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)

 

 

On $ L\sb{p}$-spectra of the laplacian on a Lie group with polynomial growth


Author: A. Hulanicki
Journal: Proc. Amer. Math. Soc. 44 (1974), 482-484
MSC: Primary 22E30
MathSciNet review: 0360931
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The following theorem is proved: If $ G$ is a Lie group with polynomial growth (a compact extension of a nilpotent group, e.g.) and $ \Delta = X_1^2 + \cdots + X_n^2$, where $ {X_1}, \cdots ,{X_n}$ is a basis of the Lie algebra of $ G$, then for all $ p,1 \leqq p < \infty $, the operator $ \Delta $ has the same spectrum on all $ {L_p}(G)$.


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

  • [1] Jacques Dixmier, Opérateurs de rang fini dans les représentations unitaires, Inst. Hautes Études Sci. Publ. Math. 6 (1960), 13–25 (French). MR 0136684
  • [2] Lars Gȧrding, Vecteurs analytiques dans les représentations des groups de Lie, Bull. Soc. Math. France 88 (1960), 73–93 (French). MR 0119104
  • [3] Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften, Band 152, Springer-Verlag, New York-Berlin, 1970. MR 0262773
  • [4] Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. MR 0089373
  • [5] A. Hulanicki, On the spectrum of convolution operators on groups with polynomial growth, Invent. Math. 17 (1972), 135–142. MR 0323951
  • [6] J. W. Jenkins, Growth of connected locally compact groups, J. Functional Analysis 12 (1973), 113–127. MR 0349895
  • [7] Edward Nelson, Analytic vectors, Ann. of Math. (2) 70 (1959), 572–615. MR 0107176
  • [8] Edward Nelson and W. Forrest Stinespring, Representation of elliptic operators in an enveloping algebra, Amer. J. Math. 81 (1959), 547–560. MR 0110024
  • [9] Elias M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory., Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. MR 0252961
  • [10] Kôsaku Yosida, Functional analysis, Die Grundlehren der Mathematischen Wissenschaften, Band 123, Academic Press, Inc., New York; Springer-Verlag, Berlin, 1965. MR 0180824

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 22E30

Retrieve articles in all journals with MSC: 22E30


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0360931-2
Article copyright: © Copyright 1974 American Mathematical Society