Uniform Ergodicity of Lotz–Räbiger Nets of Markov Operators on Abstract State Spaces

In order to successfully explore quantum systems which are perturbations of simple models, it is essential to understand the complexity of perturbation bounds. We must ask ourselves: How quantum many-body systems can be artificially engineered to produce the needed behavior. Therefore, it is convenient to make use of abstract framework to better understand classical and quantum systems. Thus, our investigation's purpose is to explore stability and perturbation bounds of positive C0-semigroups on abstract state spaces using the Dobrushin's ergodicity coefficient. Consequently, we obtain a linear relation between the stability of the semigroup and the sensitivity of its fixed point with respect to perturbations of C0-Markov semigroups. Our investigation leads to the discovery of perturbation bounds for the time averages of uniform asymptotically stable semigroups. A noteworthy mention is that we also prove the equivalence of uniform and weak ergodicities of the time averages Markov operators in terms of the ergodicity coefficient, which shines a new light onto this specific topic. Lastly, in terms of weighted averages, unique ergodicity of semigroups is also studied. Emphasis is put on the newly obtained results which is a new discovery in classical and non-commutative settings.

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“…We notice that in [12] an analogous result for discrete semigroup of Markov operators has been proved. Corollary 4.3.…”

Section: Uniform Mean Ergodicity Of Cesaro Averages Of C 0 -Markov Se...mentioning

confidence: 70%

Stability Estimates of Markov Semigroups on Abstract States Spaces

Erkurşun-Özcan

Mukhamedov

2020

Mediterr. J. Math.

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“…Investigations of this type were performed, for example, in a more general setting of ordered Banach spaces with generating cone in [12], where almost periodicity and mean ergodicity are related with asymptotic domination. In a similar setting of base norm spaces, uniform asymptotic stability, and mean ergodicity, where the appropriate limits are taken in the operator norm, were investigated in [10]. The next result is well-known in the theory of von Neumann algebras.…”

Section: Now We Prove (Ii)mentioning

confidence: 92%

Ergodic properties of Markov semigroups in von Neumann algebras

Kielanowicz¹,

Łuczak²

2020

Publicacions Matemàtiques

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We investigate ergodic properties of Markov semigroups in von Neumann algebras with the help of the notion of constrictor, which expresses the idea of closeness of the orbits of the semigroup to some set, as well as the notion of 'generalised averages', which generalises to arbitrary abelian semigroups the classical notions of Cesàro, Borel, or Abel means. In particular, mean ergodicity, asymptotic stability, and structure properties of the fixed-point space are analysed in some detail.2010 Mathematics Subject Classification: Primary: 46L55; Secondary: 47A35.

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“…[ [27,Section 4]) and results results relying on Dobrushin's ergodicity coefficient (see e.g. [19] and the references therein).…”

Section: 2mentioning

confidence: 99%

Almost interior points in ordered Banach spaces and the long-term behaviour of strongly positive operator semigroups

Glück

Weber

2020

Studia Math.

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The first part of this article is a brief survey of the properties of so-called almost interior points in ordered Banach spaces. Those vectors can be seen as a generalization of "functions which are strictly positive almost everywhere" on L p -spaces and of "quasi-interior points" in Banach lattices.In the second part we study the long-term behaviour of strongly positive operator semigroups on ordered Banach spaces; these are semigroups which, in a sense, map every non-zero positive vector to an almost interior point. Using the Jacobs-de Leeuw-Glicksberg decomposition together with the theory presented in the first part of the paper we deduce sufficiency criteria for such semigroups to converge (strongly or in operator norm) as time tends to infinity. This generalises known results for semigroups on Banach lattices as well as on normally ordered Banach spaces with unit.

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Uniform Ergodicity of Lotz–Räbiger Nets of Markov Operators on Abstract State Spaces

Cited by 7 publications

References 23 publications

Stability Estimates of Markov Semigroups on Abstract States Spaces

Stability Estimates of Markov Semigroups on Abstract States Spaces

Ergodic properties of Markov semigroups in von Neumann algebras

Almost interior points in ordered Banach spaces and the long-term behaviour of strongly positive operator semigroups

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