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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Local spectra and individual stability of uniformly bounded $C_{0}$-semigroups


Authors: Charles J. K. Batty, Jan van Neerven and Frank Räbiger
Journal: Trans. Amer. Math. Soc. 350 (1998), 2071-2085
MSC (1991): Primary 47D03
MathSciNet review: 1422890
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Abstract: We study the asymptotic behaviour of individual orbits $T(\cdot )x$ of a uniformly bounded $C_{0}$-semigroup $\{T(t)\}_{t\ge 0}$ with generator $A$ in terms of the singularities of the local resolvent $(\lambda -A)^{-1}x$ on the imaginary axis. Among other things we prove individual versions of the Arendt-Batty-Lyubich-Vu theorem and the Katznelson-Tzafriri theorem.


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Additional Information

Charles J. K. Batty
Affiliation: St. John’s College, Oxford OX1 3JP, England
Email: charles.batty@sjc.ox.ac.uk

Jan van Neerven
Affiliation: Department of Mathematics, Delft Technical University, P. O. Box 356, 2600 AJ Delft, The Netherlands
Email: J.vanNeerven@twi.tudelft.nl

Frank Räbiger
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany
Email: frra@michelangelo.mathematik.uni-tuebingen.de

DOI: http://dx.doi.org/10.1090/S0002-9947-98-01919-9
PII: S 0002-9947(98)01919-9
Keywords: Laplace transform, singular set, countable, $C_{0}$-semigroup, stability, local spectrum, orbit
Received by editor(s): February 12, 1996
Received by editor(s) in revised form: September 6, 1996
Additional Notes: The work on this paper was done during a two-year stay at the University of Tübingen. Support by an Individual Fellowship from the Human Capital and Mobility Programme of the European Community is gratefully acknowledged. I warmly thank Professor Rainer Nagel and the members of his group for their hospitality (second author). It is part of a research project supported by the Deutsche Forschungsgemeinschaft DFG (third author). Work in Oxford was also supported by an EPSRC Visiting Fellowship Research Grant (first and third authors)
Article copyright: © Copyright 1998 American Mathematical Society