Asymptotic properties of the iterates of stochastic operators on (AL) Banach lattices

Bartoszek, Wojciech

doi:10.4064/ap-52-2-165-173

Cited by 25 publications

(18 citation statements)

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“…Note that the proved theorem is a non-associative version Bartoszek's result [7]. A similar result has been obtained in [8,27] without using Dobrushin ergodicity coefficient, when A is a self-adjoint part of von Neumman algebra with a finite trace.…”

Section: Uniform Ergodicitysupporting

confidence: 78%

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On Dobrushin Ergodicity Coefficient and weak ergodicity of Markov Chains on Jordan Algebras

Mukhamedov

2013

J. Phys.: Conf. Ser.

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Abstract. In this paper we study certain properties of Dobrushin's ergodicity coefficient for stochastic operators defined on non-associative L 1 -spaces associated with semi-finite JBWalgebras. Such results extends the well-known classical ones to a non-associative setting. This allows us to investigate the weak ergodicity of nonhomogeneous discrete Markov processes (NDMP) by means of the ergodicity coefficient. We provide a necessary and sufficient conditions for such processes to satisfy the weak ergodicity. IntroductionIt is known (see [23]) that the investigations of asymptotical behavior of iterations of Markov operators on commutative L 1 -spaces are very important. On the other hand, these investigations are related with several notions of ergodicity of L 1 -contractions of measure spaces. To the investigation of such ergodic properties of Markov operators were devoted lots of papers (see for example, [7,23]). On the other hand, such kind of operators were studied in noncommutative settings. Since, the study of quantum dynamical systems has had an impetuous growth in the last years, in view of natural applications to various field of mathematics and physics. It is then of interest to understand among the various ergodic properties, which ones survive and are meaningful by passing from the classical to the quantum case. Due to noncommutativity, the latter situation is much more complicated than the former. The reader is referred e.g. to [2,15,16,18,29] for further details relative to some differences between the classical and the quantum situations. It is therefore natural to study the possible generalizations to quantum case of the various ergodic properties known for classical dynamical systems. One of the generalizations of noncommutative algebras is Jordan algebras. Note that Jordan Banach algebras [3,6] are a non-associative real analogue of von Neumann algebras. The existence of exceptional JBW -algebras does not allow one to use the ideas and methods from von Neumann algebras. The motivation of these investigations arose in quantum statistical mechanics and quantum field theory (see, [9]). Mostly, in those investigations homogeneous Markov processes were considered. Many ergodic type theorems have been proved for Markov operators acting in nonassociative and noncommutative L p -spaces (see for example, [4, 5, 20, 21, 8, 18]) On the other hand, nonhomogeneous Markov processes with general state space have become a subject of interest due to their applications in many branches of mathematics and natural sciences. In many papers (see for example, [24,17,31]) the weak ergodicity of nonhomogeneous Markov process are given in terms of Dobrushin's ergodicity coefficient [12]. In [33] some

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Section: Uniform Ergodicitysupporting

confidence: 78%

“…On the other hand, these investigations are related with several notions of ergodicity of L 1 -contractions of measure spaces. To the investigation of such ergodic properties of Markov operators were devoted lots of papers (see for example, [7,23]). On the other hand, such kind of operators were studied in noncommutative settings.…”

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confidence: 99%

On Dobrushin Ergodicity Coefficient and weak ergodicity of Markov Chains on Jordan Algebras

Mukhamedov

2013

J. Phys.: Conf. Ser.

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“…In this part of our proof we repeat some arguments used in [3] and now adapted to the noncommutative case.…”

Section: Norm Residualitymentioning

confidence: 99%

“…Moreover, the convergence holds with an exponential rate. Even though the ideas for our first result come from [19] (see also [3] and [4]), for the completeness of the paper (and convenience of the reader) we have decided to include a detailed proof.…”

Section: Norm Residualitymentioning

confidence: 99%

On residualities in the set of Markov operators on 𝒞₁

Bartoszek¹,

Kuna²

2005

Proc. Amer. Math. Soc.

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Abstract. We show that the set of those Markov operators on the Schatten class C 1 such that limn→∞ P n − Q = 0, where Q is one-dimensional projection, is norm open and dense. If we require that the limit projections must be on strictly positive states, then such operators P form a norm dense G δ . Surprisingly, for the strong operator topology operators the situation is quite the opposite.

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“…The investigation of asymptotic behavior of this operators reviles several ergodic properties of the Markov process (see [5,23,44]). When we look at quantum systems, it is important to study associated quantum dynamical systems, which has had an impetuous growth due to natural applications in various fields of mathematics and physics [35,45].…”

Section: Introductionmentioning

confidence: 99%

Uniform stability and weak ergodicity of nonhomogeneous Markov chains defined on ordered Banach spaces with a base

Mukhamedov

2015

Positivity

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In the present paper, we define an ergodicity coefficient of a positive mapping defined on ordered Banach space with a base , and study its properties. The defined coefficient is a generalization of the well-known the Dobrushin's ergodicity coefficient. By means of the ergodicity coefficient we provide uniform asymptotical stability conditions for nonhomogeneous discrete Markov chains (NDMC). These results are even new in case of von Neumann algebras. Moreover, we find necessary and sufficient conditions for the weak ergodicity of NDMC. Certain relations between uniform asymptotical stability and weak ergodicity are considered.

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Asymptotic properties of the iterates of stochastic operators on (AL) Banach lattices

Cited by 25 publications

References 0 publications

On Dobrushin Ergodicity Coefficient and weak ergodicity of Markov Chains on Jordan Algebras

On Dobrushin Ergodicity Coefficient and weak ergodicity of Markov Chains on Jordan Algebras

On residualities in the set of Markov operators on 𝒞₁

Uniform stability and weak ergodicity of nonhomogeneous Markov chains defined on ordered Banach spaces with a base

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