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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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$C^{2}$-preserving strongly continuous Markovian semigroups
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by W. M. Priestley PDF
Trans. Amer. Math. Soc. 180 (1973), 359-365 Request permission

Abstract:

Let $X$ be a compact ${C^2}$-manifold. Let $||\;||,||\;||’$ 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 $||\;||$. Theorem. Assume that $\{ {P^t}\}$ preserves ${C^2}$-functions and that the restriction of $\{ {P^t}\}$ to ${C^2}(X),||\;||’$ is strongly continuous. Then ${C^2}(X) \subset \mathcal {D}$ and $A$ is a bounded operator from ${C^2}(X),||\;||’$ to $C(X),||\;||$. 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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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 180 (1973), 359-365
  • MSC: Primary 47D05; Secondary 58G99, 60J35
  • DOI: https://doi.org/10.1090/S0002-9947-1973-0324481-5
  • MathSciNet review: 0324481