the products $$AC$$ and $$BA$$, J. Math. Anal. Appl., 427, 830-840, (2015) · Zbl 1327.47001 [21] Zguitti, H, A note on Drazin invertibility for upper triangular block operators, Mediterr. J. Math., 10, 1497-1507, (2013) · Zbl 1304.47005
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