# ON RELATIVE ESSENTIAL SPECTRA OF BLOCK OPERATOR MATRICES AND APPLICATION

• Charfi, Salma (Faculty of Sciences of Sfax Department of Mathematics) ;
• Walha, Ines (Faculty of Sciences of Sfax Department of Mathematics)
• Published : 2016.05.31

#### Abstract

In this paper, we investigate relative essential spectra of $2{\times}2$ block operator matrix using the Fredholm perturbation theory. Furthermore, an example for two-group transport equations is presented to illustrate the validity of the main results.

#### References

1. F. V. Atkinson, H. Langer, R. Mennicken, and A. A. Shkalikov, The essential spectrum of some matrix operators, Math. Nachr. 167 (1994), 5-20. https://doi.org/10.1002/mana.19941670102
2. A. Batkai, P. Binding, A. Dijksma, R. Hryniv, and H. Langer, Spectral problems for operator matrices, Math. Nachr. 278 (2005), no. 12-13, 1408-1429. https://doi.org/10.1002/mana.200310313
3. S. Charfi and A. Jeribi, On a characterization of the essential spectra of some matrix operators and applications to two-group transport operators, Math. Z. 262 (2009), no. 4, 775-794. https://doi.org/10.1007/s00209-008-0399-1
4. S. Charfi, A. Jeribi, and R. Moalla, Essential spectra of operator matrices and applications, Math. Methods Appl. Sci. 37 (2014), no. 4, 597-608. https://doi.org/10.1002/mma.2819
5. S. Charfi, A. Jeribi, and I. Walha, Essential spectra, matrix operator and applications, Acta Appl. Math. 111 (2010), no. 3, 319-337. https://doi.org/10.1007/s10440-009-9547-9
6. R. Dautray and J. L. Lions, Analyse mathematique et calcul numerique, Tome 9, Masson, Paris, 1988.
7. N. Dunford and J. T. Schwartz, Linear operators. I. General Theory, Interscience New York, 1958.
8. M. Faierman, R. Mennicken, and M. Moller, A boundary eigenvalue problem for a system of partial differential operators occurring in magnetohydrodynamics, Math. Nachr. 173 (1995), 141-167. https://doi.org/10.1002/mana.19951730110
9. I. C. Gohberg, A. S. Markus, and I. A. Feldman, Normally solvable operators and ideals associated with them, Amer. Math. Soc. Transl. Ser. 2 61 (1967), 63-84.
10. A. Jeribi, Sepctral Theory and Applications of Linear Operators and Block Operator Matrices, Spinger-Verlag, New-York, 2015.
11. A. Jeribi, N. Moalla, and I. Walha, Spectra of some block operator matrices and application to transport operators, J. Math. Anal. Appl. 351 (2009), no. 1, 315-325. https://doi.org/10.1016/j.jmaa.2008.09.074
12. A. Jeribi, N. Moalla, and S. Yengui, S-essential spectra and application to an example of transport operators, Math. Methods Appl. Sci. 37 (2014), no. 16, 2341-2353. https://doi.org/10.1002/mma.1564
13. A. Jeribi, N. Moalla, and S. Yengui, Some results on perturbation theory of matrix operators, M-essential spectra of matrix operators and application to an example of transport operators, submitted.
14. A. Jeribi and I. Walha, Gustafson, Weidmann, Kato, Wolf, Schechter and Browder essential spectra of some matrix operator and application to two-group transport equations, Math. Nachr. 284 (2011), no. 1, 67-86. https://doi.org/10.1002/mana.200710125
15. T. Kato, Perturbation theory for nullity deficiency and other quantities of linear operators, J. Anal. Math. 6 (1958), 261-322. https://doi.org/10.1007/BF02790238
16. A. Y. Konstantinov, Spectral theory of some matrix differential operators of mixed order, Ukrain. Math. Zh. 50 (1998), no. 8, 1064-1072.
17. K. Latrach and A. Dehici, Relatively strictly singular perturbations, essential spectra and application, J. Math. Anal. Appl. 252 (2000), no. 2, 767-789. https://doi.org/10.1006/jmaa.2000.7121
18. A. S. Markus, Introduction to the spectral theory of polynomial operator pencils, Amer. Math. Soc., Providence, 1988.
19. M. Mokhtar-Kharroubi, Time asymptotic behaviour and compactness in neutron transport theory, Euro. Jour. Mech. B Fluid 11 (1992), 39-68.
20. V. Muller, Spectral theory of linear operator and spectral system in Banach algebras, Operator Theory: Advances and Applications, 139. Birkhauser Verlag, Basel, 2003.
21. R. D. Nussbaum, Spectral mapping theorems and perturbation theorem for Browding essential spectrum, Trans. Amer. Math. Soc. 150 (1970), 445-455.
22. A. Pelczynski, On strictly singular and strictly cosingular operators. I. Strictlly singular and cosingular operators on C($\Omega$) spaces, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 31-36.
23. M. Schechter, Basic theory of Fredholm operators, Ann. Scuola Norm. Sup. Pisa (3) 21 (1967), 261-280.
24. A. A. Shkalikov, On the essential spectrum of some matrix operators, Math. Notes 58 (1995), no. 6, 1359-1362. https://doi.org/10.1007/BF02304901
25. I. Walha, On the M-essential spectra of two-group transport equations, Math. Methods Appl. Sci. 37 (2014), no. 14, 2135-2149. https://doi.org/10.1002/mma.2961
26. F. Wolf, On the invariance of the essential spectrum under a change of the boundary conditions of partial differential operators, Indag. Math. 21 (1959), 142-147.