Similarity results for operators of class C_0

TL;DR

This paper investigates the similarity and quasisimilarity relations between certain contraction operators of class C_0 and their associated Jordan blocks, especially under conditions involving Blaschke products and Carleson sequences.

Contribution

It establishes conditions under which a multiplicity-free contraction of class C_0 is similar or quasisimilar to its Jordan model, extending known results to more general root multiplicities.

Findings

T is quasisimilar to S(m_T) for multiplicity-free contractions.

Under Carleson sequence conditions, T is similar to S(m_T).

The similarity result extends to cases with bounded root multiplicities.

Abstract

If T is a multiplicity-free contraction of class C_0 with minimal function m_T, then it is quasisimilar to the Jordan block S(m_T). In case m_T is a Blaschke product with simple roots forming a Carleson sequence, we show that the relation between T and S(m_T) can be strengthened to similarity. Under the additional assumption that u(T) has closed range for every inner divisor u of m_T, the result also holds in the more general setting where the roots have bounded multiplicities.