Convergence of positive operator semigroups
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- by Moritz Gerlach and Jochen Glück PDF
- Trans. Amer. Math. Soc. 372 (2019), 6603-6627 Request permission
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
- MR Author ID: 962946
- 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
- Received by editor(s): September 18, 2017
- Received by editor(s) in revised form: January 26, 2019
- Published electronically: June 17, 2019
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 6603-6627
- MSC (2010): Primary 47D03; Secondary 20M30, 47B65, 47B34
- DOI: https://doi.org/10.1090/tran/7836
- MathSciNet review: 4024532