The S-Jeribi essential spectrum

C. Belabbaci

doi:10.37863/umzh.v73i3.163

C. Belabbaci Laboratory Pure and Appl. Math., Dep. Math., Univ. Laghouat, Algeria

DOI: https://doi.org/10.37863/umzh.v73i3.163

Keywords: S-essential spectra, Jeribi essential spectrum, Fredholm operators

Abstract

UDC 517.9
We study some properties and results on the S-Jeribi essential spectrum of linear bounded operators on a Banach space. In particular, we give some criteria for coincidence of this spectrum for two linear operators and the relation of this type of spectrum with the well-known S-Schechter essential spectrum.

References

F. Abdmouleh, A. Ammar, A. Jeribi, Stability of the $s$-essential spectra on a Banach space, Math. Slovaca, 63, № 2, 299 – 320 (2013), https://doi.org/10.2478/s12175-012-0099-5 DOI: https://doi.org/10.2478/s12175-012-0099-5

Y. A. Abramovich, C. D. Aliprantis, An invitation to operator theory, Grad. Stud. Math., vol. 50, Amer. Math. Soc., Providence (2002), https://doi.org/10.1090/gsm/050 DOI: https://doi.org/10.1090/gsm/050

P. Aiena, Fredholm and local spectral theory, with applications to multipliers, Springer Sci. & Business Media (2004).

A. Ammar, B. Boukettaya, A. Jeribi, Stability of the $S$-left and $S$-right essential spectra of a linear operator, Acta Math. Sci. Ser. A (Chin. Ed.), 34, № 6, 1922 – 1934 (2014), https://doi.org/10.1016/S0252-9602(14)60135-1 DOI: https://doi.org/10.1016/S0252-9602(14)60135-1

A. Ammar, M. Zerai Dhahri, A. Jeribi, A characterization of $S$- essential spectrum by means of measure of non-strictsingularity and application, Azerb. J. Math., 5, № 1 (2015).

C. Belabbaci, M. Aissani, M. Terbeche, $S$-essential spectra and measure of noncompactness, Math. Slovaca, 67, № 5, 1203 – 1212 (2017), https://doi.org/10.1515/ms-2017-0043 DOI: https://doi.org/10.1515/ms-2017-0043

I. C. Gohberg, A. S. Markus, I. A. Feldman, Normally solvable operators and ideals associated with them, Amer. Math. Soc. Trans. Ser., 2, № 61, 63 – 84 (1967).

S. Goldberg, Unbounded linear operators, McGraw-Hill, New York (1966).

A. Jeribi, Spectral theory and applications of linear operators and block operator matrices, Springer (2015), https://doi.org/10.1007/978-3-319-17566-9 DOI: https://doi.org/10.1007/978-3-319-17566-9

A. Jeribi, K. Latrach, Quelques remarques sur le spectre essentiel et application à l'équation de transport. (French), Compt. Rend. Acad. Sci. S´er. I, Math. 323, № 5 (1996).

A. Jeribi, N. Moalla, S. Yengui, $S$-essential spectra and applica- tion to an example of transport operators, Math. Methods Appl. Sci., 37, 2341 – 2353 (2014), https://doi.org/10.1002/mma.1564 DOI: https://doi.org/10.1002/mma.1564

T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Anal. Math., 6, № 1, 261 – 322 (1958).

K. Latrach, A. Jeribi, On the essential spectrum of transport operators on $L_1$-spaces, J. Math. Phys., 37, № 12, 6486 – 6494 (1996), https://doi.org/10.1063/1.531748 DOI: https://doi.org/10.1063/1.531748

V. Muller, Spectral theory of linear operators and spectral systems in Banach algebras, 2nd ed., Oper. Theory: Adv. and Appl., vol. 139, Birkh¨auser, Basel (2007).

A. Pełczyński, On strictly singular and strictly cosingular operators. I. Strictly singular and strictly cosingular operators in $C(S)$-spaces, Bull. Acad. Polon. Sci., 13, № 1, 31 – 36, 37 – 41 (1965).

M. Schechter, Spectra of partial differential operators, vol. 14, North-Holland, Amsterdam (1971).

M. Schechter, Principles of functional analysis, Amer. Math. Soc. (2002).