Hilbert-Schmidt and Trace class pseudo-differential operators on the abstract Heisenberg group

In this paper we introduce and study pseudo-differential operators with operator valued symbols on the abstract Heisenberg group H(G) := G x (G) over cap x T, where G a locally compact abelian group with its dual group (G) over cap. We obtain a necessary and sufficient condition on symbols for which these operators are in the class of Hilbert-Schmidt operators. As a key step in proving this we derive a trace formula for the trace class j-Weyl transform, j is an element of Z*. with symbols in L-2 (G x (G) over cap). We go on to present a characterization of the trace class pseudo-differential operators on H(G). Finally, we also give a trace formula for these trace class operators. (C) 2020 Elsevier Inc. All rights reserved.