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Some spectral results for demicompact operators and their restrictions with an application to transport equations


In this paper, we investigate the \(S_{0_n}\)-demicompactness of the restriction \(T_n\) of a bounded or unbounded linear operator T to \(\mathcal {R}(T^n)\), where \(S_{0_n}\) is the restriction of a given bounded linear operator \(S_0\) to \(\mathcal {R}(T^n)\). The results are formulated in terms of a condition of primality and the closedness of certain ranges. Moreover, we set forward some results on upper semi-Fredholm operators involving weak \(S_0\)-demicompactness class. In particular, we give a new characterization of the \(S_0\)-essential radius and localizations results of some \(S_0\)-essential spectra of T. An example of operator equations arising in transport theory is developed.

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  1. Abramovich, Y., Aliprantis, C. D.: Invitation to operator theory. Grad. Stud. Math. 50, Amer. Math. Soc., Providence, (2002)

  2. Aiena, P.: Semi Fredholm Operators. Perturbation Theory and Localized Svep, IVIC (2007)

  3. Banas, J., Rivero, J.: On measures of weak noncompactness. Ann. Mat. Pura Appl. 151, 213–224 (1988)

    MathSciNet  Article  Google Scholar 

  4. Ben Brahim, F., Jeribi, A., Krichen, B.: Spectral theory for polynomially demicompact operators. Filomat 33(7), 2017–2030 (2019)

    MathSciNet  Article  Google Scholar 

  5. Belabbaci, C., Aissani, M., Terbeche, M.: S-essential spectra and measure of noncompactness. Math. Slovaca 67(5), 1203–1212 (2017)

    MathSciNet  Article  Google Scholar 

  6. Berkani, M.: On a class of quasi-Fredholm operators. Integ. Eq. Op. Theory 34(2), 244–249 (1999)

    MathSciNet  Article  Google Scholar 

  7. Berkani, M., Sarih, M.: An Atkinson-type theorem for \(B\)-Fredholm operators. Stud. Math 148(3), 251–257 (2001)

    MathSciNet  Article  Google Scholar 

  8. Berkani, M., Sarih, M.: On semi B-Fredholm operators. Glasg. Math. J 43(3), 457–465 (2001)

    MathSciNet  Article  Google Scholar 

  9. Berkani, M., Koliha, J.J.: Weyl type theorems for bounded linear operators. Acta Sci. Math. (Szeged) 69(1–2), 359–376 (2003)

    MathSciNet  MATH  Google Scholar 

  10. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011)

    Book  Google Scholar 

  11. Chafika, B., Mouloud, A., Mekki, T.: S-essential spectra and measure of noncompactness. Math. Slovaca 67(5), 1203–1212 (2017)

    MathSciNet  Article  Google Scholar 

  12. Chaker, W., Jeribi, A., Krichen, B.: Demicompact linear operators, essential epectrum and some perturbations results. Math. Nachr. 288(13), 1476–1486 (2015)

    MathSciNet  Article  Google Scholar 

  13. De Blasi, F.S.: On a property of the unit sphere in a Banach space. Bull. Math. Soc. Sci. Math. R.S. Roumanie N.S 21, 259–262 (1977)

    MathSciNet  MATH  Google Scholar 

  14. Diestel, J.: Geometry of Banach Spacesselected Topics. Springer-Verlag, Berlin-New York (1975)

    Book  Google Scholar 

  15. Diestel, J., J. J., Jr. Uhl, Vector measures. American Mathematical Society, Providence, R.I., (1977)

  16. Dunford, N., Schwartz, J.T.: Linear Operators. I General Theory. Pure and Applied Mathematics, vol. 7. Interscience Publishers Inc., New York (1958)

    MATH  Google Scholar 

  17. Edmunds, D.E., Evans, W.D.: Spectral Theory and Differential Operators. Oxford University Press, Oxford (1987)

    MATH  Google Scholar 

  18. Goldberg, S.: Unbounded Linear Operators. McGraw-Hill, New York (1966)

    MATH  Google Scholar 

  19. Jeribi, A.: Spectral Theory and Applications of linear Operators and Block Operator Matrices. Springer, New York (2015)

    Book  Google Scholar 

  20. Jeribi, A., Krichen, B., Salhi, M.: Characterization of relatively demicompact operators by means of measures of noncompactness. J. Korean Math. Soc. 55(4), 877–895 (2018)

    MathSciNet  MATH  Google Scholar 

  21. Kharroubi, M.M.: Time asymptotic behaviour and compactness in transport theory. Eur. J. Mech. B Fluids 11(1), 39–68 (1992)

    MathSciNet  Google Scholar 

  22. Krichen, B.: Relative essential spectra involving relative demicompact unbounded linear operators, Acta Math. Sci. Ser. B Engl. Ed. 2, 546–556 (2014)

    MATH  Google Scholar 

  23. Krichen, B., O’Regan, D.: On the class of relatively weakly demicompact nonlinear operators. Fixed Point Theory 19(2), 625–630 (2018)

    MathSciNet  Article  Google Scholar 

  24. Krichen, B., O’Regan, D.: Weakly demicompact linear operators and axiomatic measures of weak noncompactness. Math. Slov. 69(6), 1403–1412 (2019)

    MathSciNet  Article  Google Scholar 

  25. Latrach, K., Jeribi, A.: Some results on Fredholm oprators, essential spectra and application. I. J. Math. Anal. Appl. 225, 461–485 (1998)

    Article  Google Scholar 

  26. Li, C., Deng, W.: Remarks on fractional derivatives. Appl. Math. Comput. 187(2), 777–784 (2007)

    MathSciNet  MATH  Google Scholar 

  27. Müller, V.: Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras. Oper. Theory Adv. Appl. 139, 2nd edition, Birkhäuser, Basel, Boston, Berlin, (2007)

  28. Petryshyn, W.V.: Construction of fixed points Of demicompact mappings in Hilbert spaces. J. Funct. Anal. Appl. 14, 276–284 (1966)

    MathSciNet  MATH  Google Scholar 

  29. Schechter, M.: Principles of Functional Analysis. Grad. Stud. Math. 36, 2nd edition, Amer. Math. soc., Providence, Rhode Island, (2002)

  30. Taylor, A.E., Lay, D.C.: Introduction to Functional Analysis, 2nd edn. Wiley, Hoboken (1980)

    MATH  Google Scholar 

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The authors wish to thank Pr. M. Berkani for helpful discussions and beneficial comments that motivated many of the results of Sect. 3.

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Correspondence to Bilel Krichen.

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Krichen, B., Trabelsi, B. Some spectral results for demicompact operators and their restrictions with an application to transport equations. Ricerche mat (2022).

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  • Weakly \(S_0\)-demicompact operator
  • Quasi Fredholm operators
  • Measure of noncompactness
  • S-essential radius

Mathematics Subject Classification

  • 47A53
  • 47A55