Remarks on spectra and 𝐿¹ multipliers for convolution operators

Ba̧k, Włodzimierz; Hulanicki, Andrzej

doi:10.1090/S0002-9939-05-08159-1

Remarks on spectra and $L^1$ multipliers for convolution operators
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by Włodzimierz Ba̧k and Andrzej Hulanicki PDF

Proc. Amer. Math. Soc. 134 (2006), 1467-1472 Request permission

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 $L^1$.

References

Bruce A. Barnes, When is the spectrum of a convolution operator on $L^p$ independent of $p$?, Proc. Edinburgh Math. Soc. (2) 33 (1990), no. 2, 327–332. MR 1057760, DOI 10.1017/S0013091500018241
Bruce A. Barnes, A note on invariance of spectrum for symmetric Banach $^*$-algebras, Proc. Amer. Math. Soc. 126 (1998), no. 12, 3545–3547. MR 1473655, DOI 10.1090/S0002-9939-98-04620-6
Ralph Chill and Eva Fašangová, Equality of two spectra arising in harmonic analysis and semigroup theory, Proc. Amer. Math. Soc. 130 (2002), no. 3, 675–681. MR 1866019, DOI 10.1090/S0002-9939-01-06146-9
M. Christ and D. Müller, On $L^p$ spectral multipliers for a solvable Lie group, Geom. Funct. Anal. 6 (1996), no. 5, 860–876. MR 1415763, DOI 10.1007/BF02246787
Michael Cowling, Saverio Giulini, Andrzej Hulanicki, and Giancarlo Mauceri, Spectral multipliers for a distinguished Laplacian on certain groups of exponential growth, Studia Math. 111 (1994), no. 2, 103–121. MR 1301761, DOI 10.4064/sm-111-2-103-121
E. Fasangá, Spectral mapping theorems and spectral space-independence, Proc. Amer. Math. Soc.
G. Fendler, K. Gröchenig, M. Leinert, J. Ludwig, and C. Molitor-Braun, Weighted Group algebras on Groups of Polynomial Growth, Preprint, 2004.
G. B. Folland and Elias M. Stein, Hardy spaces on homogeneous groups, Mathematical Notes, vol. 28, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. MR 657581
Rostislav I. Grigorchuk and Andrzej Żuk, The lamplighter group as a group generated by a 2-state automaton, and its spectrum, Geom. Dedicata 87 (2001), no. 1-3, 209–244. MR 1866850, DOI 10.1023/A:1012061801279
Waldemar Hebisch, Boundedness of $L^1$ spectral multipliers for an exponential solvable Lie group, Colloq. Math. 73 (1997), no. 1, 155–164. MR 1436956, DOI 10.4064/cm-73-1-155-164
Waldemar Hebisch, Spectral multipliers on exponential growth solvable Lie groups, Math. Z. 229 (1998), no. 3, 435–441. MR 1658573, DOI 10.1007/PL00004662
Waldemar Hebisch, Spectral multipliers on metabelian groups, Rev. Mat. Iberoamericana 16 (2000), no. 3, 597–604. MR 1813330, DOI 10.4171/RMI/285
W. Hebisch, J. Ludwig, and D. Müller, Sublaplacians of holomorphic $L^p$-type on exponential solvable groups, Preprint, 2001.
A. Hulanicki, On symmetry in group algebras, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 11 (1963), 1–2. MR 149323
A. Hulanicki, On $L_{p}$-spectra of the laplacian on a Lie group with polynomial growth, Proc. Amer. Math. Soc. 44 (1974), 482–484. MR 360931, DOI 10.1090/S0002-9939-1974-0360931-2
A. Hulanicki, Subalgebra of $L_{1}(G)$ associated with Laplacian on a Lie group, Colloq. Math. 31 (1974), 259–287. MR 372536, DOI 10.4064/cm-31-2-259-287
Andrzej Hulanicki, A functional calculus for Rockland operators on nilpotent Lie groups, Studia Math. 78 (1984), no. 3, 253–266. MR 782662, DOI 10.4064/sm-78-3-253-266
Jean-Pierre Kahane, Séries de Fourier absolument convergentes, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 50, Springer-Verlag, Berlin-New York, 1970 (French). MR 0275043, DOI 10.1007/978-3-662-59158-1
Y. Katznelson, An introuction to harmonic analysis, New York - London, 1968.
Harry Kesten, Full Banach mean values on countable groups, Math. Scand. 7 (1959), 146–156. MR 112053, DOI 10.7146/math.scand.a-10568
J. Ludwig, A. Müller, and S. Souaifi, Holomorphic $L^p$-type for sub-Laplacians on connected Lie groups, preprint, 2004.
M. A. Naĭmark, Normed rings, P. Noordhoff N. V., Groningen, 1959. Translated from the first Russian edition by Leo F. Boron. MR 0110956
T. W. Palmer, Classes of nonabelian, noncompact, locally compact groups, Rocky Mountain J. Math. 8 (1978), no. 4, 683–741. MR 513952, DOI 10.1216/RMJ-1978-8-4-683
Charles E. Rickart, General theory of Banach algebras, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0115101
Satish Shirali and James W. M. Ford, Symmetry in complex involutory Banach algebras. II, Duke Math. J. 37 (1970), 275–280. MR 261362