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

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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.

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References

  1. Barnes, B.A.: Common operator properties of the linear operators \(RS\) and \(SR\). Proc. Am. Math. Soc. 126, 1055–1061 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Benhida, C., Zerouali, E.H.: Local spectral theory of linear operators \(RS\) and \(SR\). Integral Equ. Oper. Theory 54, 1–8 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berkani, M.: Index of B-Fredholm operators and generalization of a Weyl theorem. Proc. Am. Math. Soc. 130(6), 1717–1723 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berkani, M., Sarih, M.: An Atkinson-type theorem for B-Fredholm operators. Stud. Math. 148(3), 251–257 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cline, R.E.: An application of representation for the generalized inverse of a matrix, MRC Technical Report, vol. 592 (1965)

  6. Corach, G., Duggal, B., Harte, R.: Extensions of Jacobsons lemma. Commun. Algebra 41, 520–531 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Drazin, M.P.: Pseudo-inverse in associative rings and semigroups. Am. Math. Mon. 65, 506–514 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  8. Harte, R.: On quasinilpotents in rings. Panam. Math. J. 1, 10–16 (1991)

    MathSciNet  MATH  Google Scholar 

  9. Koliha, J.J.: A note on generalized Drazin inverse. Glasg. Math. J. 38, 367–381 (1996)

    Article  MATH  Google Scholar 

  10. Koliha, J.J., Patrício, P.: Elements of rings with equal spectral idempotents. J. Aust. Math. Soc. 72, 137–152 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Liao, Y.H., Chen, J.L., Cui, J.: Cline’s formula for the generalized Drazin inverse. Bull. Malays. Math. Sci. Soc. 37(2), 37–42 (2014)

    MathSciNet  MATH  Google Scholar 

  12. Lian, H., Zeng, Q.: An extension of Cline’s formula for generalized Drazin inverse. Turk. Math. J. 40, 161–165 (2016)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Patrício, P., Hartwig, R.E.: Some additive results on Drazin inverses. Appl. Math. Comput. 215, 530–538 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Wang, Z., Chen, J.L.: Pseudo Drazin inverses in associative rings and Banach algebras. Linear Algebra Appl. 437, 1332–1345 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yan, K., Fang, X.: Common properties of the operator products in spectral theory. Ann. Funct. Anal. 6(4), 60–69 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Yan, K., Fang, X.: Common properties of the operator products in local spectral theory. Acta Math. Sin. Engl. Ser. 31(11), 1715–1724 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zeng, Q.P., Zhong, H.J.: New results on common properties of the bounded linear operators \(RS\) and \(SR\). Acta Math. Sin. (Engl. Ser.) 29, 1871–1884 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zeng, Q.P., Zhong, H.J.: Common properties of bounded linear operators \(AC\) and \(BA\): local spectral theory. J. Math. Anal. Appl. 414, 553–560 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zeng, Q.P., Zhong, H.J.: New results on common properties of the products \(AC\) and \(BA\). J. Math. Anal. Appl. 427, 830–840 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zguitti, H.: A note on Drazin invertibility for upper triangular block operators. Mediterr. J. Math. 10, 1497–1507 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are grateful to the referee for helpful comments concerning this paper.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Mississippi State University, Drawer Ma, Mississippi State, MS, 39762, USA

    V. G. Miller

  2. Department of Mathematics and Computer Science, Multidisciplinary Faculty of Nador, Mohammed First University, PO Box 300, 62702, Selouane, Nador, Morocco

    H. Zguitti

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  1. V. G. Miller
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Miller, V.G., Zguitti, H. New extensions of Jacobson’s lemma and Cline’s formula. Rend. Circ. Mat. Palermo, II. Ser 67, 105–114 (2018). https://doi.org/10.1007/s12215-017-0298-6

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