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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$ C\sp{2}$-preserving strongly continuous Markovian semigroups

Author: W. M. Priestley
Journal: Trans. Amer. Math. Soc. 180 (1973), 359-365
MSC: Primary 47D05; Secondary 58G99, 60J35
MathSciNet review: 0324481
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Abstract: Let $ X$ be a compact $ {C^2}$-manifold. Let $ \vert\vert\;\vert\vert,\vert\vert\;\vert\vert'$ denote the supremum norm and the $ {C^2}$-norm, respectively, and let $ \{ {P^t}\} $ be a Markovian semigroup on $ C(X)$. The semigroup's infinitesimal generator $ A$, with domain $ \mathcal{D}$, is defined by $ Af = {\lim _{t \to 0}}{t^{ - 1}}({P^t}f - f)$, whenever the limit exists in $ \vert\vert\;\vert\vert$.

Theorem. Assume that $ \{ {P^t}\} $ preserves $ {C^2}$-functions and that the restriction of $ \{ {P^t}\} $ to $ {C^2}(X),\vert\vert\;\vert\vert'$ is strongly continuous. Then $ {C^2}(X) \subset \mathcal{D}$ and $ A$ is a bounded operator from $ {C^2}(X),\vert\vert\;\vert\vert'$ to $ C(X),\vert\vert\;\vert\vert$.

From the conclusion is obtained a representation of $ Af \cdot (x)$ as an integrodifferential operator on $ {C^2}(X)$. The representation reduces to that obtained by Hunt [Semi-groups of measures on Lie groups, Trans. Amer. Math. Soc. 81 (1956), 264-293] in case $ X$ is a Lie group and $ {P^t}$ commutes with translations.

Actually, a stronger result is proved having the above theorem among its corollaries.

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Keywords: $ {P^t}$-function, $ {P^t}$-manifold, Markovian operator, strong derivative, Banach-Steinhaus theorem, Markovian semigroup, infinitesimal generator, integro-differential operator, normal derivative of harmonic extension
Article copyright: © Copyright 1973 American Mathematical Society