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 $iA_{1}, \dots , iA_{d}$ and $iB_{1}, \dots , iB_{d}$ generate bounded $C_0$-groups satisfying the Weyl canonical commutation relations $e^{isA_j}e^{itA_k} = e^{itA_k}e^{isA_j}$, $e^{isB_j}e^{itB_k} = e^{itB_k}e^{isB_j}$, and $e^{isA_j}e^{itB_k} = e^{-ist \delta_{jk}} e^{itB_k}e^{isA_j}$ $(1\le j,k\le d)$. We show that the resulting calculus $a\mapsto a(A,B) \in \mathscr{L}(X)$, initially defined for Schwartz functions $a\in \mathscr{S}(\mathbb{R}^{2d})$, 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 $C_{\widehat{a}}$ 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 H\"ormander calculus), for the abstract harmonic oscillator $L = \frac12\sum_{j=1}^d (A_j^2+B_j^2)-\frac12d$.