The weyl calculus for group generators satisfying the canonical commutation relations

van Neerven, J.M.A.M.; Portal, Pierre

doi:10.7900/jot.2018jun13.2250

The weyl calculus for group generators satisfying the canonical commutation relations

Journal article (2020)

Authors

J.M.A.M. van Neerven Analysis -

Pierre Portal Australian National University

Research Group

Analysis () (TU Delft)

DOI: https://doi.org/10.7900/jot.2018jun13.2250

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

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Published Date

2020

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Analysis

Abstract

We generalise the classical Weyl pseudo-differential calculus on R^d to the setting of two d-tuples of operators A = (A₁,..., A_d) and B = (B₁,..., B_d) 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.

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