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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Convergence of positive operator semigroups


Authors: Moritz Gerlach and Jochen Glück
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 47D03; Secondary 20M30, 47B65, 47B34
DOI: https://doi.org/10.1090/tran/7836
Published electronically: June 17, 2019
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Abstract: We present new conditions for semigroups of positive operators to converge strongly as time tends to infinity. Our proofs are based on a novel approach combining the well-known splitting theorem by Jacobs, de Leeuw, and Glicksberg with a purely algebraic result about positive group representations. Thus, we obtain convergence theorems not only for one-parameter semigroups but also for a much larger class of semigroup representations.

Our results allow for a unified treatment of various theorems from the literature that, under technical assumptions, a bounded positive $ C_0$-semigroup containing or dominating a kernel operator converges strongly as $ t \to \infty $. We gain new insights into the structure theoretical background of those theorems and generalize them in several respects; especially we drop any kind of continuity or regularity assumption with respect to the time parameter.


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Additional Information

Moritz Gerlach
Affiliation: Universität Potsdam, Institut für Mathematik, Karl-Liebknecht-Straße 24–25, 14476 Potsdam, Germany
Email: moritz.gerlach@uni-potsdam.de

Jochen Glück
Affiliation: Universität Ulm, Institut für Angewandte Analysis, 89069 Ulm, Germany
Email: jochen.glueck@alumni.uni-ulm.de

DOI: https://doi.org/10.1090/tran/7836
Keywords: Positive semigroups, semigroup representations, asymptotic behavior, kernel operator
Received by editor(s): September 18, 2017
Received by editor(s) in revised form: January 26, 2019
Published electronically: June 17, 2019
Article copyright: © Copyright 2019 American Mathematical Society