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Journal of Operator Theory

Volume 83, Issue 2, Spring 2020  pp. 253-298.

The Weyl calculus for group generators satisfying the canonical commutation relations

Authors:  Jan van Neerven (1), Pierre Portal (2)
Author institution:(1) Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands
(2) The Australian National University, Mathematical Sciences Institute, Hanna Neumann Building, Acton ACT 0200, Australia

Summary: We generalise the classical Weyl pseudo-differential calculus on $\mathbb{R}^{d}$ to the setting of two $d$-tuples of operators $A=(A_{1}, \dots , A_{d})$ and $B=(B_{1}, \dots , B_{d})$ acting on a Banach space generating bounded $C_0$-groups satisfying the Weyl canonical commutation relations. We show that the resulting Weyl calculus extends to symbols in the standard symbol class $S^{0}$ provided appropriate bounds can be established. Using transference techniques we obtain boundedness of the $H^{\infty}$-functional calculus (and even the H\"ormander calculus), for the abstract harmonic oscillator $L = \frac12\sum\limits_{j=1}^d (A_j^2+B_j^2)-\frac12d$.

Keywords: Weyl pairs, canonical commutation relations, pseudo-differential calculus, twisted convolution, transference of $C_0$-groups, UMD spaces, $H^\infty$-functional calculus, spectral multipliers

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