Abstract
In this paper we introduce and study pseudo-differential operators with operator valued symbols on the abstract Heisenberg group H(G):=G×Gˆ×T, where G a locally compact abelian group with its dual group Gˆ. 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∈Z⁎ with symbols in L2(G×Gˆ). 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.
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