Lower bounds and the asymptotic behaviour of positive operator semigroups

Gerlach, Moritz; Glück, Jochen

doi:10.1017/etds.2017.9

Cited by 13 publications

(19 citation statements)

References 35 publications

(37 reference statements)

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“…Let us now proceed towards our goal to prove convergence results for positive operator semigroups in case that they admit certain lower bounds. Our results, in particular Theorem 4.4.5 and Corollary 4.4.6, substantially generalise results for AL-Banach lattices that were recently obtain by M. Gerlach and the first author [19] as well as a result of Ayupov, Sarymsakov and Grabarnik which was proved in [1] for universal lower bounds on the predual of a von Neumann algebra (see also [13, For the proof of the non-trivial implication in Theorem 4.4.1 we borrow an essential idea from the proof of [20,Theorem 4.4]. However, the proof of the latter result cannot be completely transferred to our situation since, at some steps, it heavily uses the lattice structure of the underlying space.…”

Section: 4supporting

confidence: 55%

“…Ding's results can for instance be employed in the study of certain mixing properties of dynamical systems, see [19,Section 2]. Individual lower bounds for more general positive semigroups on AL-Banach lattices were in the focus of a recent paper [20] by Gerlach and the first of the present authors. On AL-Banach lattices mean lower bounds were for instance considered in [12] and on preduals of von Neumann algebras they were studied in [16].…”

Section: Individual and Universal Lower Boundsmentioning

confidence: 95%

“…Apply the preceding Proposition 4.3.5 to ϕ = 1 where 1 denotes the norm functional on X. In case that X is a Banach lattice, the above results were implicitly contained in the proofs of Corollaries 3.5 and 4.5 of [20].…”

Section: Individual and Universal Lower Boundsmentioning

confidence: 99%

“…We point out that the condition inf{ h f : f ∈ X + , f = 1} > 0 in hypothesis (i) of the above theorem cannot be dropped, in general -even if X is an L 1 -space; a counterexample can be found in [20,Example 4.3]. Before we prove Theorem 4.4.5 we present a couple of important special cases: Now, let J = [0, ∞).…”

Section: 4mentioning

confidence: 99%

“…For each t > 0 the discrete semigroup (T nt ) n∈N0 also fulfils the hypotheses of the theorem; hence, by what we have just proved, this semigroup is strongly convergent. According to [20,Theorem 2.1] this implies that the semigroup (T t ) t∈[0,∞) is strongly convergent, too.…”

Section: 4mentioning

confidence: 99%

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Long-term analysis of positive operator semigroups via asymptotic domination

Glück

Wolff

2019

Positivity

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We consider positive operator semigroups on ordered Banach spaces and study the relation of their long time behaviour to two different domination properties. First, we analyse under which conditions almost periodicity and mean ergodicity of a semigroup T are inherited by other semigroups which are asymptotically dominated by T . Then, we consider semigroups whose orbits asymptotically dominate a positive vector and show that this assumption is often sufficient to conclude strong convergence of the semigroup as time tends to infinity.Our theorems are applicable to time-discrete as well as time-continuous semigroups. They generalise several results from the literature to considerably larger classes of ordered Banach spaces.

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Section: 4supporting

confidence: 55%

Section: Individual and Universal Lower Boundsmentioning

confidence: 95%

Section: Individual and Universal Lower Boundsmentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

See 3 more Smart Citations

Long-term analysis of positive operator semigroups via asymptotic domination

Glück

Wolff

2019

Positivity

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Uniform convergence of operator semigroups without time regularity

Alexander

Glück

2021

J. Evol. Equ.

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When we are interested in the long-term behaviour of solutions to linear evolution equations, a large variety of techniques from the theory of $$C_0$$ C 0 -semigroups is at our disposal. However, if we consider for instance parabolic equations with unbounded coefficients on $${\mathbb {R}}^d$$ R d , the solution semigroup will not be strongly continuous, in general. For such semigroups many tools that can be used to investigate the asymptotic behaviour of $$C_0$$ C 0 -semigroups are not available anymore and, hence, much less is known about their long-time behaviour. Motivated by this observation, we prove new characterisations of the operator norm convergence of general semigroup representations—without any time regularity assumptions—by adapting the concept of the “semigroup at infinity”, recently introduced by M. Haase and the second named author. Besides its independence of time regularity, our approach also allows us to treat the discrete-time case (i.e. powers of a single operator) and even more abstract semigroup representations within the same unified setting. As an application of our results, we prove a convergence theorem for solutions to systems of parabolic equations with the aforementioned properties.

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On the asymptotic behaviour of semigroups for flows in infinite networks

Alexander

2022

Semigroup Forum

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We study transport processes on infinite networks. The solution of these processes can be modeled by an operator semigroup on a suitable Banach space. Classically, such semigroups are strongly continuous and therefore their asymptotic behaviour is quite well understood. However, recently new examples of transport processes emerged where the corresponding semigroup is not strongly continuous. Due to this lack of strong continuity, there are currently only few results on the long-term behaviour of these semigroups. In this paper, we discuss the asymptotic behaviour for a certain class of these transport processes. In particular, it is proved that the solution semigroups behave asymptotically periodic with respect to the operator norm as a consequence of a more general result on the long-term behaviour by positive semigroups containing a multiplication operator. Furthermore, we revisit known results on the asymptotic behaviour of transport processes on infinite networks and prove the asymptotic periodicity of their extensions to the space of bounded measures.

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Lower bounds and the asymptotic behaviour of positive operator semigroups

Cited by 13 publications

References 35 publications

Long-term analysis of positive operator semigroups via asymptotic domination

Long-term analysis of positive operator semigroups via asymptotic domination

Uniform convergence of operator semigroups without time regularity

On the asymptotic behaviour of semigroups for flows in infinite networks

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