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