Spectral properties for \(\gamma \)-diagonally dominant operator matrices using demicompactness classes and applications

  • Aref Jeribi
  • Bilel KrichenEmail author
  • Ali Zitouni
Original Paper


In this article, we use the concept of demicompact operators in order to investigate the stability of essential spectra of closed operators and we establish some perturbation results for \(\gamma \)-diagonally dominant operator matrices acting on Banach spaces. Our theoretical results will be illustrated by investigating the essential spectra of operators in Sturm–Liouville problems and in transport equations.


Essential spectra Demicompact linear operator Operator matrix Diagonally dominant matrix 

Mathematics Subject Classification

47A55 47A53 47H08 



  1. 1.
    Adamjan, V.M., Langer, H.: Spectral properties of a class of rational operator valued functions. J. Oper. Theory 33, 259–277 (1995)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Akashi, W.Y.: On the perturbation theory for Fredholm operators. Osaka J. Math. 21(3), 603–612 (1984)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Banas, J., Goebel, K.: Measures of noncompactness in Banach spaces. Vol. 60, Lecture Notes in pure and applied mathematics. New York (NY): Marcel Dekker (1980)Google Scholar
  4. 4.
    Chaker, W., Jeribi, A., Krichen, B.: Demicompact linear operators, essential spectrum and some perturbation results. Math. Nachr. 288(13), 1476–1486 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chaker, W., Jeribi, A., Krichen, B.: Some Fredholm results arround relative demicompactness concept. Preprint (2019)Google Scholar
  6. 6.
    Jeribi, A., Mnif, M.: Fredholm operators, essential spectra and application to transport equations. Acta Appl. Math. 89(1–3), 155–176 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Edmunds, D.E., Evans, W.D.: Spectral theory and differential operators. Oxford Science Publications, Oxford (1987)zbMATHGoogle Scholar
  8. 8.
    Gustafson, K., Weidmann, J.: On the essential spectrum. J. Math. Anal. Appl. 25, 121–127 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jeribi, A.: Spectral theory and applications of linear operator and block operator matrices. Springer, New York (2015)CrossRefzbMATHGoogle Scholar
  10. 10.
    Jeribi, A.: Linear operators and their essential pseudospectra. CRC Press, Boca Raton (2018)CrossRefzbMATHGoogle Scholar
  11. 11.
    Jeribi, A., Krichen, B., Dhahri, M.Z.: Essential spectra of some matrix operators involving \(\gamma \)-relatively bounded entries and an application. Linear Multilinear Algebra 64(8), 1654–1668 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jeribi, A., Krichen, B., Zitouni, A.: Properties of demicompact operators, essential spectra and some perturbation results for block operator matrices with applications. Preprint (2019)Google Scholar
  13. 13.
    Kato, T.: Perturbation theory for linear operators. Springer, New York (1966)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kato, T.: Perturbation theory for nullity, deficiency and other quantities of linear operators. J. Anal. Math. 6, 261–322 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Langer, H., Markus, A., Matsaev, V., Tretter, C.: Self-adjoint block operator matrices with non-separated diagonal entries and their Schur complements. J. Funct. Anal. 199, 427–451 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Latrach, K., Dehici, A.: Fredholm, semi-Fredholm perturbations and essential spectra. J. Math. Anal. Appl. 259, 227–301 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Latrach, K., Jeribi, A.: Some results on Fredholm operators, essential spectra and application. J. Math. Anal. Appl. 225, 461–485 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Moalla, N., Dammak, M., Jeribi, A.: Essential spectra of some matrix operator and applicatio to two-group transport operatorwith general boundary conditions. J. Math. Anal. Appl 323, 1071–1090 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mokhtar-Kharroubi, M.: Time asymptotic bahaviour and compactness in neutron transport theory. Eur. J. Mech. B Fluid 11, 39–68 (1992)Google Scholar
  20. 20.
    Milovanovi’c-Arandjelovi’c, M.M.: Measures of noncompactness on uniform spaces the axiomatic approach, in IMC ’Filomat 2001’, Nis, pp. 221–225 (2001)Google Scholar
  21. 21.
    Petryshyn, W.V.: Construction of fixed points of demicompact mappings in Hilbert space. J. Math. Anal. Appl. 14, 276–284 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rakoc̆evic̀, V.: On one subset of M. Schechter’s essential spectrum. Mat. Vesnik 5(18)(33)(4), 389-391 (1981)Google Scholar
  23. 23.
    Schechter, M.: On the essential spectrum of an arbitrary operator. J. Math. Anal. Appl. 13, 205–215 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Schechter, M.: Principles of Functional Analysis. Graduate Studies in Mathematics, vol. 36 (American Mathematical Society, Providence (2002)Google Scholar
  25. 25.
    Schmoeger, C.: The spectral mapping theorem for the essential approximate point spectrum. Colloq. Math. 74(2), 167–176 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Tretter, C.: Spectral theory block operator matrices and applications. Imperial College Press, London (2008)CrossRefzbMATHGoogle Scholar
  27. 27.
    Wolf, F.: On the invariance of the essential spectrum under a change of the boundary conditions of partial differential operators. Indag. Math. 21, 142–147 (1959)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Sciences of SfaxUniversity of SfaxSfaxTunisia

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