Measure of non strict singularity of Schechter essential spectrum of two bounded operators and application

Document Type: Research Paper

Authors

1 Institut Suprieur de Gestion Industrielle de Sfax, University‎ ‎of Sfax‎, ‎Route de Tunis km 10‎, ‎Cite El Ons B.P 1164, 3021‎, ‎Sfax‎, ‎Tunisie.

2 Department of‎ ‎Mathematics‎, ‎Faculty of Sciences of Sfax, University of Sfax‎, ‎Route de Soukra‎, ‎Km 3.5‎, ‎P.O. Box‎ ‎1171, 3000‎, ‎Sfax‎, ‎Tunisie.

Abstract

In this paper‎, ‎we discuss the essential spectrum of sum of two bounded operators‎ ‎using measure of non strict singularity‎. ‎Based on this new investigation‎, ‎a problem of one-speed neutron transport operator is presented‎.

Keywords

Main Subjects


F. Abdmouleh and A. Jeribi, Symmetric family of Fredholm operators of indices zero, stability of essential spectra and application to transport operators, J. Math. Anal. Appl. 364 (2010) 414--423.

J. Banas and K. Geobel, Measures of Noncompactness in Banasch Spaces, Lecture Notes in Pure and Applied Mathematics 60, Marcel Dekker Inc. New York, 1980.

A. Ben Amar, A. Jeribi and M. Mnif, Some results on Fredholm and semi-Fredholm perturbations, Arab J. Math. 3 (2014) 313--323.

R. Dautray and J.L. Lions, Analyse Mathematique et Calcul numerique, Masson, Paris, 1988.

N. Dunford and J.T. Schwartz, Linear Operators, Part I: General Theory, Interscience Publ. New York, 1958.

I. Gohberg, A. Markus and I.A. Feldman, Normally solvable operators and ideals associated with them, Amer. Math. Soc. Tran. Ser. 61 (1967) 63--84.

S. Goldberg, Unbounded Linear Operators, McGraw-Hill, New York, 1966.

K. Gustafson and J. Weidmann, On the essential spectrum, J. Math. Anal. Appl. 25 (1969) 121--127.

A. Jeribi and M. Mnif, Fredholm operators, essential spectra and application to transport equations, Acta Appl. Math. 89 (2005) 155--176.

A. Jeribi and I. Walha, Gustafson, Weidmann, Kato, Wolf, Schechter and Browder essential spectra of some matrix operator and application to a two-group transport equations, Math. Nachr. 284 (2011) 67--86.

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.

T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Anal. Math. 6 (1958) 261--322.

K. Kuratawski, Sur les espaces complets, Fund. Math. 15 (1930) 301--309.

K. Latrach and A. Dehici, Fredholm, semi-Fredholm perturbations and essential spectra, J. Math. Anal. Appl. 259 (2001) 277--301.

K. Latrach and J.M. Paoli, Perturbation results for linear operators and application to the transport equation, Rocky Mountain J. Math. 38 (2008) 955--978.

B. Lods, On linear kinetic equations involving unbounded cross-sections, Math. Methods Appl. Sci. 27 (2004) 1049--1075.

N. Moalla, A characterization of Schechter's essential spectrum by mean of measure of non-strict-singularity and application to matrix operator, Acta Math. Sci. Ser. B 32 (2012), no. 6, 2329--2340.

M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Transport Theory: New Aspects, Series on Advances in Mathematics for Applied Sciences 46, World Scientific, River Edge, NJ, 1997.

V. Muller, Spectral theory of linear operators and spectral system in Banach algebras, Operator Theory: Advances and Applications 139, Birkhuser Verlag, Basel, 2003.

A. Pelczynski, On strictly singular and strictly cosingular operators. I. Strictlly singular and cosingular operators on C(S) spaces, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965) 31--36.

V. Rakocevic, Measures of noncompactness and some applications, Filomat 12(2) (1998) 87--120.

M. Schechter, Riesz operators and Fredholm perturbations, Bull. Amer. Math. Soc. 74 (1968) 1139--1144.

M. Schechter, Quantities related to strictly singular operators, Indiana Univ. Math. J. 21 (1972) 1061--1071.

M. Schechter, Principles of Functional Analysis, Grad. Stud. Math. 36, Amer. Math. Soc. Providence, RI, 2002.

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.


Volume 43, Issue 5
September and October 2017
Pages 1543-1558
  • Receive Date: 05 May 2016
  • Revise Date: 02 September 2016
  • Accept Date: 02 September 2016