On the Characterization of Trace Class Representations and Schwartz Operators

In this note we collect several characterizations of unitary representations (, 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 (f ) is trace class.In particular we derive the new result that, for some m N , all operators (f ) , f C m c (G) , are trace class.As a consequence the corresponding distribution character is of finite order.We further show 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 Frchet 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 Frchet-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.