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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Analyticity and quasi-analyticity for one-parameter semigroups
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by J. W. Neuberger
Proc. Amer. Math. Soc. 25 (1970), 488-494
DOI: https://doi.org/10.1090/S0002-9939-1970-0259661-3

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 $(0,\;\infty )$.
References
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Bibliographic Information
  • © Copyright 1970 American Mathematical Society
  • 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