Unitary equivalence and similarity to Jordan models for weak contractions of class $C_0$

TL;DR

This paper investigates when weak contractions of class C_0 are unitarily equivalent to their Jordan models, using boundary representation theory, and extends results to cases with arbitrary finite multiplicity.

Contribution

It provides new criteria for unitary equivalence of weak contractions to Jordan models, especially for finite multiplicity, utilizing boundary representations.

Findings

Established conditions for unitary equivalence to Jordan models.

Extended results to arbitrary finite multiplicity cases.

Improved understanding of similarity when minimal functions are Blaschke products.

Abstract

We obtain results on the unitary equivalence of weak contractions of class $C_0$ to their Jordan models under an assumption on their commutants. In particular, our work addresses the case of arbitrary finite multiplicity. The main tool is the theory of boundary representations due to Arveson. We also generalize and improve previously known results concerning unitary equivalence and similarity to Jordan models when the minimal function is a Blaschke product.