TY - JOUR A1 - Gerlach, Moritz Reinhardt A1 - Glück, Jochen T1 - Convergence of positive operator semigroups JF - Transactions of the American Mathematical Society N2 - 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 ->infinity. 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. KW - Positive semigroups KW - semigroup representations KW - asymptotic behavior KW - kernel operator Y1 - 2019 U6 - https://doi.org/10.1090/tran/7836 SN - 0002-9947 SN - 1088-6850 VL - 372 IS - 9 SP - 6603 EP - 6627 PB - American Mathematical Soc. CY - Providence ER - TY - JOUR A1 - Edeko, Nikolai A1 - Gerlach, Moritz Reinhardt A1 - Kühner, Viktoria T1 - Measure-preserving semiflows and one-parameter Koopman semigroups JF - Semigroup forum N2 - For a finite measure space X, we characterize strongly continuous Markov lattice semigroups on Lp(X) by showing that their generator A acts as a derivation on the dense subspace D(A)L(X). We then use this to characterize Koopman semigroups on Lp(X) if X is a standard probability space. In addition, we show that every measurable and measure-preserving flow on a standard probability space is isomorphic to a continuous flow on a compact Borel probability space. KW - Measure-preserving semiflow KW - Koopman semigroup KW - Derivation KW - Topological model Y1 - 2019 U6 - https://doi.org/10.1007/s00233-018-9960-3 SN - 0037-1912 SN - 1432-2137 VL - 98 IS - 1 SP - 48 EP - 63 PB - Springer CY - New York ER - TY - JOUR A1 - Gerlach, Moritz Reinhardt A1 - Glück, Jochen T1 - Mean ergodicity vs weak almost periodicity JF - Studia mathematica N2 - We provide explicit examples of positive and power-bounded operators on c(0) and l(infinity) which are mean ergodic but not weakly almost periodic. As a consequence we prove that a countably order complete Banach lattice on which every positive and power-bounded mean ergodic operator is weakly almost periodic is necessarily a KB-space. This answers several open questions from the literature. Finally, we prove that if T is a positive mean ergodic operator with zero fixed space on an arbitrary Banach lattice, then so is every power of T . KW - positive operators KW - weakly almost periodic KW - order continuous norm KW - KB-space KW - mean ergodic Y1 - 2019 U6 - https://doi.org/10.4064/sm170918-20-3 SN - 0039-3223 SN - 1730-6337 VL - 248 IS - 1 SP - 45 EP - 56 PB - Polska Akademia Nauk, Instytut Matematyczny CY - Warszawa ER - TY - JOUR A1 - Gerlach, Moritz Reinhardt A1 - Glück, Jochen T1 - Lower bounds and the asymptotic behaviour of positive operator semigroups JF - Ergodic theory and dynamical systems N2 - If (T-t) is a semigroup of Markov operators on an L-1-space that admits a nontrivial lower bound, then a well-known theorem of Lasota and Yorke asserts that the semigroup is strongly convergent as t -> infinity. In this article we generalize and improve this result in several respects. First, we give a new and very simple proof for the fact that the same conclusion also holds if the semigroup is merely assumed to be bounded instead of Markov. As a main result, we then prove a version of this theorem for semigroups which only admit certain individual lower bounds. Moreover, we generalize a theorem of Ding on semigroups of Frobenius-Perron operators. We also demonstrate how our results can be adapted to the setting of general Banach lattices and we give some counterexamples to show optimality of our results. Our methods combine some rather concrete estimates and approximation arguments with abstract functional analytical tools. One of these tools is a theorem which relates the convergence of a time-continuous operator semigroup to the convergence of embedded discrete semigroups. Y1 - 2017 U6 - https://doi.org/10.1017/etds.2017.9 SN - 0143-3857 SN - 1469-4417 VL - 38 SP - 3012 EP - 3041 PB - Cambridge Univ. Press CY - New York ER - TY - JOUR A1 - Gerlach, Moritz Reinhardt T1 - Convergence of dynamics and the Perron-Frobenius operator JF - Israel Journal of Mathematics N2 - We complete the picture how the asymptotic behavior of a dynamical system is reflected by properties of the associated Perron-Frobenius operator. Our main result states that strong convergence of the powers of the Perron-Frobenius operator is equivalent to setwise convergence of the underlying dynamic in the measure algebra. This situation is furthermore characterized by uniform mixing-like properties of the system. Y1 - 2018 U6 - https://doi.org/10.1007/s11856-018-1671-7 SN - 0021-2172 SN - 1565-8511 VL - 225 IS - 1 SP - 451 EP - 463 PB - Hebrew univ magnes press CY - Jerusalem ER - TY - JOUR A1 - Gerlach, Moritz Reinhardt A1 - Glück, Jochen T1 - On a convergence theorem for semigroups of positive integral operators JF - Comptes Rendus Mathematique N2 - 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. Y1 - 2017 U6 - https://doi.org/10.1016/j.crma.2017.07.017 SN - 1631-073X SN - 1778-3569 VL - 355 SP - 973 EP - 976 PB - Elsevier CY - Paris ER -