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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}}\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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  • [1] A. Beurling, On analytic extension of semi-groups of operators, J. Functional Analysis (to appear). MR 0282248 (43:7960)
  • [2] T. Kato, A characterization of holomorphic semigroups, Proc. Amer. Math. Soc. (to appear). MR 0264456 (41:9050)
  • [3] D. G. Kendall, Some recent developments in the theory of denumerable Markov processes, Trans. Fourth Prague Conf. on Information Theory, Statistical Decision Functions, Random Processes (Prague, 1965), Academia, Prague, 1967, pp. 11-27. MR 36 #974. MR 0217885 (36:974)
  • [4] E. Hille and R. S. Phillips, Functional analysis and semi-groups, rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, R.I., 1957. MR 19, 664. MR 0089373 (19:664d)
  • [5] J. W. Neuberger, A quasi-analyticity condition in terms of finite differences, Proc. London Math. Soc. (3). 14 (1964), 245-259. MR 28 #3130. MR 0159914 (28:3130)
  • [6] -, Quasi-analyticity of trajectories of semi-groups of bounded linear transformations, Notices Amer. Math. Soc. 12 (1965), 815. Abstract #65T-454.
  • [7] -, An exponential formula for one-parameter semi-groups of nonlinear transformations, J. Math. Soc. Japan 18 (1966), 154-157. MR 34 #622. MR 0200734 (34:622)
  • [8] -, Quasi-analytic collections containing Fourier series which are not infinitely differentiable, J. London Math. Soc. 43 (1968), 612-616. MR 37 #5344. MR 0229778 (37:5344)
  • [9] D. Williams, On operator semigroups and Markov groups, (to appear). MR 0254920 (40:8127)

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Additional Information

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

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