Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Published electronically: June 19, 2001
MathSciNet review: 1866019
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that a new notion of a spectrum of a function $u\in L^\infty({\mathbb R}_+,X)$ ($X$ is a Banach space), defined by B. Basit and the first author, coincides with the Arveson spectrum of some shift group, provided $u$ is uniformly continuous. We apply this result to prove a new version of a tauberian theorem.

References [Enhancements On Off] (What's this?)

  • 1. W. Arendt and C. J. K. Batty, Almost periodic solutions of first- and second-order Cauchy problems, J. Differential Equations 137 (1997), no. 2, 363–383. MR 1456602,
  • 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. William Arveson, The harmonic analysis of automorphism groups, Operator algebras and applications, Part I (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 199–269. MR 679706
  • 5. Bolis Basit, Some problems concerning different types of vector valued almost periodic functions, Dissertationes Math. (Rozprawy Mat.) 338 (1995), 26. MR 1316273
  • 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. Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. MR 1721989
  • 9. Yitzhak Katznelson, An introduction to harmonic analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0248482
  • 10. L. H. Loomis, The spectral characterization of a class of almost periodic functions., Ann. of Math. (2) 72 (1960), 362–368. MR 0120502,
  • 11. Hans Reiter, Classical harmonic analysis and locally compact groups, Clarendon Press, Oxford, 1968. MR 0306811

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47D03, 47A10, 40E05

Retrieve articles in all journals with MSC (2000): 47D03, 47A10, 40E05

Additional Information

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

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

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