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

Let G be a step two nilpotent Lie group. In this paper, we give necessary and sufficient conditions on the operator valued symbols sigma such that the associated pseudo-differential operators T-sigma on G are in the class of Hilbert-Schmidt operators. As a key step to prove this, we define (mu, v)-Weyl transform on G and derive a trace formula for (mu, v)-Weyl transform with symbols in L-2 (R-2n). We show that Hilbert-Schmidt pseudo-differential operators on L-2 (G) are same as Hilbert-Schmidt (mu, v)-Weyl transform with symbol in L-2 (R2n+r+k x 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. (C) 2021 Elsevier Masson SAS. All rights reserved.

Automated summary

This paper examines the pseudo-differential operators in the class of Hilbert-Schmidt operators on a step two nilpotent Lie group, defining the (mu, v)-Weyl transform and presenting a trace formula for it. The study also explores the equivalence between Hilbert-Schmidt pseudo-differential operators and (mu, v)-Weyl transform with certain symbols. Additionally, the paper provides a characterization and trace formula for trace class pseudo-differential operators on the Lie group G.