Reflection positivity and Hankel operators -- the multiplicity free case

TL;DR

This paper explores the relationship between reflection positive representations and Hankel operators, introducing the concept of Hankel positive representations and demonstrating how they can be adapted to reflection positivity through scalar product adjustments.

Contribution

It introduces Hankel positive representations for triples involving groups and involutions, linking positive Hankel operators with reflection positivity in a novel framework.

Findings

Every Hankel positive representation can be made reflection positive with a scalar product change.

A measure derived from Hankel operators defines a Pick function serving as an operator symbol.

The approach connects Hankel operators, Carleson measures, and reflection positivity in a unified theory.

Abstract

We analyze reflection positive representations in terms of positive Hankel operators. This is motivated by the fact that positive Hankel operators are described in terms of their Carleson measures, whereas the compatibility condition between representations and reflection positive Hilbert spaces is quite intricate. This leads us to the concept of a Hankel positive representation of triples $(G,S,\tau)$, where $G$ is a group, $\tau$ an involutive automorphism of $G$ and $S \subseteq G$ a subsemigroup with $\tau(S) = S^{-1}$. For the triples $(\mathbb Z,\mathbb N,-id_{\mathbb Z})$, corresponding to reflection positive operators, and $(\mathbb R,\mathbb R_+,-id_{\mathbb R})$, corresponding to reflection positive one-parameter groups, we show that every Hankel positive representation can be made reflection positive by a slight change of the scalar product. A key method consists in using the measure $\mu_H$ on $\mathbb R_+$ defined by a positive Hankel operator $H$ on $H^2(\mathbb C_+)$ to define a Pick function whose imaginary part, restricted to the imaginary axis, provides an operator symbol for $H$.