Spectral Properties Involving Generalized Weakly Demicompact Operators

  • Imen Ferjani
  • Aref Jeribi
  • Bilel KrichenEmail author


In this paper, we investigate the concept of generalized weakly demicompact operators with respect to weakly closed densely defined linear operators. We give their relationship with Fredholm and upper semi-Fredholm operators. In particular a characterization by means of upper semi-Browder spectrum is given. Moreover, we provide some sufficient conditions on the inputs of a closable block operator matrix to ensure the generalized weak demicompactness of its closure. Our results generalize many known ones in the literature.


Generalized weakly demicompact operator Fredholm and semi-Fredholm operators matrix operator essential spectrum 

Mathematics Subject Classification

47A53 47A10 



  1. 1.
    Akashi, W.Y.: On the perturbation theory for Fredholm operators. Osaka J. Math. 21, 603–612 (1984)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Astala, K.: On measures of noncompactness and ideal variations in Banach spaces. Ann. Acad. Sci. Fenn. Ser. A.I. Math. Diss. 29 (1980)Google Scholar
  3. 3.
    Atkinson, F.V., Langer, H., Mennicken, R., Shkalikov, A.A.: The essential spectrum of some matrix operators. Math. Nachr. 167, 5–20 (1994)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Banaś, J., Martinón, A.: On measures of weak noncompactness in Banach sequence spaces. Port. Math. 52, 131–138 (1995)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Banaś, J., Rivero, J.: On measures of weak noncompactness. Ann. Mat. Pura Appl. 151, 213–262 (1988)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011)zbMATHGoogle Scholar
  8. 8.
    Chaker, W., Jeribi, A., Krichen, B.: Demicompact linear operators, essential spectrum and some perturbation results. Math. Nachr. 1–11 (2015)Google Scholar
  9. 9.
    Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. The International Series of Monographs on Physics. Clarendon Press, Oxford (1961)Google Scholar
  10. 10.
    Curtain, R.F., Zwart, H.: An Introduction to Infnite-dimensional Linear Systems Theory. Applied Mathematics. Springer, New York (1995)CrossRefGoogle Scholar
  11. 11.
    De Blasi, F.S.: On a property of the unit sphere in a Banach space. Bull. Math. Soc. Sci. Math. R. S. Roum. (N.S) 21, 259–262 (1977)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Dunford, N., Schwartz, J.T.: Linear operations on summable functions. Trans. Am. Math. Soc. 47, 323–392 (1940)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dunford, N., Schwartz, J.T.: Linear operators, Part I. General Theory, Interscience. New York. (1958)Google Scholar
  14. 14.
    Emmanuele, G.: Measure of weak noncompactness and fixed point theorems. Boll. Math. Soc. Sci. Math. R.S. Roum. 25, 353–358 (1981)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Goldberg, S.: Unbounded Linear Operators, Theory and Applications. McGraw-Hill Book Co., New York (1966)zbMATHGoogle Scholar
  16. 16.
    Jeribi, A.: Spectral Theory and Applications of Linear Operators and Block Operator Matrices. Springer, New-York (2015)CrossRefGoogle Scholar
  17. 17.
    Jeribi, A., Moalla, N.: Fredholm operators and Riesz theory for polynomially compact operators. Acta Appl. Math. 90, 227–245 (2006)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Jeribi, A., Krichen, B.: Nonlinear Functional Analysis in Banach Spaces and Banach Algebras: Fixed Point Theory under Weak Topology for Nonlinear Operators and Block Operator Matrices with Applications, Monographs and Research Notes in Mathematics. CRC Press Taylor and Francis, Boca Raton (2015)CrossRefGoogle Scholar
  19. 19.
    Jeribi, A., Krichen, B., Salhi, M.: Characterization of relatively demicompact operators by means of measures of noncompactness. J. Korean Math. Soc. 1–19 (2018)Google Scholar
  20. 20.
    Krichen, B.: Relative essential spectra involving relative demicompact unbounded linear operators. Acta Math. Sci. Ser. B Engl. Ed. 34, 546–556 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Krichen, B., O’Regan, D.: On the class of relatively weakly demicompact nonlinear operators. Fixed Point Theory 19, 625–630 (2018)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Li, C., Deng, X.: Remarks on fractional derivatives. Appl. Math. Comput. 187, 777–784 (2007)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Lifschitz, A.E.: Magnetohydrodynamics and Spectral Theory. Developments in Electromagnetic Theory and Applications. Kluwer Academic Publishers Group, Dordrecht (1989)CrossRefGoogle Scholar
  24. 24.
    Müller, V.: Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras. Birkhäuser, Basel (2007)Google Scholar
  25. 25.
    Petryshyn, W.V.: Remarks on condensing and k-set contractive mappings I. J. Math. Anal. Appl. 39, 717–741 (1972)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Rakočević, V.: Approximate point spectrum and commuting compact perturbations. Glasgow Math. J. 28, 193–8 (1986)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Schechter, M.: Principles of Functional Analysis. Academic Press, New York (1971)zbMATHGoogle Scholar
  28. 28.
    Shkalikov, A.A.: On the essential spectrum of some matrix operators. Math. Notes 58, 1359–1362 (1995)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Thaller, B.: The Dirac Equation. Texts and Monographs in Physics. Springer, Berlin (1992)Google Scholar
  30. 30.
    Tretter, C.: Spectral Theory of Block Operator Matrices and Applications. Copyright by Imperial College Press (2008)Google Scholar
  31. 31.
    Williams, V.: Closed Fredholm and semi-Fredholm operators, essential spectra and perturbations. J. Funct. Anal. 20, 1–25 (1975)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Živković-Zlatanović, S.Č., Djordjević, D.S., Harte, R.E.: On left and right Browder operators. J. Korean Math. Soc. 485, 1053–1063 (2011)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Živković-Zlatanović, S.Č., Djordjević, D.S., Harte, R.E.: Polynomially Riesz perturbations. J. Math. Anal. Appl. 408, 442–451 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics. University of Sfax. Faculty of Sciences of SfaxSfaxTunisia

Personalised recommendations