Abstract
We present sufficient conditions on a Gaussian Mehler semigroup on a reflexive Banach space Eto be induced by a single positive symmetric operator Q \in \(Q \in \mathcal{L}(E^* ,E)\), and give a counterexample which shows that this representation theorem is false in every nonreflexive Banach space with a Schauder basis. We also show that the transition semigroup of a Gaussian Mehler semigroup on a separable Banach space Eacts in a pointwise continuous way, uniformly on compact subsets of E, in the space BUC(E) of bounded uniformly continuous real-valued funtions on E. The transition semigroup is shown to be strongly continuous on BUC(E) if and only if S(t) = Ifor all t⩽ 0
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Neerven, J.v. Continuity and Representation of Gaussian Mehler Semigroups. Potential Analysis 13, 199–211 (2000). https://doi.org/10.1023/A:1008799213878
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DOI: https://doi.org/10.1023/A:1008799213878