Abstract
Dynamical semigroups constitute a quantum-mechanical generalization of Markov semigroups, a concept familiar from the theory of stochastic processes. Let ℋ be a Hilbert space andA a von Neumann algebra. A dynamical semigroup Pt is a σ-weakly continuous one-parameter semigroup of completely positive maps ofA into itself. A semigroup Pt possessing the property of preserving the identityI∈A is said to be conservative and its infinitesimal operator L[·] is said to be regular. The present paper studies necessary and sufficient conditions for strongly continuous dynamical semigroups to be conservative. It is shown that under certain additional assumptions one can formulate necessary and sufficient conditions which are analogous to Feller's condition for regularity of a diffusion process: the equation P=L[P] has no solutions inA +. Using a Jensen-type inequality for completely positive maps, constructive sufficient conditions are obtained for conservativeness, in the form of inequalities for commutators. The restriction of a dynamical subgroup to an Abelian subalgebra ofℒ ∞ (R n) yields a series of new regularity conditions for both diffusion and jump processes.
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 36, pp. 149–184, 1990.
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Chebotarev, A.M. Necessary and sufficient conditions for conservativeness of dynamical semigroups. J Math Sci 56, 2697–2719 (1991). https://doi.org/10.1007/BF01095977
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DOI: https://doi.org/10.1007/BF01095977