# CHARACTERIZATION OF RELATIVELY DEMICOMPACT OPERATORS BY MEANS OF MEASURES OF NONCOMPACTNESS

• Jeribi, Aref (Department of Mathematics Faculty of Sciences of Sfax University of Sfax) ;
• Krichen, Bilel (Department of Mathematics Faculty of Sciences of Sfax University of Sfax) ;
• Salhi, Makrem (Department of Mathematics Faculty of Sciences of Sfax University of Sfax)
• Accepted : 2018.01.16
• Published : 2018.07.01

#### Abstract

In this paper, we show that an unbounded $S_0$-demicompact linear operator T with respect to a bounded linear operator $S_0$, acting on a Banach space, can be characterized by the Kuratowskii measure of noncompactness. Moreover, some other quantities related to this measure provide sufficient conditions to the operator T to be $S_0$-demicompact. The obtained results are used to discuss the connection with Fredholm and upper Semi-Fredholm operators.

#### References

1. P. Aiena, Semi-Fredholm operators, perturbation theory and localized SVEP, IVIC, 2007.
2. W. Y. Akashi, On the perturbation theory for Fredholm operators, Osaka J. Math. 21 (1984), no. 3, 603-612.
3. J. Appell, Measures of noncompactness, condensing operators and fixed points: an application-oriented survey, Fixed Point Theory 6 (2005), no. 2, 157-229.
4. K. Astala, On measures of noncompactness and ideal variations in Banach spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes No. 29 (1980), 42 pp.
5. S. Axler, N. Jewell, and A. Shields, The essential norm of an operator and its adjoint, Trans. Amer. Math. Soc. 261 (1980), no. 1, 159-167. https://doi.org/10.1090/S0002-9947-1980-0576869-9
6. J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, 60, Marcel Dekker, Inc., New York, 1980.
7. W. Chaker, A. Jeribi, and B. Krichen, Demicompact linear operators, essential spectrum and some perturbation results, Math. Nachr. 288 (2015), no. 13, 1476-1486. https://doi.org/10.1002/mana.201200007
8. J. B. Conway, A Course in Functional Analysis, second edition, Graduate Texts in Mathematics, 96, Springer-Verlag, New York, 1990.
9. H. O. Cordes and J. P. Labrousse, The invariance of the index in the metric space of closed operators, J. Math. Mech. 12 (1963), 693-719.
10. D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1987.
11. F. Galaz-Fontes, Measures of noncompactness and upper semi-Fredholm perturbation theorems, Proc. Amer. Math. Soc. 118 (1993), no. 3, 891-897. https://doi.org/10.1090/S0002-9939-1993-1151810-3
12. T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966.
13. B. Krichen, Relative essential spectra involving relative demicompact unbounded linear operators, Acta Math. Sci. Ser. B Engl. Ed. 34 (2014), no. 2, 546-556.
14. B. Krichen and D. O'Regan, On the class of relatively weakly demicompact nonlinear operators fixed point theory, to appear.
15. C. Kuratowski, Topologie. I. Espaces Metrisables, Espaces Complets, Monografie Matematyczne, vol. 20, Warszawa-Wroc law, 1948.
16. W. V. Petryshyn, Construction of fixed points of demicompact mappings in Hilbert space, J. Math. Anal. Appl. 14 (1966), 276-284. https://doi.org/10.1016/0022-247X(66)90027-8
17. W. V. Petryshyn, Remarks on condensing and k-set-contractive mappings, J. Math. Anal. Appl. 39 (1972), 717-741. https://doi.org/10.1016/0022-247X(72)90194-1
18. V. Rakocevic, Measures of noncompactnessand some applications, University of Nis, Faculty of Sciences and Mathematics 12 (1998), 87-120.
19. M. Schechter, Principles of Functional Analysis, second edition, Graduate Studies in Mathematics, 36, American Mathematical Society, Providence, RI, 2002.
20. V. Williams, Closed Fredholm and semi-Fredholm operators, essential spectra and perturbations, J. Functional Analysis 20 (1975), no. 1, 1-25. https://doi.org/10.1016/0022-1236(75)90050-6