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In this paper, we study some operator theoretical properties of pseudo-differential operators with operator-valued symbols on the Heisenberg motion group. Specifically, we investigate–boundedness of pseudo-differential operators on the Heisenberg motion group for the range. We also provide a necessary and sufficient condition on the operator-valued symbols in terms of  λ -Weyl transforms such that the corresponding pseudo-differential operators on the Heisenberg motion group are in the class of Hilbert–Schmidt operators. As a consequence, we obtain a characterization of the trace class pseudo-differential operators on the Heisenberg motion group and provide a trace formula for these trace class operators.

L 2 -L p estimates and Hilbert–Schmidt pseudo differential operators on the Heisenberg motion group