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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9947-98-01919-9
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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  • 1. W. Arendt and C.J.K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc. 306 (1988), 837-852. MR 89g:47053
  • 2. W. Arendt and J. Prüss, Vector-valued Tauberian theorems and asymptotic behavior of linear Volterra equations, SIAM J. Math. Anal. 23 (1992), 412-448. MR 92m:47150
  • 3. C.J.K. Batty, Asymptotic behaviour of semigroups of operators, Functional analysis and operator theory (J. Zemánek, ed.), vol. 30, Banach Center Publ., Warsaw, 1994, pp. 35-52. MR 95g:47058
  • 4. C.J.K. Batty, Tauberian theorems for the Laplace-Stieltjes transform, Trans. Amer. Math. Soc. 322 (1990), 783-804. MR 91c:44001
  • 5. C.J.K. Batty, Spectral conditions for stability of one-parameter semigroups, J. Diff. Equations 127 (1996), 87-96. MR 97e:47060
  • 6. C.J.K. Batty, Z. Brzezniak and D.A. Greenfield, A quantitative asymptotic theorem for contraction semigroups with countable unitary spectrum, Studia Math. 121 (1996), 167-183. CMP 97:03
  • 7. C.J.K. Batty, J. van Neerven and F. Räbiger, Tauberian theorems and stability of solutions of the Cauchy problem, Trans. Amer. Math. Soc. 350 (1998), 2087-2103.
  • 8. C.J.K. Batty and Vu Quôc Phóng, Stability of individual elements under one-parameter semigroups, Trans. Amer. Math. Soc. 322 (1990), 805-818. MR 91c:47072
  • 9. R. deLaubenfels and Vu Quôc Phóng, Stability and almost periodicity of solutions of ill-posed abstract Cauchy problems, Proc. Amer. Math. Soc. 125 (1997), 235-241. MR 97c:34123
  • 10. J. Esterle, E. Strouse and F. Zouakia, Stabilité asymptotique de certains semigroupes d'opérateurs et idéaux primaires de $L^{1}({\mathbb R}_{+})$, J. Operator Theory 28 (1992), 203-227. MR 95f:43001
  • 11. D.E. Evans, On the spectrum of a one-parameter strongly continuous representation, Math. Scand. 39 (1976), 80-82. MR 55:3873
  • 12. Falun Huang, Spectral properties and stability of one-parameter semigroups, J. Diff. Equations 104 (1993), 182-195. MR 94f:47047
  • 13. J.P. Kahane and Y. Katznelson, Sur les algèbres de restrictions des séries de Taylor absolument convergentes à un fermé du cercle, J. Analyse Math. 23 (1970), 185-197. MR 42:8179
  • 14. Y. Katznelson, An Introduction to Harmonic Analysis, Dover Publications, 2nd ed., New York, 1976. MR 54:10976
  • 15. Y. Katznelson and L. Tzafriri, On power bounded operators, J. Functional Anal. 68 (1986), 313-328. MR 88e:47006
  • 16. U. Krengel, Ergodic theorems, De Gruyter, Berlin, 1985. MR 87i:28001
  • 17. Yu. I. Lyubich and Vu Quôc Phóng, Asymptotic stability of linear differential equations on Banach spaces, Studia Math. 88 (1988), 37-42. MR 89e:47062
  • 18. R. Nagel (ed.), One-parameter Semigroups of Positive Operators, Springer Lect. Notes in Math. 1184, 1986. MR 88i:47022
  • 19. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983. MR 85g:47061
  • 20. W. Rudin, Fourier Analysis on Groups, Interscience, New York, 1962. MR 27:2808
  • 21. Vu Quôc Phóng, Theorems of Katznelson-Tzafriri type for semigroups of operators, J. Functional Anal. 103 (1992), 74-84. MR 93e:47050
  • 22. Vu Quôc Phóng and Yu.I. Lyubich, A spectral criterion for asymptotic almost periodicity of uniformly continuous representations of abelian semigroups, J. Soviet Math. 48 (1990), 644-647. MR 90e:22004 (Russian original)

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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: https://doi.org/10.1090/S0002-9947-98-01919-9
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

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