On the characterization of trace class representations and Schwartz operators
van Dijk G, Neeb KH, Salmasian H, Zellner C (2016)
Publication Type: Journal article, Original article
Publication year: 2016
Journal
Publisher: Heldermann Verlag
Book Volume: 26
Pages Range: 787 - 805
Journal Issue: 3
URI: https://arxiv.org/abs/1512.02451
Abstract
In this note we collect several characterizations of unitary representations (\pi, \mathcal{H}) of a finite dimensional Lie group G which are trace class, i.e., for each compactly supported smooth function f on G, the operator \pi(f) is trace class. In particular we derive the new result that, for some m \in \mathbb{N}, all operators \pi(f), f \in C^m_c(G), are trace class. As a consequence the corresponding distribution character \theta_\pi is of finite order. We further show \pi is trace class if and only if every operator A, which is smoothing in the sense that A\mathcal{H}\subseteq \mathcal{H}^\infty, is trace class and that this in turn is equivalent to the Fr\'echet space \mathcal{H}^\infty being nuclear, which in turn is equivalent to the realizability of the Gaussian measure of \mathcal{H} on the space \mathcal{H}^{-\infty} of distribution vectors. Finally we show that, even for infinite dimensional Fr\'echet-Lie groups, A and A^* are smoothing if and only if A is a Schwartz operator, i.e., all products of A with operators from the derived representation are bounded.
Authors with CRIS profile
Karl Hermann Neeb
Lehrstuhl für Mathematik (Lie-Gruppen und Darstellungstheorie)
Christoph Zellner
Department Mathematik
Involved external institutions
Netherlands (NL)How to cite
APA:
van Dijk, G., Neeb, K.H., Salmasian, H., & Zellner, C. (2016). On the characterization of trace class representations and Schwartz operators. Journal of Lie Theory, 26(3), 787 - 805.
MLA:
van Dijk, Gerrit, et al. "On the characterization of trace class representations and Schwartz operators." Journal of Lie Theory 26.3 (2016): 787 - 805.
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