Lower bounds and the asymptotic behaviour of positive operator semigroups
- 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 theIf (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.…
Verfasserangaben: | Moritz Reinhardt GerlachORCiDGND, Jochen GlückORCiDGND |
---|---|
DOI: | https://doi.org/10.1017/etds.2017.9 |
ISSN: | 0143-3857 |
ISSN: | 1469-4417 |
Titel des übergeordneten Werks (Englisch): | Ergodic theory and dynamical systems |
Verlag: | Cambridge Univ. Press |
Verlagsort: | New York |
Publikationstyp: | Wissenschaftlicher Artikel |
Sprache: | Englisch |
Datum der Erstveröffentlichung: | 02.05.2017 |
Erscheinungsjahr: | 2018 |
Datum der Freischaltung: | 25.06.2021 |
Band: | 38 |
Seitenanzahl: | 30 |
Erste Seite: | 3012 |
Letzte Seite: | 3041 |
Fördernde Institution: | LGFG Baden-Wurttemberg, Germany [1301 LGFG-E] |
Organisationseinheiten: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
DDC-Klassifikation: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
Peer Review: | Referiert |
Publikationsweg: | Open Access / Green Open-Access |