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

Abstract

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 T σ 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 L 2 ( R 2 n ) . We show that Hilbert-Schmidt pseudo-differential operators on L 2 ( G ) are same as Hilbert-Schmidt ( μ , ν ) -Weyl transform with symbol in L 2 ( R 2 n + r + k × R 2 n + 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.