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 $\pi(f)$, $f \in C^m_c(G)$, 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.