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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Analyticity and quasi-analyticity for one-parameter semigroups


Author: J. W. Neuberger
Journal: Proc. Amer. Math. Soc. 25 (1970), 488-494
MSC: Primary 47.50
MathSciNet review: 0259661
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Abstract: Suppose that $ T$ is a strongly continuous (even at 0) one-parameter semigroup of bounded linear transformations on a real Banach space $ S$ and $ T$ has generator $ A$. Theorem A. If $ \lim {\sup _{x \to 0}}\vert T(x) - I\vert < 2$ then $ AT(x)$ is bounded for all $ x > 0$. Suppose $ \{ {\delta _q}\} _{q = 1}^\infty $ is a sequence of positive numbers convergent to 0 and each of $ N(q),\;q = 1,\;2, \cdots $ is an increasing sequence of positive integers. Denote by $ Q$ the collection consisting of (1) all real analytic functions on $ (0,\;\infty )$ and (2) all $ h$ on $ (0,\;\infty )$ for which there is a Banach space $ S$, a member $ p$ of $ S$, a member $ f$ of $ {S^{\ast}}$ and a strongly continuous semigroup $ L$ of bounded linear transformations so that $ h(x) = f[L(x)p]$ for all $ x > 0$ where $ L$ satisfies $ \lim {\sup _{n \to \infty \,(n \in N(q))}}\vert L({\delta _q}/n) - I\vert < 2,\;q = 1,\;2,\; \cdots $. Theorem B. No two members of $ Q$ agree on an open subset of $ (0,\;\infty )$.


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DOI: https://doi.org/10.1090/S0002-9939-1970-0259661-3
Keywords: Quasi-analytic, analytic, semigroup of bounded linear transformations
Article copyright: © Copyright 1970 American Mathematical Society