$H^\infty$-functional calculus and models of Nagy-Foia\c{s} type for sectorial operators

We prove that a sectorial operator admits an H-infty - functional calculus if and only if it has a functional model of Nagy-Foias type. Furthermore, we give a concrete formula for the characteristic function (in a generalized sense) of such an operator. More generally, this approach applies to any sectorial operator by passing to a different norm (the McIntosh square function norm). We also show that this quadratic norm is close to the original one, in the sense that there is only a logarithmic gap between them.