Extended g-Drazin Inverse in a Banach Algebra

  • Dijana MosićEmail author


We introduce and investigate a new generalized inverse as an extension of the g-Drazin inverse in a Banach algebra. This new inverse will be called an extended g-Drazin inverse. Using idempotents, we characterize this inverse and give some its representations. Also, we prove generalizations of Cline’s formula for extended g-Drazin inverse. As a consequence of our results, we present the definition and characterizations of an extended Drazin inverse.


g-Drazin inverse Idempotent Cline’s formula Banach algebra 

Mathematics Subject Classification

46H05 46H99 15A09 



The author is grateful to the referee for careful reading of the paper.


  1. 1.
    Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, Berlin (2003)zbMATHGoogle Scholar
  2. 2.
    Campbell, S.L.: Singular Systems of Differential Equations. Pitman, San Francisco (1980)zbMATHGoogle Scholar
  3. 3.
    Castro-González, N., Dopazo, E.: Representations of the Drazin inverse for a class of block matrices. Linear Algebra Appl. 400, 253–269 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Castro-González, N., Koliha, J.J.: New additive results for the \(g\)-Drazin inverse. Proc. R. Soc. Edinb. Sect. A 134, 1085–1097 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cline, R. E.: An application of representation for the generalized inverse of a matrix, MRC Technical Report 592 (1965)Google Scholar
  6. 6.
    Corach, G., Duggal, B., Harte, R.E.: Extensions of Jacobson’s Lemma. Commun. Algebra 41, 520–531 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Deng, C., Wei, Y.: A note on the Drazin inverse of an anti-triangular matrix. Linear Algebra Appl. 431, 1910–1922 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Drazin, M.P.: Pseudoinverse in associative rings and semigroups. Am. Math. Month. 65, 506–514 (1958)CrossRefzbMATHGoogle Scholar
  9. 9.
    Harte, R.E.: On quasinilpotents in rings. PanAm. Math. J. 1, 10–16 (1991)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Koliha, J.J.: A generalized Drazin inverse. Glasg. Math. J. 38, 367–381 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Koliha, J.J., Tran, T.D.: Semistable operators and singularly perturbed differential equations. J. Math. Anal. Appl. 231, 446–458 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lian, H., Zeng, Q.: An extension of Cline’s formula for generalized Drazin inverse. Turk. Math. J. 40, 161–165 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Liao, Y., Chen, J., Cui, J.: Cline’s formula for the generalized Drazin inverse. Bull. Malays. Math. Sci. Soc. (2) 37(1), 37–42 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Mbekhta, M., Oudghiri, M., Souilah, K.: Additive maps preserving Drazin invertible operators of index \(n\). Banach J. Math. Anal. 11(2), 416–437 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mbekhta, M., Oudghiri, M., Souilah, K.: Additive maps preserving Drazin invertible operators of index one. Math. Proc. R. Ir. Acad. 116A(1), 19–34 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mosić, D.: Additive results for the generalized Drazin inverse in a Banach algebra. Bull. Malays. Math. Sci. Soc. 40(4), 1465–1478 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mosić, D.: A note on Cline’s formula for the generalized Drazin inverse. Linear Multilinear Algebra 63(6), 1106–1110 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mosić, D.: Generalized Inverses, Faculty of Sciences and Mathematics. University of Niš, Niš (2018)Google Scholar
  19. 19.
    Mosić, D.: Reverse order laws on the conditions of the commutativity up to a factor. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A. Mat. 111, 685–695 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mosić, D., Zou, H., Chen, J.: The generalized Drazin inverse of the sum in a Banach algebra. Ann. Funct. Anal. 8(1), 90–105 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Müller, V.: Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras. Birkhauser, Basel (2007)zbMATHGoogle Scholar
  22. 22.
    Patrício, P., Hartwig, R.E.: Some additive results on Drazin inverses. Appl. Math. Comput. 215(2), 530–538 (2009)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Yan, K., Fang, X.C.: Common properties of the operator products in local spectral theory. Acta Math. Sin. (Engl. Ser.) 31, 1715–1724 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Zeng, Q., Wu, Z., Wen, Y.: New extensions of Cline’s formula for generalized inverses. Filomat 31(7), 1973–1980 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Zeng, Q., Zhong, H.: New results on common properties of the products \(AC\) and \(BA\). J. Math. Anal. Appl. 427(2), 830–840 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  1. 1.Faculty of Sciences and MathematicsUniversity of NišNišSerbia

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