On the lattice structure of kernel operators

Gerlach, Moritz; Kunze, Markus

doi:10.1002/mana.201300218

Cited by 7 publications

(5 citation statements)

References 21 publications

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“…It was observed only recently that this result can also be used to give an analytic proof of a famous theorem by Doob about the asymptotic behaviour of oneparameter Markov semigroups, see [22] and [21,Sec 4]. During the last two decades several new versions of the above theorem were discovered.…”

Section: Introductionmentioning

confidence: 90%

Convergence of positive operator semigroups

Gerlach

Glück

2019

Trans. Amer. Math. Soc.

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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 → ∞. 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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Section: Introductionmentioning

confidence: 90%

Convergence of positive operator semigroups

Gerlach

Glück

2019

Trans. Amer. Math. Soc.

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“…[10, Thm VI.1.1] and [40,Thm 4.2]) while such criteria are more delicate in the infinite dimensional case, and in particular on non-discrete state spaces (see e.g. [6,Sec 4.2] and [20,Thm 4.4] for different versions and proofs of a classical theorem of Doob which addresses this issue; see also [19,Thm 3.6] for a related Tauberian theorem).…”

Section: Introductionmentioning

confidence: 99%

Lower bounds and the asymptotic behaviour of positive operator semigroups

Gerlach¹,

Glück²

2017

Ergod. Th. Dynam. Sys.

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If (Tt) is a semigroup of Markov operators on an L 1 -space that admits a non-trivial lower bound, then a well-known theorem of Lasota and Yorke asserts that the semigroup is strongly convergent as t → ∞. In this article we generalise 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 generalise 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.

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“…It follows from T n ≤ M for all n that k(x, E) ≤ M for all x ∈ E. Moreover, k is again a kernel, cf. [15,Lem. 3.5].…”

Section: Lemma 31 There Exists a Countable Setmentioning

confidence: 99%

A monotone convergence theorem for strong Feller semigroups

Budde¹,

Alexander²,

Glück³

et al. 2022

Preprint

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For an increasing sequence (Tn) of one-parameter semigroups of sub Markovian kernel operators over a Polish space, we study the limit semigroup and prove sufficient conditions for it to be strongly Feller. In particular, we show that the strong Feller property carries over from the approximating semigroups to the limit semigroup if the resolvent of the latter maps 1 to a continuous function.This is instrumental in the study of elliptic operators on R d with unbounded coefficients: our abstract result enables us to assign a semigroup to such an operator and to show that the semigroup is strongly Feller under very mild regularity assumptions on the coefficients.We also provide counterexamples to demonstrate that the assumptions in our main result are close to optimal.

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On the lattice structure of kernel operators

Cited by 7 publications

References 21 publications

Convergence of positive operator semigroups

Convergence of positive operator semigroups

Lower bounds and the asymptotic behaviour of positive operator semigroups

A monotone convergence theorem for strong Feller semigroups

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