We give a new and very short proof of a theorem of Greiner asserting that a positive and contractive -semigroup on an -space is strongly convergent in case it has a strictly positive fixed point and contains an integral operator. Our proof is a streamlined version of a much more general approach to the asymptotic theory of positive semigroups developed recently by the authors. Under the assumptions of Greiner's theorem, this approach becomes particularly elegant and simple. We also give an outlook on several generalisations of this result.
Nous présentons une nouvelle preuve très courte d'un théorème de Greiner qui dit qu'un semi-groupe de contractions positives sur un espace converge fortement au cas où il contiendrait un opérateur intégral et posséderait un point fixe positif presque partout. Notre preuve est une version simplifiée d'une approche plus générale de la théorie asymptotique des semi-groupes positifs développée récemment par les auteurs. Dans la situation du théorème de Greiner, cette approche est particulièrement élégante et simple. Finalement, on présente un bref aperçu de plusieurs généralisations de ce résultat.
Accepted:
Published online:
Moritz Gerlach 1; Jochen Glück 2
@article{CRMATH_2017__355_9_973_0,
author = {Moritz Gerlach and Jochen Gl\"uck},
title = {On a convergence theorem for semigroups of positive integral operators},
journal = {Comptes Rendus. Math\'ematique},
pages = {973--976},
publisher = {Elsevier},
volume = {355},
number = {9},
year = {2017},
doi = {10.1016/j.crma.2017.07.017},
language = {en},
}
Moritz Gerlach; Jochen Glück. On a convergence theorem for semigroups of positive integral operators. Comptes Rendus. Mathématique, Volume 355 (2017) no. 9, pp. 973-976. doi : 10.1016/j.crma.2017.07.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.07.017/
[1] Locally Solid Riesz Spaces, Pure Appl. Math., vol. 76, Academic Press [Harcourt Brace Jovanovich, Publishers], New York, London, 1978
[2] Positive semigroups of kernel operators, Positivity, Volume 12 (2008) no. 1, pp. 25-44
[3] Integral representations of resolvents and semigroups, Forum Math., Volume 6 (1994) no. 1, pp. 111-135
[4] One-Parameter Semigroups of Positive Operators, Lect. Notes Math., vol. 1184, Springer-Verlag, Berlin, 1986
[5] Asynchronous exponential growth of a general structured population model, Acta Appl. Math., Volume 119 (2012), pp. 149-166
[6] Asymptotic behavior of distributions of mRNA and protein levels in a model of stochastic gene expression, J. Math. Anal. Appl., Volume 333 (2007) no. 2, pp. 753-769
[7] Triviality of the peripheral point spectrum, J. Evol. Equ., Volume 5 (2005) no. 3, pp. 407-415
[8] Dynamics of Kolmogorov systems of competitive type under the telegraph noise, J. Differ. Equ., Volume 250 (2011) no. 1, pp. 386-409
[9] Operator Theoretic Aspects of Ergodic Theory, Grad. Texts Math., vol. 272, Springer-Verlag, New York, 2015
[10] One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts Math., vol. 194, Springer-Verlag, New York, 2000
[11] On the peripheral point spectrum and the asymptotic behavior of irreducible semigroups of Harris operators, Positivity, Volume 17 (2013) no. 3, pp. 875-898
[12] Convergence of positive operator semigroups, 2017 (preprint) | arXiv
[13] On the lattice structure of kernel operators, Math. Nachr., Volume 288 (2015) no. 5–6, pp. 584-592
[14] A new proof of Doob's theorem, J. Math. Anal. Appl., Volume 388 (2012) no. 2, pp. 763-774
[15] Spektrum und Asymptotik stark stetiger Halbgruppenpositiver Operatoren, Sitzungsber. Heidelb. Akad. Wiss., Math. Naturwiss. Kl. (1982), pp. 55-80
[16] On the peripheral spectrum of bounded positive semigroups on atomic Banach lattices, Arch. Math. (Basel), Volume 87 (2006) no. 4, pp. 359-367
[17] Ergodic Theorems, De Gruyter Stud. Math., vol. 6, Walter de Gruyter & Co., Berlin, 1985
[18] Dynamics and density evolution in piecewise deterministic growth processes, Ann. Pol. Math., Volume 94 (2008) no. 2, pp. 111-129
[19] Dynamic behavior of stochastic gene expression models in the presence of bursting, SIAM J. Appl. Math., Volume 73 (2013) no. 5, pp. 1830-1852
[20] Banach Lattices, Universitext, Springer-Verlag, Berlin, 1991
[21] Asymptotic decomposition of substochastic semigroups and applications, Stoch. Dyn. (2017) (in press) | DOI
[22] Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl., Volume 249 (2000) no. 2, pp. 668-685
[23] Asymptotic decomposition of substochastic operators and semigroups, J. Math. Anal. Appl., Volume 436 (2016) no. 1, pp. 305-321
[24] Long-time behaviour of a stochastic prey–predator model, Stoch. Process. Appl., Volume 108 (2003) no. 1, pp. 93-107
[25] , Springer Berlin Heidelberg, Berlin, Heidelberg (2002), pp. 215-238
[26] Banach Lattices and Positive Operators, Grundlehren Math. Wiss., vol. 215, Springer-Verlag, New York, 1974
[27] Triviality of the peripheral point spectrum of positive semigroups on atomic Banach lattices, Positivity, Volume 12 (2008) no. 1, pp. 185-192
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