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 $$\mathbb{H}$(G):=G \times \widehat{G} \times $\mathbb{T}$,$where$G$ a locally compact abelian group with its dual group $\widehat{G}$. 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, $j\in {\mathbb{Z}}^{\ast }$ with symbols in $L^2(G\times \widehat{G})\mathrm{.}$ We go on to present a characterization of the trace class pseudo-differential operators on $\mathbb{H}(G)$. Finally, we also give a trace formula for these trace class operators.