Abstract
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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References
Abramovich, Y., Aliprantis, C. D.: Invitation to operator theory. Grad. Stud. Math. 50, Amer. Math. Soc., Providence, (2002)
Aiena, P.: Semi Fredholm Operators. Perturbation Theory and Localized Svep, IVIC (2007)
Banas, J., Rivero, J.: On measures of weak noncompactness. Ann. Mat. Pura Appl. 151, 213–224 (1988)
Ben Brahim, F., Jeribi, A., Krichen, B.: Spectral theory for polynomially demicompact operators. Filomat 33(7), 2017–2030 (2019)
Belabbaci, C., Aissani, M., Terbeche, M.: S-essential spectra and measure of noncompactness. Math. Slovaca 67(5), 1203–1212 (2017)
Berkani, M.: On a class of quasi-Fredholm operators. Integ. Eq. Op. Theory 34(2), 244–249 (1999)
Berkani, M., Sarih, M.: An Atkinson-type theorem for \(B\)-Fredholm operators. Stud. Math 148(3), 251–257 (2001)
Berkani, M., Sarih, M.: On semi B-Fredholm operators. Glasg. Math. J 43(3), 457–465 (2001)
Berkani, M., Koliha, J.J.: Weyl type theorems for bounded linear operators. Acta Sci. Math. (Szeged) 69(1–2), 359–376 (2003)
Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011)
Chafika, B., Mouloud, A., Mekki, T.: S-essential spectra and measure of noncompactness. Math. Slovaca 67(5), 1203–1212 (2017)
Chaker, W., Jeribi, A., Krichen, B.: Demicompact linear operators, essential epectrum and some perturbations results. Math. Nachr. 288(13), 1476–1486 (2015)
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)
Diestel, J.: Geometry of Banach Spacesselected Topics. Springer-Verlag, Berlin-New York (1975)
Diestel, J., J. J., Jr. Uhl, Vector measures. American Mathematical Society, Providence, R.I., (1977)
Dunford, N., Schwartz, J.T.: Linear Operators. I General Theory. Pure and Applied Mathematics, vol. 7. Interscience Publishers Inc., New York (1958)
Edmunds, D.E., Evans, W.D.: Spectral Theory and Differential Operators. Oxford University Press, Oxford (1987)
Goldberg, S.: Unbounded Linear Operators. McGraw-Hill, New York (1966)
Jeribi, A.: Spectral Theory and Applications of linear Operators and Block Operator Matrices. Springer, New York (2015)
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)
Kharroubi, M.M.: Time asymptotic behaviour and compactness in transport theory. Eur. J. Mech. B Fluids 11(1), 39–68 (1992)
Krichen, B.: Relative essential spectra involving relative demicompact unbounded linear operators, Acta Math. Sci. Ser. B Engl. Ed. 2, 546–556 (2014)
Krichen, B., O’Regan, D.: On the class of relatively weakly demicompact nonlinear operators. Fixed Point Theory 19(2), 625–630 (2018)
Krichen, B., O’Regan, D.: Weakly demicompact linear operators and axiomatic measures of weak noncompactness. Math. Slov. 69(6), 1403–1412 (2019)
Latrach, K., Jeribi, A.: Some results on Fredholm oprators, essential spectra and application. I. J. Math. Anal. Appl. 225, 461–485 (1998)
Li, C., Deng, W.: Remarks on fractional derivatives. Appl. Math. Comput. 187(2), 777–784 (2007)
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)
Petryshyn, W.V.: Construction of fixed points Of demicompact mappings in Hilbert spaces. J. Funct. Anal. Appl. 14, 276–284 (1966)
Schechter, M.: Principles of Functional Analysis. Grad. Stud. Math. 36, 2nd edition, Amer. Math. soc., Providence, Rhode Island, (2002)
Taylor, A.E., Lay, D.C.: Introduction to Functional Analysis, 2nd edn. Wiley, Hoboken (1980)
Acknowledgements
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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Krichen, B., Trabelsi, B. Some spectral results for demicompact operators and their restrictions with an application to transport equations. Ricerche mat (2022). https://doi.org/10.1007/s11587-022-00688-3
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DOI: https://doi.org/10.1007/s11587-022-00688-3
Keywords
- Weakly \(S_0\)-demicompact operator
- Quasi Fredholm operators
- Measure of noncompactness
- S-essential radius