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

 

Decomposition of $ C\sp{\infty }$ intertwining operators for Lie groups


Author: R. Penney
Journal: Proc. Amer. Math. Soc. 54 (1976), 368-370
MSC: Primary 22E45
MathSciNet review: 0404531
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Abstract: Let $ U$ be a unitary representation of a Lie group $ G$ in a Hilbert space $ \mathcal{K}$ and let $ {C^\infty }(U)$ denote the space of differentiable vectors for $ U$ given its usual topology. A continuous operator on $ {C^\infty }(U)$ is said to be a $ {C^\infty }$ intertwining operator for $ U$ if it commutes with $ U$. It is shown that if one decomposes $ U$ via a central decomposition into a direct integral of unitary representations, then every $ {C^\infty }$ intertwining operator decomposes into a direct integral of unique $ {C^\infty }$ intertwining operators. Furthermore, it is shown that if $ U$ is type I and primary, then every $ {C^\infty }$ intertwining operator extends to unique bounded (in the sense of $ \mathcal{K}$) intertwining operator defined on all of $ \mathcal{K}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0404531-6
PII: S 0002-9939(1976)0404531-6
Keywords: $ {C^\infty }$ vector
Article copyright: © Copyright 1976 American Mathematical Society