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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
DOI: https://doi.org/10.1090/S0002-9939-1970-0259661-3
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 $\lim {\sup _{x \to 0}}|T(x) - I| < 2$ 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 $\lim {\sup _{n \to \infty (n \in N(q))}}|L({\delta _q}/n) - I| < 2,\;q = 1,\;2,\; \cdots$. Theorem B. No two members of $Q$ agree on an open subset of $(0,\;\infty )$.


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Keywords: Quasi-analytic, analytic, semigroup of bounded linear transformations
Article copyright: © Copyright 1970 American Mathematical Society