Completely non unitary contractions and Analyticity

TL;DR

This paper classifies completely non-unitary contractions with finite defect spaces satisfying a Hardy shift property, showing they are analytic iff they lack non-zero eigenvalues and characterizing their hyponormality.

Contribution

It provides a complete classification of such contractions and characterizes their analyticity and hyponormality based on eigenvalues and defect space inclusion.

Findings

Contractions are analytic iff they have no non-zero eigenvalues.

Characterization of hyponormality for these contractions.

Complete classification under the Hardy shift property.

Abstract

We study completely non-unitary contractions $T$ with finite dimensional defect spaces $\mathcal{D}_T$ and $\mathcal{D}_{T^*}$. We present a complete classification of all such contractions $T$ that satisfy a generalized property of Hardy shift operator: $\mathcal{D}_T\subseteq\mathcal{D}_{T^*}$. We show that $T$ is analytic if and only if it has no non-zero eigenvalue. Furthermore, we characterize the hyponormality of all those $T$.