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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Remarks on spectra and $ L^1$ multipliers for convolution operators


Authors: Wlodzimierz Bak and Andrzej Hulanicki
Journal: Proc. Amer. Math. Soc. 134 (2006), 1467-1472
MSC (2000): Primary 43A10, 43A20
Published electronically: October 18, 2005
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Abstract: We prove that the spectrum of a convolution operator on a locally compact group $ G$ by a self-adjoint $ L^1$-function $ f$ is the same on $ L^1(G)$ and $ L^2(G)$ and consequently on all $ L^p$ spaces, $ 1\leq p<\infty ,$ if and only if a Beurling algebra contains non-analytic functions on $ {\mathbb{R}}$ operating on $ f$ into $ L^1$.


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Wlodzimierz Bak
Affiliation: Instytut Matematyczny, Uniwersytet Wrocławski, Plac Grunwaldzki 2/4, 50-384 Wrocław, Poland
Email: bak@math.uni.wroc.pl

Andrzej Hulanicki
Affiliation: Instytut Matematyczny, Uniwersytet Wrocławski, Plac Grunwaldzki 2/4, 50-384 Wrocław, Poland
Email: hulanick@math.uni.wroc.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08159-1
PII: S 0002-9939(05)08159-1
Received by editor(s): September 1, 2004
Received by editor(s) in revised form: December 21, 2004
Published electronically: October 18, 2005
Additional Notes: This work was partially done within the project TMR Network “Harmonic Analysis”, contract no. ERB FMRX-CT97-0159.
This research was partially financed by the European Commission IHP Network 2002–2006 Harmonic Analysis and Related Problems (Contract Number: HPRN-CT-2001-00273-HARP) and by KBN grant 1 P03A 018 26
Communicated by: David R. Larson
Article copyright: © Copyright 2005 American Mathematical Society