Equality of two spectra arising in harmonic analysis and semigroup theory

Authors:
Ralph Chill and Eva Fasangová

Journal:
Proc. Amer. Math. Soc. **130** (2002), 675-681

MSC (2000):
Primary 47D03; Secondary 47A10, 40E05

DOI:
https://doi.org/10.1090/S0002-9939-01-06146-9

Published electronically:
June 19, 2001

MathSciNet review:
1866019

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that a new notion of a spectrum of a function ( is a Banach space), defined by B. Basit and the first author, coincides with the Arveson spectrum of some shift group, provided is uniformly continuous. We apply this result to prove a new version of a tauberian theorem.

**1.**W. Arendt and C.J.K. Batty. Almost periodic solutions of first- and second-order Cauchy problems.*J. Diff. Eq.***137**, 363-383, 1997. MR**98g:34099****2.**W. Arendt and C.J.K. Batty. Asymptotically almost periodic solutions of inhomogeneous Cauchy problems on the half-line.*Bull. London Math. Soc.***31**, 291-304, 1999. CMP**99:09****3.**W. Arendt, C.J.K. Batty, M. Hieber and F. Neubrander. Vector-valued Laplace transforms and Cauchy problems. Monographs in Mathematics, 96, Birkhäuser Verlag, Basel, 2001.**4.**W. Arveson. The harmonic analysis of automorphism groups. Operator Algebras and Applications. Proc. Symp. Pure Math.**38**, Kingston/Ont. (1980), 199-269, 1982. MR**84m:46085****5.**B. Basit. Some problems concerning different types of vector valued almost periodic functions.*Dissertationes Math.***338**, 1995. MR**96d:43007****6.**C.J.K. Batty, W. Hutter, F. Räbiger. Almost periodicity of mild solutions of inhomogeneous periodic Cauchy problems.*J. Diff. Eq.***156**, 309-327, 1999. CMP**99:17****7.**R. Chill. Fourier transforms and asymptotics of evolution equations. PhD Thesis, Universität Ulm, 1998.**8.**K.-J. Engel and R. Nagel. One-Parameter Semigroups for Linear Evolution Equations. Springer Verlag, New York, Berlin, Heidelberg, 1999. MR**2000i:47075****9.**Y. Katznelson. An Introduction to Harmonic Analysis. Wiley, New York, London, 1968. MR**40:1734****10.**L. H. Loomis. The spectral characterization of a class of almost periodic functions.*Ann. of Math.***72**, 362-368, 1960. MR**22:11255****11.**H. Reiter. Classical Harmonic Analysis and Locally Compact Groups. Oxford Mathematical Monographs, Oxford, At the Clarendon Press, 1968. MR**46:5933**

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Additional Information

**Ralph Chill**

Affiliation:
Abteilung Angewandte Analysis, Universität Ulm, 89069 Ulm, Germany

Email:
chill@mathematik.uni-ulm.de

**Eva Fasangová**

Affiliation:
Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

Email:
fasanga@karlin.mff.cuni.cz

DOI:
https://doi.org/10.1090/S0002-9939-01-06146-9

Keywords:
Spectrum of a function,
Arveson spectrum,
$C_0$-group,
Fourier transform,
tauberian theorem

Received by editor(s):
February 7, 2000

Received by editor(s) in revised form:
August 2, 2000

Published electronically:
June 19, 2001

Additional Notes:
Part of this work was done while the first author visited the Charles University of Prague. He is grateful for the warm hospitality and the financial support. The second author is supported by grant No. 201/98/1450 of the Grant Agency of the Czech Republic, grant No. 166/1999 of the Grant Agency of Charles University, and grant No. CEZ J 13/98113200007.

Communicated by:
David R. Larson

Article copyright:
© Copyright 2001
American Mathematical Society