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Relatively compact-like perturbations, essential spectra and application

Part of: Integral, integro-differential, and pseudodifferential operators General theory of linear operators

Published online by Cambridge University Press:  09 April 2009

Khalid Latrach  and
J. Martin Paoli
Khalid Latrach
Affiliation:
Départment de Mathématiques, Université de Corse, Quartier Grossetti, BP. 52, 20250 Corte, France e-mail: [email protected]
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Abstract

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The purpose of this paper is to provide a detailed treatment of the behaviour of essential spectra of closed densely defined linear operators subjected to additive perturbations not necessarily belonging to any ideal of the algebra of bounded linear operators. If A denotes a closed densely defined linear operator on a Banach space X, our approach consists principally in considering the class of A-closable operators which, regarded as operators in ℒ(XA, X) (where XA denotes the domain of A equipped with the graph norm), are contained in the set of A-Fredholm perturbations (see Definition 1.2). Our results are used to describe the essential spectra of singular neutron transport equations in bounded geometries.

MSC classification

Secondary: 47A53: (Semi-) Fredholm operators; index theories 47A55: Perturbation theory 47G20: Integro-differential operators
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 77 , Issue 1 , August 2004 , pp. 73 - 90
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Calkin, J. W., ‘Two-sided ideals and congruences in the ring of bounded operators in Hubert spaces’, Ann. of Math. 42 (1941), 839–873.CrossRefGoogle Scholar
[2]Chabi, M. and Mokhtar-Kharroubi, M., ‘Singular neutron transport equations in L 1 spaces’, preprint, 1995.Google Scholar
[3]Dautray, R. and Lions, J. L., Analyse mathématique et calcul numérique: Tome 9 (Masson, Paris, 1988).Google Scholar
[4]Demeru, M. L. and Montagnini, B., ‘Complete continuity of the free gas scattering operator in neutron thermalization theory’, J. Math. Anal. Appl. 12 (1965), 49–57.Google Scholar
[5]Diestel, J. and Uhl, J. J., Vector measures, Math. Surveys 15 (Amer. Math. Soc., Providence, R.I., 1977).CrossRefGoogle Scholar
[6]Dunford, N. and Schwartz, J. T., Linear operators. Part I (Intersciences, New York, 1958).Google Scholar
[7]Gohberg, I. and Krein, G., ‘Fundamental theorems on deficiency numbers, root numbers and indices of linear operators’, Amer. Math. Soc. Transl. Ser. 2 13 (1960), 185–264.Google Scholar
[8]Gohberg, I., Markus, A. and Feldman, I. A., ‘Normally solvable operators and ideals associated with them’, Amer. Math. Soc. Transl. Ser. 2 61 (1967), 63–84.Google Scholar
[9]Greenberg, W., Van der Mee, C. and Protopopescu, V., Boundary value problems in abstract kinetic theory (Birkhäuser, Basel, 1987).Google Scholar
[10]Gustafson, K. and Weidmann, J., ‘On the essential spectrum’, J. Math. Anal. Appl. 25 (1969), 121–127.Google Scholar
[11]Kaper, H. G., Lekkerkerker, C. G. and Hejtmanek, J., Spectral methods in linear transport theory (Birkhäuser, Basel, 1982).Google Scholar
[12]Kato, T., ‘Perturbation theory for nullity, deficiency and other quantities of linear operators’, J. Anal. Math. 6 (1958), 261–322.CrossRefGoogle Scholar
[13]Kato, T., Perturbation theory for linear operators (Springer, New York, 1966).Google Scholar
[14]Krasnosel'skii, M. A. et al. , Integral operators in space of summable functions (Noordhoff, Leyden, 1976).Google Scholar
[15]Latrach, K., ‘Essential spectra on spaces with the Dunford-Pettis property’, J. Math. Anal. Appl. 233 (1999), 607–622.Google Scholar
[16]Latrach, K. and Dehici, A., ‘Relatively strictly singular perturbations, essential spectra and application’, J. Math. Anal. Appl. 252 (2000), 767–789.Google Scholar
[17]Latrach, K., ‘Fredholm, semi-Fredholm perturbations and essential spectra’, J. Math. Anal. Appl. 259 (2001), 277–301.Google Scholar
[18]Lods, B., ‘On singular neutron transport equations’, preprint, 2000.Google Scholar
[19]Mokhtar-Kharroubi, M., Mathematical topics in neutron transport theory, new aspects, Adv. Math. Sci. Appl. 46 (World Scientific, Singapore, 1997).Google Scholar
[20]Nagy, B. S., ‘On the stability of the index of unbounded linear transformations’, Acta Math. Hungar. 3 (1952), 49–52.Google Scholar
[21]Pelczynski, A., ‘Strictly singular and strictly cosingular operators’, Bull. Acad. Polon. Sci. 13 (1965), 31–41.Google Scholar
[22]Reed, M. and Simon, B., Methods of modern mathematical physics. IV Analysis of operators (Academic Press, New York, 1978).Google Scholar
[23]Schechter, M., ‘On the essential spectrum of an arbitrary operator, I’, J. Math. Anal. Appl. 13 (1966), 205–215.Google Scholar
[24]Schechter, M., Principles of functional analysis (Academic Press, New York, 1971).Google Scholar
[25]Suhadoic, A., ‘Linearized Boltzmann equation in L1 space’, J. Math. Anal. Appl. 35 (1971), 1–13.Google Scholar
[26]Vidav, I., ‘Existence and uniqueness of nonnegative eigenfunction of the Boltzmann operator’, J. Math. Anal. Appl. 22 (1968), 144–155.Google Scholar
[27]Vladimirskii, J. I., ‘Strictly cosingular operators’, Soviet Math. Dokl. 8 (1967), 739–740.Google Scholar
[28]Voigt, J., ‘On substochastic c0-semigroups and their generators’, in: Semesterbericht Funktionalanalysis. Tübingen. Wintersemester (1984/1985) pp. 1–15.Google Scholar
[29]Whitley, R. J., ‘Strictly singular operators and their conjugates’, Trans. Amer Math. Soc. 18 (1964), 252–261.CrossRefGoogle Scholar
[30]Wolf, F., ‘On the invariance of the essential spectrum under a change of the boundary conditions of partial differential operators’, Indag. Math. 21 (1959), 142–147.Google Scholar