On the characterization of trace class representations and Schwartz operators

TL;DR

This paper characterizes trace class unitary representations of finite-dimensional Lie groups, linking trace class properties to smooth operators, nuclearity of the smooth vectors space, and Schwartz operators, with implications for distribution characters.

Contribution

It provides new characterizations of trace class representations, including conditions involving smoothing operators, nuclearity, and Schwartz operators, extending to infinite-dimensional Lie groups.

Findings

Operators $ ext{pi}(f)$ are trace class for some $m$, with $f$ in $C^m_c(G)$.

Distribution character $ heta_ ext{pi}$ has finite order.

Trace class property is equivalent to the nuclearity of the smooth vectors space.

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.