Differentiability of functions of contractions

Details Differentiability of functions of contractions Differentiability of functions of contractions

2008-05-28

arxiv.org/abs/0805.4370v1

Mathematics

Functional Analysis

26 pages

Scientific paper

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.

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