Reflection Positivity and Hankel Operators- the Multiplicity Free Case

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,τ), where G is a group, τ an involutive automorphism of G and S⊆G a subsemigroup with τ(S)=S⁻¹. For the triples (Z,N,−idZ), corresponding to reflection positive operators, and (R,R+,−idR), 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 μH on R+ defined by a positive Hankel operator H on H²(C+) to define a Pick function whose imaginary part, restricted to the imaginary axis, provides an operator symbol for H.