The weyl calculus for group generators satisfying the canonical commutation relations

Jan Van Neerven; Pierre Portal

doi:10.7900/jot.2018jun13.2250

The weyl calculus for group generators satisfying the canonical commutation relations

Jan Van Neerven
, Pierre Portal

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

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.

Original language	English
Pages (from-to)	253-298
Number of pages	46
Journal	Journal of Operator Theory
Volume	83
Issue number	2
DOIs	https://doi.org/10.7900/jot.2018jun13.2250
Publication status	Published - 1 Mar 2020

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title = "The weyl calculus for group generators satisfying the canonical commutation relations",

abstract = "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.",

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

author = "\{Van Neerven\}, Jan and Pierre Portal",

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year = "2020",

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doi = "10.7900/jot.2018jun13.2250",

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T1 - The weyl calculus for group generators satisfying the canonical commutation relations

AU - Van Neerven, Jan

AU - Portal, Pierre

N1 - Publisher Copyright: © by THETA, 2020.

PY - 2020/3/1

Y1 - 2020/3/1

N2 - 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.

AB - 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.

KW - Canonical commutation relations

KW - H¥-functional calculus

KW - Pseudo-differential calculus

KW - Spectral multipliers

KW - Transference of C-groups

KW - Twisted convolution

KW - UMD spaces

KW - Weyl pairs

UR - https://www.scopus.com/pages/publications/85088237609

U2 - 10.7900/jot.2018jun13.2250

DO - 10.7900/jot.2018jun13.2250

M3 - Article

SN - 0379-4024

VL - 83

SP - 253

EP - 298

JO - Journal of Operator Theory

JF - Journal of Operator Theory

IS - 2

ER -

The weyl calculus for group generators satisfying the canonical commutation relations

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