On the characterization of trace class representations and Schwartz operators

Original · EN

In this note we collect several characterizations of unitary representations (π, 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 π(f) is trace class. In particular we derive the new result that, for some m ∈ N, all operators π(f), f ∈ Cᵐc(G), are trace class. As a consequence the corresponding distribution character θπ is of finite order. We further show π is trace class if and only if every operator A, which is smoothing in the sense that AH H∞, is trace class and that this in turn is equivalent to the Fréchet space H∞ being nuclear, which in turn is equivalent to the realizability of the Gaussian measure of H on the space H⁻∞ of distribution vectors. Finally we show that, even for infinite dimensional Fréchet-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.