On the Characterization of Trace Class Representations and Schwartz Operators

Gerrit van Dijk¹ ; Karl-Hermann Neeb² ; Hadi Salmasian³ ; Christoph Zellner²

¹Mathematical Institute, Leiden University, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands
²Department Mathematik, FAU Erlangen-Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany
³Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Ave., Ottawa Ont. K1N 6N5, Canada

Journal of Lie Theory, Tome 26 (2016) no. 3, pp. 787-805

Résumé

\def\cH{{\cal H}} We collect several characterizations of unitary representations $(\pi, \cH)$ 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 $A\cH\subseteq \cH^\infty$, is trace class and that this in turn is equivalent to the Fr\'echet space $\cH^\infty$ being nuclear, which in turn is equivalent to the realizability of the Gaussian measure of $\cH$ on the space $\cH^{-\infty}$ 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.

Détail

Classification : 22E45, 22E66
Mots-clés : Trace class representation, smoothing operator, Schwartz operator, unitary representation