Trace class and Hilbert-Schmidt pseudo differential operators on step two nilpotent Lie groups

Abstract

Let G be a step two nilpotent Lie group. In this paper, we give necessary and sufficient conditions on the operator valued symbols σ such that the associated pseudo-differential operators Tσ on G are in the class of Hilbert-Schmidt operators. As a key step to prove this, we define (μ,ν)-Weyl transform on G and derive a trace formula for (μ,ν)-Weyl transform with symbols in L2(R2n). We show that Hilbert-Schmidt pseudo-differential operators on L2(G) are same as Hilbert-Schmidt (μ,ν)-Weyl transform with symbol in L2(R2n+r+k×R2n+r+k). Further, we present a characterization of the trace class pseudo-differential operators on G and provide a trace formula for these trace class operators.