[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 = {Gerlach, Moritz and Gl\"uck, Jochen},
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},
url = {http://www.numdam.org/articles/10.1016/j.crma.2017.07.017/}
}
TY - JOUR AU - Gerlach, Moritz AU - Glück, Jochen TI - On a convergence theorem for semigroups of positive integral operators JO - Comptes Rendus. Mathématique PY - 2017 SP - 973 EP - 976 VL - 355 IS - 9 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2017.07.017/ DO - 10.1016/j.crma.2017.07.017 LA - en ID - CRMATH_2017__355_9_973_0 ER -
%0 Journal Article %A Gerlach, Moritz %A Glück, Jochen %T On a convergence theorem for semigroups of positive integral operators %J Comptes Rendus. Mathématique %D 2017 %P 973-976 %V 355 %N 9 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2017.07.017/ %R 10.1016/j.crma.2017.07.017 %G en %F CRMATH_2017__355_9_973_0
Gerlach, Moritz; Glück, Jochen. On a convergence theorem for semigroups of positive integral operators. Comptes Rendus. Mathématique, Tome 355 (2017) no. 9, pp. 973-976. doi : 10.1016/j.crma.2017.07.017. http://www.numdam.org/articles/10.1016/j.crma.2017.07.017/
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