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Proceedings of the American Mathematical Society

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Interpolation spaces and unitary representations


Author: Roe Goodman
Journal: Proc. Amer. Math. Soc. 83 (1981), 153-158
MSC: Primary 22E25; Secondary 22E15, 46L99, 46M35
MathSciNet review: 620003
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Abstract: Let $ G$ be a Lie group, $ \pi $ a unitary representation of $ G$ on a Hilbert space $ \mathcal{H}(\pi )$, and $ {\mathcal{H}^k}(\pi )$ the subspace of $ {C^k}$ vectors for $ \pi $. By quadratic interpolation there is a continuous scale $ {\mathcal{H}^s}(\pi )$, $ s > 0$, of $ G$-invariant Hilbert spaces. When $ G = H \cdot K$ is a semidirect product of closed subgroups, then it is proved that $ {\mathcal{H}^s}(\pi ) = {\mathcal{H}^s}({\left. \pi \right\vert _H}) \cap {\mathcal{H}^s}({\left. \pi \right\vert _K})$ for $ s > 0$. For solvable $ G$ this gives a characterisation of $ {\mathcal{H}^s}(\pi )$ in terms of smoothness along one-parameter subgroups, and an elliptic regularity result.


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  • [1] Lipman Bers, Fritz John, and Martin Schechter, Partial differential equations, Lectures in Applied Mathematics, Vol. III, Interscience Publishers John Wiley & Sons, Inc. New York-London-Sydney, 1964. MR 0163043
  • [2] Paul L. Butzer and Hubert Berens, Semi-groups of operators and approximation, Die Grundlehren der mathematischen Wissenschaften, Band 145, Springer-Verlag New York Inc., New York, 1967. MR 0230022
  • [3] Roe W. Goodman, One-parameter groups generated by operators in an enveloping algebra., J. Functional Analysis 6 (1970), 218–236. MR 0268330
  • [4] Roe W. Goodman, Some regularity theorems for operators in an enveloping algebra, J. Differential Equations 10 (1971), 448–470. MR 0289716
  • [5] Roe Goodman, Elliptic and subelliptic estimates for operators in an enveloping algebra, Duke Math. J. 47 (1980), no. 4, 819–833. MR 596116
  • [6] Pierre Grisvard, Commutativité de deux foncteurs d’interpolation et applications, J. Math. Pures Appl. (9) 45 (1966), 143–206 (French). MR 0221309
  • [7] Pierre Grisvard, Interpolation non commutative, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 52 (1972), 11–15 (French, with Italian summary). MR 0358337
  • [8] A. Hulanicki, Subalgebra of 𝐿₁(𝐺) associated with Laplacian on a Lie group, Colloq. Math. 31 (1974), 259–287. MR 0372536
  • [9] Richard S. Palais, Seminar on the Atiyah-Singer index theorem, With contributions by M. F. Atiyah, A. Borel, E. E. Floyd, R. T. Seeley, W. Shih and R. Solovay. Annals of Mathematics Studies, No. 57, Princeton University Press, Princeton, N.J., 1965. MR 0198494
  • [10] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0620003-X
Article copyright: © Copyright 1981 American Mathematical Society