Jeribi, Aref:
A characterization of the essential spectrum and applications
Bollettino dell'Unione Matematica Italiana Serie 8 5-B (2002), fasc. n.3, p. 805-825, Unione Matematica Italiana (English)
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Sunto
In questo articolo lo spettro essenziale di operatori lineari chiusi e densamente definiti è caratterizzato in una grande classe degli spazi, che possiedono la proprietà di Dunford-Pettis o che sono isomorfi ad uno degli spazi $L_{p}(\Omega)$$p>1$. È dato un test di verifica pratico che garantisce la sua stabilità, per gli operatori perturbati. Inoltre applichiamo i risultati ottenuti per studiare lo spettro essenziale dell'equazione unidimensionale di trasporto con gli stati di contorno generali. Per concludere, sono discusse le condizioni sufficienti in termini di contorno e di operatori di scontro che assicurano l'invarianza dello spettro essenziale dell'operatore di flusso continuo.
Referenze Bibliografiche
[1]
M. BOULANOUAR-
L. LEBOUCHER,
Une équation de transport dans la dynamique des populations cellulaires,
C. R. Acad. Sci. Paris, Serie I,
321 (
1995), 305-308. |
MR 1346131 |
Zbl 0835.92025[2] R. DAUTRAY-J. L. LIONS, Analyse Mathématique et Calcul Numérique, Tome 9, Masson, Paris, 1988.
[3]
J. DIESTEL,
A survey of results related to Dunfor-Pettis property, in «
Contemporary Math. 2,
Amer. Math. Soc. of
Conf. on Integration, Topology and Geometry in Linear Spaces» (
1980), 15-60. |
MR 621850 |
Zbl 0571.46013[4]
N. DUNFORD AND
PETTIS,
Linear operations on summable functions,
Trans. Amer. Math. Soc.,
47 (
1940), 323-392. |
MR 2020 |
Jbk 66.0556.01[5]
N. DUNFORD-
J. T. SCHWARTZ,
Linears operators, Part 1,
Interscience Publishers Inc., New York,
1958. |
MR 117523 |
Zbl 0084.10402[6]
I. BOHBERG-
A. MARKUS-
I. A. FELDMAN,
Normally solvable operators and ideals associated with them,
Amer. Math. Soc. Transl. ser. 2,
61 (
1967), 63-84. |
Zbl 0181.40601[7]
S. GOLDBERT,
Unbounded Linear Operators,
McGraw-Hill, New-York,
1966. |
Zbl 0148.12501[8]
W. GREENBERG-
G. VAN DER MEE-
V. PROTOPOPESCU,
Boundary Value Problems in Abstract Kinetic Theory,
Birkhauser, Basel,
1987. |
MR 896904 |
Zbl 0624.35003[9]
G. GREINER,
Spectral properties and asymptotic behaviour of the linear transport equation,
Math. Z.,
185 (
1984), 167-177. |
MR 731337 |
Zbl 0567.45001[10]
A. GROTHENDIECK,
Sur les applications linéaires faiblement compactes d'espaces du type $C(K)$,
Canad. J. Math.,
5 (
1953), 129-173. |
MR 58866 |
Zbl 0050.10902[11]
K. GUSTAFSON-
J. WEIDMANN,
On the essential spectrum,
J. Math. Anal. Appl.,
25, 6 (
1969), 121-127. |
MR 242004 |
Zbl 0189.44104[12]
A. JERIBI,
Quelques remarques sur les opérateurs de Frédholm et application à l'équation de transport,
C. R. Acad. Sci. Paris, Série I,
325 (
1997), 43-48. |
MR 1461395 |
Zbl 0883.47006[13]
A. JERIBI,
Quelques remarques sur le spectre de Weyl et applications,
C. R. Acad. Sci. Paris, Série I,
327 (
1998), 485-490. |
MR 1652578 |
Zbl 0932.47002[14]
A. JERIBI-
K. LATRACH,
Quelques remarques sur le spectre essentiel et application à l'équation de transport,
C. R. Acad. Sci. Paris, Série I,
323 (
1996), 469-474. |
MR 1408978 |
Zbl 0861.47001[15] K. JÖRGEN, Linear integral operator, Pitman. Advanced publishing program (1982).
[16]
H. G. KAPER-
C. G. LEKKERKERKER-
J. HEJTMANEK,
Spetral Methods in Linear Transport Theory,
Birkhauser, Basel,
1982. |
MR 685594 |
Zbl 0498.47001[17]
T. KATO,
Perturbation theory for nullity, deficiency and other quantities of linear operators,
J. Anal. Math.,
6 (
1958), 261-322. |
MR 107819 |
Zbl 0090.09003[18]
K. LATRACH,
Compactness properties for linear transport operator with abstract boundary conditions in slab geometry,
Trans. Theor. Stat. Phys.,
22 (
1993), 39-65. |
MR 1198878 |
Zbl 0774.45006[19]
K. LATRACH,
Some remarks on the essential spectrum of transport operators with abstract boundary conditions,
J. Math. Phys.,
35 (
1994), 6199-6212. |
MR 1299947 |
Zbl 0819.47068[20]
K. LATRACH,
Time asymptotic behaviour for linear transport equation with abstract boundary conditions in slab geometry,
Trans. Theor. Stat. Phys.,
23 (
1994), 633-670. |
MR 1272677 |
Zbl 0816.45008[21]
K. LATRACH,
Description of the real point spectrum for a class of neutron transport operators,
Trans. Theor. Stat. Phys.,
23 (
1993), 595-630. |
MR 1229293 |
Zbl 0786.45010[22]
K. LATRACH,
Quelques propriétés spectrales d'opérateurs de transport avec des conditions aux limites abstraites,
C. R. Acad. Sci. Paris, Série I,
320 (
1995), 809-814. |
MR 1326687 |
Zbl 0827.35108[23]
K. LATRACH,
Essential spectra on spaces with the Dunfor-Pettis property,
J. Math. Anal. Appl.,
233 (
1999), 607-722. |
MR 1689610 |
Zbl 0930.47008[24]
K. LATRACH-
A. JERIBI,
On the essential spectrum of transport operators on L1- spaces,
J. Math. Phys.,
37 (
1996), 6486-6494. |
MR 1419181 |
Zbl 0876.47002[25]
K. LATRACH-
A. JERIBI,
Sur une équation de transport intervenanten dynamique des populations,
C. R. Acad. Sci. Paris, Série I,
325 (
1997), 1087-1109. |
MR 1614016 |
Zbl 0897.35020[26]
K. LATRACH A. JERIBI,
A nonlinear boundary value problem arising in growing cell populations,
Nonli. Anal., Ser. A, Theory methods,
36 (7) (
1999), 843-862. |
MR 1682848 |
Zbl 0935.35170[27]
K. LATRACH-
A. JERIBI,
Some results on Fredholm operators, essential spectra, and application,
J. Math. Anal.,
225 (
1998), 461-485. |
MR 1644272 |
Zbl 0927.47007[28]
J. L. LEBOWITZ-
S. I. RUBINOW,
A theory for the age and generation time distribution of a microbial population,
J. Math. Biol.,
1 (
1974), 17-36. |
MR 496810 |
Zbl 0402.92023[29]
J. LINDENSTRAUSS-
L. TZAFRIRI,
Classical Banach Spaces I,
Springer-Verlag, Berlin-Heidelberg-New York,
1977. |
MR 500056 |
Zbl 0362.46013[30]
V. D. MILMAN,
Some properties of strictly singular operators,
Functional Ana. Appl.,
3 (
1969), 77-78. |
MR 241997 |
Zbl 0179.17801[31]
M. MOKHTAR-KHARROUBI,
Spectral theory of the neutron transport operator in bounded geometries,
Trans. Theor. Stat. Phys.,
16 (
1987), 467-502. |
MR 906915 |
Zbl 0629.45011[32]
M. MOKHTAR-KHARROUBI,
Time asymptotic behaviour and compactness in neutron transport theory,
Europ. J. of Mech., B Fluid,
11 (
1992), 39-68. |
MR 1151539[33]
A. PALCZEWSKI,
Spectral properties of the space nonhomogeneous linearized Boltzmann operator,
Transp. Theor. Stat. Phys.,
13 (
1984), 409-430. |
MR 759864 |
Zbl 0585.47023[34]
A. PELCZYNSKI,
Strictly singular and cosingular operators,
Bull. Acad. Polon. Sci.,
13 (
1965), 31-41. |
MR 177300 |
Zbl 0138.38604[35]
M. ROTENBERG,
Transport theory for growing cell populations,
J. Theor. Biol.,
103 (
1983), 181-199. |
MR 713945[37]
M. SCHECHTER,
Spectra of Partial Differential Operators,
Horth-Holland, Amsterdam,
1971. |
MR 447834 |
Zbl 0225.35001[38]
M. SCHECHTER,
On the essential spectrum of an arbitrary operator,
J. Math. Anal. Appl.,
13 (
1965), 205-215. |
MR 188798 |
Zbl 0147.12101[39]
C. VAN DER MEE-
P. ZWEIFEL,
A. Fokker-Plank equation for growing cell populations,
J. Math. Biol.,
25 (
1987), 61-72. |
MR 886612 |
Zbl 0644.92019[40]
I. VIDAV,
Existence and uniqueness of nonnegative eigenfunction of the Boltzmann operator,
J. Math. Anal. Appl.,
22 (
1968), 144-155. |
MR 230531 |
Zbl 0155.19203[41]
J. VOIGT,
Positivity in time dependent transport theory,
Acta Aplicandae Mathematicae,
2 (
1984), 311-331. |
MR 753698 |
Zbl 0579.47040[42]
L. WEIS,
On perturbation of Fredholm operators in $L^p$-spaces,
Proc. Amer. Math. Soc.,
67 (
1977), 87-92. |
MR 467377 |
Zbl 0377.46016[43]
F. WOLF,
On the essential spectrum of partial differential boundary problems,
Comm. Pure Appl. Math.,
12 (
1959), 211-228. |
MR 107750 |
Zbl 0087.30501[44]
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. |
MR 107751 |
Zbl 0086.08203
Con il contributo del Ministero per i Beni e le Attività Culturali