2003
DOI: 10.1007/s00440-003-0268-0
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Transience and recurrence of quantum Markov semigroups

Abstract: This article introduces a concept of transience and recurrence for a Quantum Markov Semigroup and explores its main properties via the associated potential. We show that an irreducible semigroup is either recurrent or transient and characterize transient semigroups by means of the existence of non trivial superharmonic operators.

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Cited by 43 publications
(50 citation statements)
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“…The proved theorem is a noncommutative analog of Theorem 1.1. Certain similar results have been obtained in [19], [10] for quantum dynamical semigroups in von Neumann algebras.…”
Section: Mixing and Completely Mixing Contractionssupporting
confidence: 83%
“…The proved theorem is a noncommutative analog of Theorem 1.1. Certain similar results have been obtained in [19], [10] for quantum dynamical semigroups in von Neumann algebras.…”
Section: Mixing and Completely Mixing Contractionssupporting
confidence: 83%
“…8 implies that, when the von Neumdun algebra A is σ -finite, the definition of transient QMS is equivalent with the one given in [11]; instead, it is not yet clear if the same holds for the recurrent QMS. In order to prove this, starting with a positive element x such that U(x)[u] > 0 for some u ∈ D(U(x)) we have to construct a non-zero integrable element.…”
Section: Definition 6 the Projectionmentioning
confidence: 90%
“…it is a QMS). In the work [11] transience and recurrence are defined as the natural extension of the corresponding classical concepts and irreducible semigroups are shown to be either transient or recurrent. Our intention here is to find the decomposition of a QMS into "sub"-semigroups corresponding to classes of transient and recurrent states.…”
Section: Introductionmentioning
confidence: 99%
See 1 more Smart Citation
“…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.…”
mentioning
confidence: 99%