A Cline's formula for the generalized Drazin-Riesz inverses



Let X be Banach space, A,B,C be bounded linear operators on X satisfying operator equation ABA = ACA. In this note, we show that AC is generalized Drazin-Riesz invertible if and only if BA is generalized Drazin Riesz invertible. So, we generalize Cline’s formula to the case of the generalized Drazin-Riesz invertibility.

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P.Aiena, Fredholm and Local Spectral Theory with Applications to Multipliers, Kluwer.Acad.Press, 2004. [2] B. A. Barnes, Common operator properties of the linear operators RS and SR, Proc. Am. Math. Soc. 126(1998), 1055-1061. [3] R E. Cline, An application of representation for the generalized inverse of a matrix, MRC Technical Report 592, 1965. [4] G. Corach, B. Duggal, R. Harte, Extensions of Jacobsons lemma, Commun. Algebra 41(2013), 520-531. [5] M P. Drazin, Pseudo-inverse in associative rings and semigroups, Amer. Math. Monthly, 65(1958), 506-514. [6] JJ Koliha, A generalized Drazin inverse, Glasgow Math. J, 38(1996), 367-81. [7] H. Lian, Q. Zeng, An extension of Cline’s formula for generalized Drazin inverse, Turk. Math. J. 40(2016), 161-165. [8] Q P. Zeng, H J. Zhong, Common properties of bounded linear operators AC and BA: spectral theory, Math. Nachr. 287(2014) 717-725. [9] Q P. Zeng, H J. Zhong, New results on common properties of the products AC and BA, J. Math. Anal. Appl. 427 (2015), 830-840.


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