On the characterization of trace class representations and Schwartz operators

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 $\pi (f)$, $f\in C_c^m(G)$, 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 $A\mathcal{H}\subseteq {\mathcal{H}}^{\mathrm{\infty }}$, is trace class and that this in turn is equivalent to the Fr'echet space ${\mathcal{H}}^{\mathrm{\infty }}$ being nuclear, which in turn is equivalent to the realizability of the Gaussian measure of $\mathcal{H}$ on the space ${\mathcal{H}}^{-\mathrm{\infty }}$ of distribution vectors. Finally we show that, even for infinite dimensional Fr'echet-Lie groups, $A$ and $A^{\ast }$ 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.