Curvature Condition for Noncontractions does not imply Similarity to the Backward Shift

TL;DR

This paper demonstrates that satisfying the curvature condition alone does not guarantee similarity to the backward shift operator, emphasizing the necessity of the contraction assumption in such characterizations.

Contribution

It provides a counterexample showing the curvature condition is insufficient for similarity to the backward shift without contraction assumptions.

Findings

Counterexample operator satisfies curvature condition but not similar to backward shift

Contraction assumption is necessary for similarity characterization

Highlights limitations of curvature condition in operator similarity

Abstract

We give an example of an operator that satisfies the curvature condition as defined in [2], but is not similar to the backward shift S* on the Hardy class H^2. We conclude therefore that the contraction assumption in the similarity characterization given in [2] is a necessary requirement.