@article{GerlachGlueck2017,
  author    = {Gerlach, Moritz Reinhardt and Gl{\"u}ck, Jochen},
  title     = {On a convergence theorem for semigroups of positive integral operators},
  series = {Comptes Rendus Mathematique},
  volume    = {355},
  journal   = {Comptes Rendus Mathematique},
  publisher = {Elsevier},
  address   = {Paris},
  issn      = {1631-073X},
  doi       = {10.1016/j.crma.2017.07.017},
  pages     = {973 -- 976},
  year      = {2017},
  abstract  = {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.},
  language  = {en}
}
@article{GerlachGlueck2019,
  author    = {Gerlach, Moritz Reinhardt and Gl{\"u}ck, Jochen},
  title     = {Mean ergodicity vs weak almost periodicity},
  series = {Studia mathematica},
  volume    = {248},
  journal   = {Studia mathematica},
  number    = {1},
  publisher = {Polska Akademia Nauk, Instytut Matematyczny},
  address   = {Warszawa},
  issn      = {0039-3223},
  doi       = {10.4064/sm170918-20-3},
  pages     = {45 -- 56},
  year      = {2019},
  abstract  = {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 .},
  language  = {en}
}
@article{GerlachGlueck2018,
  author    = {Gerlach, Moritz Reinhardt and Gl{\"u}ck, Jochen},
  title     = {Lower bounds and the asymptotic behaviour of positive operator semigroups},
  series = {Ergodic theory and dynamical systems},
  volume    = {38},
  journal   = {Ergodic theory and dynamical systems},
  publisher = {Cambridge Univ. Press},
  address   = {New York},
  issn      = {0143-3857},
  doi       = {10.1017/etds.2017.9},
  pages     = {3012 -- 3041},
  year      = {2018},
  abstract  = {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.},
  language  = {en}
}
@article{GerlachGlueck2019,
  author    = {Gerlach, Moritz Reinhardt and Gl{\"u}ck, Jochen},
  title     = {Convergence of positive operator semigroups},
  series = {Transactions of the American Mathematical Society},
  volume    = {372},
  journal   = {Transactions of the American Mathematical Society},
  number    = {9},
  publisher = {American Mathematical Soc.},
  address   = {Providence},
  issn      = {0002-9947},
  doi       = {10.1090/tran/7836},
  pages     = {6603 -- 6627},
  year      = {2019},
  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 ->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.},
  language  = {en}
}