The theory of pseudo-differential operators has provided a very powerful and flexible tool for treating certain problems in linear partial differential equations. The importance of the Heisenberg group in general harmonic analysis and problems involving partial differential operators on manifolds is well established. In this talk, I will introduce the pseudo-differential operators with operator-valued symbols on the Heisenberg group. I will give the necessary and sufficient conditions on the symbols for which these operators are in the Hilbert-Schmidt class. I will identify these Hilbert-Schmidt operators with the Weyl transforms with symbols in L2(R2n+1 × R2n+1). I will also provide a characterization of trace class pseudo-differential opera- tors on the Heisenberg group. A trace formula for these trace class operators would be presented.
School of Mathematical Sciences
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