. Ce-qui-signifie-que-?-?-?-ap, Il reste à montrer que ? ap (T ) ? ? ap

?. Soit-?, T ) et (? n ) n une suite vérifiant (6.22) En opérant comme précédemment on peut trouver une sous-suite (m n ) n telle que

]. S. Bibliographie1, B. Albertoni, and . Montagnini, On the spectrum of neutron transport equation in finite bodies, J. Math. Anal. Appl, vol.13, pp.19-48, 1966.

D. Aliprantis and O. Burkinshaw, Positive operators, 1985.
DOI : 10.1007/978-1-4020-5008-4

F. Andreu and J. M. Mazon, On the boundary spectrum of dominatedC o-Semigroups, Semigroup Forum, vol.303, issue.1, pp.129-139, 1989.
DOI : 10.1007/BF02573226

F. Andreu, J. Martinez, and J. M. Mazon, A spectral mapping theorem for perturbed strongly continuous semigroups, Mathematische Annalen, vol.129, issue.1, pp.453-462, 1991.
DOI : 10.1007/BF01445219

N. Angelescu, N. Marinescu, and V. Protopopescu, On the spectrum of the linear transport operator with generalized boundary conditions, Letters in Mathematical Physics, vol.32, issue.2, pp.141-146, 1976.
DOI : 10.1007/BF00398377

W. Arendt, Kato's equality and spectral decomposition for positive C0-groups, Manuscripta Mathematica, vol.164, issue.2-3, pp.277-298, 1982.
DOI : 10.1007/BF01174880

W. Arendt, C. J. Batty, M. Hieber, and F. Neubrander, Vectorvalued Laplace Transforms and Cauchy Problems, 2001.
DOI : 10.1007/bf02774144

J. Banasiak, MATHEMATICAL PROPERTIES OF INELASTIC SCATTERING MODELS IN LINEAR KINETIC THEORY, Mathematical Models and Methods in Applied Sciences, vol.10, issue.02, pp.163-186, 2000.
DOI : 10.1142/S0218202500000112

J. Banasiak, On Well-Posedness of a Boltzmann-Like Semiconductor Model, Mathematical Models and Methods in Applied Sciences, vol.13, issue.06, pp.875-892, 2003.
DOI : 10.1142/S0218202503002751

J. Banasiak, G. Frosali, and G. Spiga, Interplay of elastic and inelastic scattering operators in extended kinetic models and their hydrodynamic limits-reference manual. Transport Theory Statist, Phys, pp.31-187, 2002.

C. Bardos, Probl??mes aux limites pour les ??quations aux d??riv??es partielles du premier ordre ?? coefficients r??els; th??or??mes d'approximation; application ?? l'??quation de transport, Annales scientifiques de l'??cole normale sup??rieure, vol.3, issue.2, pp.185-233, 1970.
DOI : 10.24033/asens.1190
URL : http://www.numdam.org/article/ASENS_1970_4_3_2_185_0.pdf

C. Bardos and M. Cessenat, Décomposition de l'opérateur de transport dans une bande, C. R. Acad. Sci. Paris, vol.289, pp.95-98, 1979.

R. Beals and V. Protopopescu, Abstract time-dependent transport equations, Journal of Mathematical Analysis and Applications, vol.121, issue.2, pp.370-405, 1987.
DOI : 10.1016/0022-247X(87)90252-6
URL : http://doi.org/10.1016/0022-247x(87)90252-6

G. I. Bell and S. Glasstone, Nuclear reactor theory, 1970.

A. Belleni-morante, The initial value problem for neutron transport in a slab with perfect reflection boundary conditions, Journal of Mathematical Analysis and Applications, vol.30, issue.2, pp.353-374, 1970.
DOI : 10.1016/0022-247X(70)90167-8

M. D. Blake, A spectral bound for asymptotically norm-continuous semigroups, J. Operator Theory, vol.45, pp.111-130, 2001.

M. D. Blake, S. Brendle, and R. Nagel, On the structure of the critical spectrum of strongly continuous semigroup. Evolution equations and their applications in physical and life sciences, Lecture Notes in Pure and Appl. Math, vol.215, pp.55-65, 1998.

M. Borysiewicz and J. Mika, Time behavior of thermal neutrons in moderating media, Journal of Mathematical Analysis and Applications, vol.26, issue.3, pp.461-478, 1969.
DOI : 10.1016/0022-247X(69)90193-0

S. Brendle, On the Asymptotic Behavior of Perturbed Strongly Continuous Semigroups, Mathematische Nachrichten, vol.116, issue.1, pp.35-47, 2001.
DOI : 10.1002/1522-2616(200106)226:1<35::AID-MANA35>3.0.CO;2-R

S. Brendle, R. Nagel, and J. Poland, On the spectral mapping theorem for perturbed strongly continuous semigroups, Archiv der Mathematik, vol.74, issue.5, pp.365-378, 2000.
DOI : 10.1007/s000130050456

H. Brezis, Analyse fonctionnelle, théorie et application, 1983.

M. Cessenat, Théorèmes de trace L p pour des espaces de fonctions de la neutronique, C. R. Acad. Sci. Paris Sér. I Math, vol.299, issue.16, pp.831-834, 1984.

M. Cessenat, Théorèmes de trace pour des espaces de fonctions de la neutronique, C. R. Acad. Sci. Paris Sér. I Math, vol.300, issue.3, pp.89-92, 1985.

. Ph, Clément et al. One parameter semigroups, 1987.

R. Dautray and J. L. Lions, Analyse mathématique et calcul numérique, 1988.

E. B. Davies, One-parameter semigroups, 1980.
DOI : 10.1017/CBO9780511618864.007

D. Doods and J. Fremlin, Compact operators in Banach lattices, Israel Journal of Mathematics, vol.10, issue.4, pp.287-320, 1979.
DOI : 10.1007/BF02760610

N. Dunford and J. T. Schwartz, Linear Operators, part I. Interscience, 1958.

D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operations, 1989.

O. El-mennaoui and K. Engel, On the characterization of eventually norm continuous semigroups in Hilbert spaces, Archiv der Mathematik, vol.116, issue.5, pp.437-440, 1994.
DOI : 10.1007/BF01196674

K. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Semigroup Forum, vol.63, issue.2, 1999.
DOI : 10.1007/s002330010042

L. Gearhart, Spectral theory for contraction semigroups on Hilbert space, Transactions of the American Mathematical Society, vol.236, pp.385-394, 1978.
DOI : 10.1090/S0002-9947-1978-0461206-1

P. Gérard, Regularization by averaging for solutions of partial differential equations. Recent developments in hyperbolic equations, Pitman Res. Notes Math. Ser, vol.183, pp.59-66, 1987.

F. Golse, P. Lions, B. Perthame, and R. Sentis, Regularity of the moments of the solution of a Transport Equation, Journal of Functional Analysis, vol.76, issue.1, pp.110-125, 1988.
DOI : 10.1016/0022-1236(88)90051-1

F. Golse, B. Perthame, and R. Sentis, Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale d'un opérateur de transport, C. R. Acad. Sci. Paris, vol.301, pp.341-344, 1985.

W. Greenberg, C. Van-der-mee, and V. Propopescu, Abstract boundary value problems from kinetic theory, Transport Theory and Statistical Physics, vol.84, issue.3-4, 1987.
DOI : 10.1080/00411458208245739

G. Greiner, Positivity in time dependent linear transport operator theory, Semesterbericht Funktionalanalysis, 1982.

G. Greiner, Spectral properties and asymptotic behavior of the linear transport equation, Mathematische Zeitschrift, vol.90, issue.2, pp.167-177, 1984.
DOI : 10.1007/BF01181687

G. Greiner and R. Nagel, On the stability of strongly continuous semigroups of positive operators on L 2 (µ), Ann. Scuaola Norm. Sup. Pisa Cl. Sci- X, issue.2, pp.257-262, 1983.

G. Greiner, J. Voigt, and M. Wolff, On the spectral bound of the generator of semigroups of positive operators, J. Operator Theory, vol.5, pp.245-256, 1981.

J. Hejtmanek and H. G. Kaper, Counterexample to the spectral mapping theorem for the exponential function, Proc. Amer, pp.563-568, 1986.
DOI : 10.1090/S0002-9939-1986-0826482-1

E. Hille and R. S. Phillips, Functional analysis and semigroups, p.31, 1957.

K. Jörgens, An asymptotic expansion in the theory of neutron transport, Communications on Pure and Applied Mathematics, vol.10, issue.2, pp.219-242, 1958.
DOI : 10.1002/cpa.3160110206

H. G. Kaper, C. G. Lekkerkerker, and J. Hejtmanek, Spectral Methods in linear transport theory, 1982.

T. Kato, Perturbation Theory for linear operators, 1966.

M. Krasnoselskii, Integral operators in spaces of summable functions. Nordhoff International publishing, 1976.

I. Ku??er, A Survey of Neutron Transport Theory, Acta. Physica Austriaca supp. X, pp.491-528, 1973.
DOI : 10.1007/978-3-7091-8336-6_20

E. W. Larsen and P. F. , On the spectrum of the linear transport operator, Journal of Mathematical Physics, vol.15, issue.11, pp.1987-1997, 1974.
DOI : 10.1063/1.1666570

K. Latrach, Théorie spectrale d'équation cinétiques, Thèse de Doctorat de l'université de Franche-Comté, 1992.

K. Latrach, Description of the real point spectrum for a class of neutron transport operators, Transport Theory and Statistical Physics, vol.8, issue.5, pp.595-629, 1993.
DOI : 10.1016/0022-247X(88)90230-2

K. Latrach, On the Spectrum of the Transport Operator with Abstract Boundary Conditions in Slab Geometry, Journal of Mathematical Analysis and Applications, vol.252, issue.1, pp.1-17, 2000.
DOI : 10.1006/jmaa.2000.5543

K. Latrach, COMPACTNESS RESULTS FOR TRANSPORT EQUATIONS AND APPLICATIONS, Mathematical Models and Methods in Applied Sciences, vol.11, issue.07, pp.1181-1202, 2001.
DOI : 10.1142/S021820250100129X

K. Latrach and B. Lods, REGULARITY AND TIME ASYMPTOTIC BEHAVIOUR OF SOLUTIONS TO TRANSPORT EQUATIONS, Transport Theory and Statistical Physics, vol.30, issue.7, pp.617-639, 2001.
DOI : 10.1081/TT-100107419

J. Lehner and G. M. Wing, On the spectrum of an unsymmetric operator arising in the transport theory of neutrons, Communications on Pure and Applied Mathematics, vol.15, issue.2, pp.217-234, 1955.
DOI : 10.1002/cpa.3160080202

J. Lehner and G. M. Wing, Solution of the linearized Boltzmann transport equation for slab geometry. Duke Math, J, vol.23, pp.125-142, 1956.

M. Li, X. Gu, and F. Huang, On unbounded perturbations of semigroups: Compactness and norm continuity, Semigroup Forum, vol.65, issue.1, pp.58-70, 2002.
DOI : 10.1007/s002330010081

B. Lods, Théorie spectrale des equations cinétiques, Thèse de doctorat de l'université de Franche-Comté, 2002.

B. Lods, On linear kinetic equations involving unbounded cross-sections, Mathematical Methods in the Applied Sciences, vol.27, issue.9, pp.1049-1075, 2004.
DOI : 10.1002/mma.485

B. Lods and M. Sbihi, Stability of the essential spectrum for 2D-transport models with Maxwell boundary conditions, Mathematical Methods in the Applied Sciences, vol.24, issue.5
DOI : 10.1002/mma.684
URL : https://hal.archives-ouvertes.fr/hal-00473900

A. Majorana and C. Milazzo, Space Homogeneous Solutions of the Linear Semiconductor Boltzmann Equation, Journal of Mathematical Analysis and Applications, vol.259, issue.2, pp.609-629, 2001.
DOI : 10.1006/jmaa.2001.7444

I. Marek, Frobenius Theory of Positive Operators: Comparison Theorems and Applications, SIAM Journal on Applied Mathematics, vol.19, issue.3, pp.607-628, 1970.
DOI : 10.1137/0119060

J. Martinez and J. M. Mazon, C0-semigroups norm continuous at infinity, Semigroup Forum, vol.106, issue.1, pp.213-224, 1996.
DOI : 10.1007/BF02574097

J. Mika, R. Stankiewicz, and T. Trombetti, Effective solution to the initial value problem in neutron thermalization theory, Journal of Mathematical Analysis and Applications, vol.25, issue.1, pp.149-161, 1969.
DOI : 10.1016/0022-247X(69)90219-4

M. Mokhtar-kharroubi, La compacité dans la théorie du transport des neutrons, C. R. Acad. Sci. Paris Sér. I Math, vol.303, issue.13, pp.617-619, 1986.

M. Mokhtar-kharroubi, Compactness properties for positive semigroups on Banach lattices and applications, Houston J. Math, vol.17, issue.1, pp.25-38, 1991.

M. Mokhtar-kharroubi, Time asymptotic behaviour and compactness in neutron transport theory, Europ. J. Mech. B Fluid, vol.11, pp.39-68, 1992.

M. Mokhtar-kharroubi, Mathematical topics in neutron transport theory, new aspects, World Scientific. Series on advances in Mathematics for applied Sciences, vol.46, 1997.

M. Mokhtar-kharroubi, On the convex compactness property for the strong operator topology and related topics, Mathematical Methods in the Applied Sciences, vol.27, issue.6, pp.687-701, 2004.
DOI : 10.1002/mma.497

M. Mokhtar-kharroubi, Optimal spectral theory of the linear Boltzmann equation, Journal of Functional Analysis, vol.226, issue.1, pp.21-47, 2005.
DOI : 10.1016/j.jfa.2005.02.014

M. Mokhtar-kharroubi, Spectral Theory for Neutron Transport, J. Diff. Int. Eq
DOI : 10.1007/978-3-319-11322-7_7
URL : https://hal.archives-ouvertes.fr/hal-00488501

M. Mokhtar-kharroubi, Spectral properties of a class of positive semigroups on Banach lattices and transport absorption operators

M. Mokhtar-kharroubi and M. Sbihi, Critical Spectrum and Spectral Mapping Theorems in Transport Theory, Semigroup Forum, vol.70, issue.3, pp.406-435, 2005.
DOI : 10.1007/s00233-004-0165-6
URL : https://hal.archives-ouvertes.fr/hal-01311998

M. Mokhtar-kharroubi and M. Sbihi, Spectral mapping theorems for neutron transport, L 1 -theory. Semigroup Forum
URL : https://hal.archives-ouvertes.fr/hal-00488501

B. Montagnini, The eigenvalue spectrum of the linear Boltzmann operator in L1(R6) and L2(R6), Meccanica, vol.22, issue.2, pp.134-144, 1979.
DOI : 10.1007/BF02128506

B. Montagnini and M. L. Demuru, Complete continuity of the free gas scattering operator in neutron thermalization theory, Journal of Mathematical Analysis and Applications, vol.12, issue.1, pp.49-57, 1965.
DOI : 10.1016/0022-247X(65)90052-1

R. Nagel, One-parameter semigroups of positive operators, Lect. Notes Math, vol.1184, 1986.

R. Nagel and J. Poland, The Critical Spectrum of a Strongly Continuous Semigroup, Advances in Mathematics, vol.152, issue.1, pp.120-133, 2000.
DOI : 10.1006/aima.1998.1893

A. Pazy, Semigroups of linear operators and applications to partial differential equations, 1983.
DOI : 10.1007/978-1-4612-5561-1

J. G. Peng, M. S. Wang, and G. T. Zhu, The distribution of the spectra of the generators of a class of perturbed semigroups and its application, Chinese) Acta Math, pp.107-113, 1997.

V. Protopopescu and A. A. Corciovei, On the spectrum of the linear transport operator with diffuse reflection, Rev. Roumaine Phys, vol.21, pp.713-719, 1976.

M. Ribaric and I. Vidav, Analytic properties of the inverse A(z)???1 of an analytic linear operator valued function A (z), Archive for Rational Mechanics and Analysis, vol.28, issue.4, pp.298-310, 1969.
DOI : 10.1007/BF00281506

M. Rotenberg, Transport theory for growing cell populations, Journal of Theoretical Biology, vol.103, issue.2, pp.181-199, 1983.
DOI : 10.1016/0022-5193(83)90024-3

M. Sbihi, A resolvant approach to the stability of essential and critical spectra of perturbed C 0 -semigroups on Hilbert spaces with applications to transport theory, 2005.

M. Sbihi, Spectral theory of neutron transport semigroups with partly elastic collision operators, Journal of Mathematical Physics, vol.47, issue.12
DOI : 10.1063/1.2397557
URL : https://hal.archives-ouvertes.fr/hal-01312519

H. H. Schaefer, Topological Vector Spaces, 1971.

M. Schechter, Spectra of Partial Differential Operators, 1971.

G. Schlüchtermann, On weakly compact operators, Mathematische Annalen, vol.129, issue.1, pp.263-266, 1992.
DOI : 10.1007/BF01444620

G. Schlüchtermann, Perturbation of linear semigroups Recent progress in operator theory, Oper. Theory Adv. Appl, vol.103, pp.263-277, 1995.

D. G. Song, A note on the growth bound ofC 0-semigroups, Semigroup Forum, vol.123, issue.1, pp.190-198, 1997.
DOI : 10.1007/BF02676601

D. G. Song, Some notes on the spectral properties of C 0 -semigroups generated by linear transport operators, Transp. Theory Stat. Phys, vol.26, pp.233-242, 1997.

A. Suhadolc, Linearized Boltzmann equation in L1 space, Journal of Mathematical Analysis and Applications, vol.35, issue.1, pp.1-13, 1971.
DOI : 10.1016/0022-247X(71)90231-9

P. Takac, A spectral, mapping theorem for the exponential function in ltnear transport theory, Transport Theory and Statistical Physics, vol.2, issue.5, pp.655-667, 1985.
DOI : 10.1080/00411458508211697

P. Takac, Two counterexamples to the spectral mapping theorem for semigroups of positive operators, Integral Equations and Operator Theory, vol.60, issue.3, pp.460-467, 1986.
DOI : 10.1007/BF01199354

S. Ukai, Eigenvalues of the neutron transport operator for a homogeneous finite moderator, Journal of Mathematical Analysis and Applications, vol.18, issue.2, pp.297-314, 1967.
DOI : 10.1016/0022-247X(67)90059-5

R. Van-norton, On the real spectrum of a mono-energetic neutron transport operator, Communications on Pure and Applied Mathematics, vol.15, issue.2, pp.149-158, 1962.
DOI : 10.1002/cpa.3160150204

I. Vidav, Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator, Journal of Mathematical Analysis and Applications, vol.22, issue.1, pp.144-155, 1968.
DOI : 10.1016/0022-247X(68)90166-2

I. Vidav, Spectra of perturbed semigroups with applications to transport theory, Journal of Mathematical Analysis and Applications, vol.30, issue.2, pp.264-279, 1970.
DOI : 10.1016/0022-247X(70)90160-5

V. S. Vladimirov, Mathematical Problems in the one velocity Theory of Particle Transport, Atomic Energy of Canada Ltd. Chalk River. Ont Report AECL, p.1661, 1963.

J. Voigt, On the perturbation theory for strongly continuous semigroups, Mathematische Annalen, vol.18, issue.2, pp.163-171, 1977.
DOI : 10.1007/BF01351602

J. Voigt, A perturbation theorem for the essential spectral radius of strongly continuous semigroups, Monatshefte f???r Mathematik, vol.22, issue.2, pp.153-161, 1980.
DOI : 10.1007/BF01303264

J. Voigt, Functional analytic treatment of the initial boundary value problem for collisionless gases, 1981.

J. Voigt, positivity in time dependent linear transport theory, Acta Applicandae Mathematicae, vol.39, issue.3-4, pp.311-331, 1984.
DOI : 10.1007/BF02280857

J. Voigt, Spectral properties of the neutron transport equation, Journal of Mathematical Analysis and Applications, vol.106, issue.1, pp.140-153, 1985.
DOI : 10.1016/0022-247X(85)90137-4

J. Voigt, On the convex compactness property for the strong operator topology, Note. di. Mat, vol.12, pp.259-269, 1992.

J. Voigt, Stability of the essential type of strongly continuous semigroups, Trans. Steklov. Math. Inst, vol.203, pp.469-477, 1994.

L. Weis, A generalization of the Vidav-Jorgens perturbation theorem for semigroups and its application to transport theory, Journal of Mathematical Analysis and Applications, vol.129, issue.1, pp.6-23, 1988.
DOI : 10.1016/0022-247X(88)90230-2

L. Weis, The Stability of Positive Semigroups on L p Spaces, Proc. Amer, pp.3089-3094, 1995.
DOI : 10.2307/2160665

M. M. Williams, Mathematical methods in particle transport theory, London Butterworths, 1971.

P. F. Yao, On the Inversion of the Laplace Transform of $C_0 $ Semigroups and Its Applications, SIAM Journal on Mathematical Analysis, vol.26, issue.5, pp.1331-1341, 1995.
DOI : 10.1137/S0036141092240035

P. You, Characteristic conditions for a $c\sb 0$-semigroup with continuity in the uniform operator topology for $t>0$ in Hilbert space, Proc. Amer, pp.991-997, 1992.
DOI : 10.1090/S0002-9939-1992-1098405-7

J. Zabczyk, A note on C 0 -semigroups, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys, vol.23, pp.895-898, 1975.

Q. C. Zhang and F. L. Huang, Compact perturbation of C 0 -operator semigroups, Chinese) Sichuan Daxue Xuebao, pp.829-833, 1998.