[Sur un théorème de convergence pour les semi-groupes d'opérateurs intégraux positifs]
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.
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.
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@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/
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