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Hilbert–Schmidt and trace class pseudo-differential operators on the Heisenberg group

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Abstract

Pseudo-differential operators with operator-valued symbols on the Heisenberg group \({\mathbb{H }}^n\) are introduced. We give necessary and sufficient conditions on the symbols for which these operators are in the Hilbert–Schmidt class. These Hilbert–Schmidt operators are then identified with Weyl transforms with symbols in \(L^2({\mathbb{R }}^{2n+1}\times {\mathbb{R }}^{2n+1}).\) We also give a characterization of trace class pseudo-differential operators on the Heisenberg group \({\mathbb{H }}^n\). A trace formula for these trace class operators is presented.

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Acknowledgments

The authors are grateful to the referee for the careful reading of the paper and the useful comments.

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Authors and Affiliations

  1. Department of Mathematics, Imperial College London, 180 Queen’s Gate, London, SW7 2AZ, UK

    Aparajita Dasgupta

  2. Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, M3J 1P3, Canada

    M. W. Wong

Authors
  1. Aparajita Dasgupta
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  2. M. W. Wong
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Corresponding author

Correspondence to M. W. Wong.

Additional information

This research has been supported by the Grace Chisholm Young Fellowship of the London Mathematical Society and the Natural Sciences and Engineering Research Council of Canada.

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Dasgupta, A., Wong, M.W. Hilbert–Schmidt and trace class pseudo-differential operators on the Heisenberg group. J. Pseudo-Differ. Oper. Appl. 4, 345–359 (2013). https://doi.org/10.1007/s11868-013-0079-8

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  • DOI: https://doi.org/10.1007/s11868-013-0079-8

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