Advertisement

Drazin inverse of multivalued operators and its applications

  • Ayoub Ghorbel
  • Maher Mnif
Article
  • 24 Downloads

Abstract

In this paper, the notion of Drazin invertibility in the case of multivalued operators is introduced. Many results from operator theory are covered. Applications of some obtained results allow to study the Drazin invertibility of a multivalued operator matrix \( M_C := \left( \begin{array}{c@{\quad }c} A &{} C \\ 0 &{} B \\ \end{array} \right) \) acting in the product of Banach or Hilbert spaces \( X \times Y \).

Keywords

Drazin invertible multivalued operators Left Drazin invertible multivalued operators Right Drazin invertible multivalued operators Upper triangular multivalued operator matrices 

Mathematics Subject Classification

47A06 47A53 

Notes

References

  1. 1.
    Aiena, P., Biondi, T.M., Carpintero, C.: On Drazin Invertibility, pp. 2839–2848. The American Mathematical Society, Providence (2008)zbMATHGoogle Scholar
  2. 2.
    Alvarez, T., Chamkha, Y., Mnif, M.: Left and right-Atkinson linear relation matrices. Mediter. J. Math. 13(4), 2039–2059 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alvarez, T., Fakhfakh, F., Mnif, M.: Left-right Fredholm and left-right Browder linear relations. Filomat 31(2), 255–271 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Arens, R.: Operational calculus of linear relation. Pac. J. Math. 11, 9–23 (1961)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Baskakov, A.G., Chernyshov, K.I.: Spectral analysis of linear relations and degenerate operator semigroups. Sb. Math. 193(11), 1573–1610 (2012)CrossRefGoogle Scholar
  6. 6.
    Berkani, M.: On a class of quasi-Fredholm operators. Integr. Equ. Oper. Theory 34(2), 244–249 (1999)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bouaniza, H., Mnif, M.: On strictly quasi-Fredholm linear relations and semi-B-Fredholm linear relation perturbations. Filomat 31(20), 6337–6355 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Caradus, S.R.: Generalized Inverses and Operator Theory. Queens Paper in Pure and Applied Mathematics. Queens University, Kingston (1978)Google Scholar
  9. 9.
    Chafai, E., Mnif, M.: Ascent and essential ascent spectrum of linear relations. Extracta Mathematicae 31(2), 145–167 (2016)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chafai, E., Mnif, M.: Descent and essential descent spectrum of linear relations. Extracta Mathematicae 29(1), 117–139 (2014)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Chamkha, Y., Mnif, M.: Browder spectra of upper triangular matrix linear relations. Publ. Math. Debrecen 82(3–4), 569–590 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chamkha, Y., Mnif, M.: The class of B-Fredholm linear relations. Complex Anal. Oper. Theory. 4(1), 1681–1699 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Coddington, E.A.: Multivalued operators and boundary value problems. In: Analytic theory of Differential Equations, Lecture Notes in Mathematics, vol. 183, p. 18. Springer, Berlin (1971)Google Scholar
  14. 14.
    Coddington, E.A., Dijksma, A.: Self-adjoint subspaces and eigenfunction expansions for ordinary differential subspaces. J. Differ. Equ. 20(2), 473–526 (1976)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Cross, R.W.: Multivalued Linear Operators. Pure and Applied Mathematics. Marcel Dekker, New York (1998)Google Scholar
  16. 16.
    Cross, R.W., Favini, A., Yakukov, Y.: Perturbation Results for Multivalued Linear Operators, Progress in Nonlinear Differential Equations and Their Applications, vol. 80, pp. 111–130. Birkhäuser/Springer Basel AG, Basel (2011)Google Scholar
  17. 17.
    Cvetković, M.D.: Generalized Drazin Invertibility of Operator Matrices. Linear and Multilinear Algebra 66(4), 692–703 (2018)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Dijksma, A., EL Sabbah, A., de Snoo, H.S.V.: Selfadjoint extensions of regular canonical systems with Stieltjes boundary conditions. J. Math. Anal. Appl. 152(2), 546–583 (1990)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Djordjević, D.S.: Perturbations of spectra of operator matrices. J. Oper. Theory 48, 467–486 (2002)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Fakhfakh, F., Mnif, M.: Perturbation theory of lower semi-Browder multivalued linear operators. Publ. Math. Debrecen 78(3/4), 595–606 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Farah, A.: Fractional powers, invertibility and singularity of linear relations. Thesis, University of Sfax (2017)Google Scholar
  22. 22.
    King, C.F.: A note on Drazin inverses. Pac. J. Math. 70(2), 383–390 (1977)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Lay, D.C.: Spectral properties of generalized inverses of linear operators. SIAM J. Appl. Math 29(1), 103–109 (1975)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Sandovici, A., Snoo, H., Winkler, H.: Ascent, descent, nullity, defect and related notions for linear relations in linear spaces. Linear Algebra Appl. 423, 456–497 (2007)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Sandovici, A., Snoo, H.: An index formula for the product of linear relations. Linear Algebra Appl. 431(11), 2160–2171 (2009)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Von Neumann, J.: Functional Operators II, The Geometry of Orthogonal Spaces, Annals of Mathematics Studies, vol. 22. Princeton University Press, Princeton (1950)Google Scholar
  27. 27.
    Wilcox, D.: Multivalued semi-Fredholm operators in normed linear spaces. Thesis, University of Cape Town (2002)Google Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Département de Mathématiques, Faculté des Sciences de SfaxUniversité de SfaxSfaxTunisia

Personalised recommendations