Similarity results for operators of class $C_0$ and the algebra $H^\infty(T)$

TL;DR

This paper investigates the conditions under which multiplicity-free operators of class C_0 with the same finite Blaschke product as minimal function are similar, and how their associated operator algebras' isomorphisms relate to this similarity.

Contribution

It establishes isomorphism and similarity criteria for operators of class C_0 with the same finite Blaschke product, extending previous results to infinite products under the generalized Carleson condition.

Findings

Operator algebras H^ty(T) are isomorphic for such operators.

Operators are similar if the algebra isomorphism's norm is controlled.

Results extend to infinite Blaschke products satisfying the generalized Carleson condition.

Abstract

Given two multiplicity-free operators $T_1$ and $T_2$ of class $C_0$ having the same finite Blaschke product as minimal function, the operator algebras $H^\infty(T_1)$ and $H^\infty(T_2)$ are isomorphic and $T_1$ is similar to $T_2$. We find conditions under which the norm of the similarity between the operators can be controlled by the norm of the algebra isomorphism. As an application, we improve upon earlier work and obtain results regarding similarity when the minimal function is an infinite product of finite Blaschke products satisfying the generalized Carleson condition.