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

TL;DR

This paper establishes a characterization of sectorial operators admitting an H-infinity functional calculus through Nagy-Foias type models, providing explicit formulas and norm comparisons.

Contribution

It introduces a new equivalence between H-infinity calculus and Nagy-Foias models for sectorial operators, with concrete formulas and norm analysis.

Findings

Characterization of H-infinity calculus via Nagy-Foias models

Explicit formula for the generalized characteristic function

Comparison of quadratic and original norms with a logarithmic gap

Abstract

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.