Hilbert–Schmidt and trace class pseudo-differential operators on the Heisenberg group

Abstract

Pseudo-differential operators with operator-valued symbols on the Heisenberg group \({\mathbb{H }}^n\) are introduced. We give necessary and sufficient conditions on the symbols for which these operators are in the Hilbert–Schmidt class. These Hilbert–Schmidt operators are then identified with Weyl transforms with symbols in \(L^2({\mathbb{R }}^{2n+1}\times {\mathbb{R }}^{2n+1}).\) We also give a characterization of trace class pseudo-differential operators on the Heisenberg group \({\mathbb{H }}^n\). A trace formula for these trace class operators is presented.