Differentiability of functions of contractions

TL;DR

This paper investigates the differentiability of functions of contractions on Hilbert spaces, providing sharp conditions and explicit formulas for derivatives using advanced operator integral techniques.

Contribution

It establishes precise differentiability criteria in terms of Besov spaces and derives explicit formulas for derivatives of functions of contractions.

Findings

Sharp Besov space conditions for differentiability

Explicit formulas for directional derivatives

Higher derivatives characterized via multiple operator integrals

Abstract

In this paper we study differentiability properties of the map $T\mapsto\phi(T)$, where $\phi$ is a given function in the disk-algebra and $T$ ranges over the set of contractions on Hilbert space. We obtain sharp conditions (in terms of Besov spaces) for differentiability and existence of higher derivatives. We also find explicit formulae for directional derivatives (and higher derivatives) in terms of double (and multiple) operator integrals with respect to semi-spectral measures.