The Weyl calculus for group generators satisfying the canonical commutation relations

Abstract

Classical pseudo-differential calculus on ${\mathbb{R}}^d$ can be viewed as a (non-commutative) functional calculus for the standard position and momentum operators $(Q_1,\dots ,Q_d)$ and $(P_1,\dots ,P_d)$. We generalise this calculus 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 $X$ such that $i{A}_1,\dots ,i{A}_d$ and $i{B}_1,\dots ,i{B}_d$ generate bounded $C_0$-groups satisfying the Weyl canonical commutation relations $e^{is{A}_j}{e}^{it{A}_k}=e^{it{A}_k}{e}^{is{A}_j}$, $e^{is{B}_j}{e}^{it{B}_k}=e^{it{B}_k}{e}^{is{B}_j}$, and $e^{is{A}_j}{e}^{it{B}_k}=e^{-ist{\delta }_{jk}}{e}^{it{B}_k}{e}^{is{A}_j}$ $(1\le j,k\le d)$. We show that the resulting calculus $a\mapsto a(A,B)\in \mathcal{L}(X)$, initially defined for Schwartz functions $a\in \mathcal{S}({\mathbb{R}}^{2d})$, extends to symbols in the standard symbol class $S^0$ of pseudo-differential calculus provided appropriate bounds can be established. We also prove a transference result that bounds the operators $a(A,B)$ in terms of the twisted convolution operators $C_{\hat{a}}$ acting on $L^2({\mathbb{R}}^{2d};X)$. We apply these results to obtain $R$-sectoriality and boundedness of the $H^{\infty }$-functional calculus (and even the Hörmander calculus), for the abstract harmonic oscillator $L=\frac{1}{2}{\sum }_{j=1}^d(A_j^2+B_j^2)-\frac{1}{2}d$.

38 pages, submitted for publication