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Spectral synthesis and applications to C0-groups

Part of: Groups and semigroups of linear operators, their generalizations and applications

Published online by Cambridge University Press:  09 April 2009

M. Zarrabi
M. Zarrabi
Affiliation:
U.F.R de Mathématiques et InformatiqueUniversité BordeauxI 351, cours de la Liberation 33405 TalenceFrance e-mail: zarrabi@math.u-bordeaux.fr
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Abstract

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Let k ≥ 0 be an integer, T = (T(t))t ∈R a C0-group of bounded operators and A the infinitesimal generator of T. We prove that if, and , and if the spectrum of A is equal to {λ}, then A is bounded and (A – λ)k+1 = 0. Examples are given to show that these conditions are, essentially, the best possible.

MSC classification

Secondary: 47D03: Groups and semigroups of linear operators
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 60 , Issue 1 , February 1996 , pp. 128 - 142
Copyright
Copyright © Australian Mathematical Society 1996

References

[1]Allan, G. R. and Ransford, T. J., ‘Power-dominated elements in a Banach algebra’, Studia Math. 94 (1989), 63–79.CrossRefGoogle Scholar
[2]Atzmon, A., ‘Operators which are annihilated by analytic functions and invariant subspaces’, Acta Math. 144 (1980), 27–63.CrossRefGoogle Scholar
[3]Beurling, A., ‘Sur les integrales de Fourier absolument convergentes et leur application à une transformation fonctionnelle’, in: Neuvième Cong. Math. Scandinaves, Helsinki 1938 (Tryekeri, Helsinki, 1939) pp. 199–210.Google Scholar
[4]Boas, R. P., Entire functions (Academic Press, New York, 1954).Google Scholar
[5]Dales, H. G. and Hayman, W. K., ‘Esterle's proof of the Tauberian theorem for Beurling algebras’, Ann. Inst. Fourier (Grenoble) 31 (1981), 141–150.CrossRefGoogle Scholar
[6]Esterle, J., ‘Distributions on Kronecker sets, strong form of uniqueness and closed ideals of A +’, J. Reine Angew. Math. 450 (1994), 43–82.Google Scholar
[7]Esterle, J., Strouse, E. and Zouakia, F., ‘Stabilité asymptotique de certains semi-groupes d'opérateurs’, J. Operator Theory 28 (1992), 203–227.Google Scholar
[8]Gurarii, V. P., ‘Harmonic analysis in spaces with weights’, Trans. Moscow Math. Soc. 35 (1979), 21–75.Google Scholar
[9]Hille, E. and Phillips, R. S., Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ. 31 (Amer. Math. Soc., Providence, 1957).Google Scholar
[10]Katznelson, Y., An introduction to harmonic analysis (Wiley, New York, 1968).Google Scholar
[11]Pazy, A., Semigroups of linear operators and applications to partial differential equations (Springer, New York, 1983).CrossRefGoogle Scholar
[12]Schaefer, H. H., Wolf, M. and Arendt, W., ‘On lattice isomorphisms with positive real spectrum and group operators’, Math. Z. 164 (1978), 115–123.CrossRefGoogle Scholar
[13]Zarrabi, M., ‘Ensembles de synthèse pour certaines algèbres de Beurling’, Rev. Roumaine Math. Pures Appl. 35 (1990), 385–396.Google Scholar
[14]Zarrabi, M., ‘Contractions à spectre dénombrable et propriétés d'unicité des fermés dénombrables du cercle’, Ann. Inst. Fourier (Grenoble) 43 (1993), 251–263.CrossRefGoogle Scholar