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

Author(s): Wlodzimierz Bak; Andrzej Hulanicki
Journal: Proc. Amer. Math. Soc. 134 (2006), 1467-1472.
MSC (2000): Primary 43A10, 43A20
Posted: 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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Additional Information:

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

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

DOI: 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
Posted: 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 \emph{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
Copyright of article: Copyright 2005, American Mathematical Society

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