The weyl calculus for group generators satisfying the canonical commutation relations

Jan Van Neerven, Pierre Portal

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We generalise the classical Weyl pseudo-differential calculus on Rd to the setting of two d-tuples of operators A = (A1,..., Ad) and B = (B1,..., Bd) acting on a Banach space generating bounded C0-groups satisfying the Weyl canonical commutation relations. We show that the resulting Weyl calculus extends to symbols in the standard symbol class S0 provided appropriate bounds can be established. Using transference techniques we obtain boundedness of the H-functional calculus (and even the Hormander calculus), for the abstract harmonic oscillator.

Original languageEnglish
Pages (from-to)253-298
Number of pages46
JournalJournal of Operator Theory
Issue number2
Publication statusPublished - 2020

Bibliographical note

Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project
Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.


  • Canonical commutation relations
  • H¥-functional calculus
  • Pseudo-differential calculus
  • Spectral multipliers
  • Transference of C-groups
  • Twisted convolution
  • UMD spaces
  • Weyl pairs


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