Convergence of Positive Operator Semigroups

Moritz Gerlach; Jochen Glück

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arXiv · 1705.01587

Convergence of Positive Operator Semigroups

Moritz Gerlach·Jochen Glück

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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 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 generalise them in several respects; especially we drop any kind of continuity or regularity assumption with respect to the time parameter.

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BibTeX RIS

Moritz Gerlach, Jochen Glück. 2017-05-03. Convergence of Positive Operator Semigroups. https://arxiv.org/abs/1705.01587

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