Convergence of positive operator semigroups

Gerlach, Moritz; Glück, Jochen

doi:10.1090/tran/7836

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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