Blaschke Products, Level Sets, and Crouzeix's Conjecture

TL;DR

This paper investigates a specialized version of Crouzeix's conjecture within the context of model spaces and finite Blaschke products, establishing new results for certain cases using geometric analysis of numerical ranges.

Contribution

It introduces the level set Crouzeix (LSC) conjecture and proves it for specific cases involving unicritical Blaschke products of degree 4.

Findings

Proved the LSC conjecture for unicritical Blaschke products of degree 4.

Established structural and uniqueness properties of level sets of finite Blaschke products.

Utilized the geometry of the numerical range to derive key results.

Abstract

We study several problems motivated by Crouzeix's conjecture, which we consider in the special setting of model spaces and compressions of the shift with finite Blaschke products as symbols. We pose a version of the conjecture in this setting, called the level set Crouzeix (LSC) conjecture, and establish structural and uniqueness properties for (open) level sets of finite Blaschke products that allow us to prove the LSC conjecture in several cases. In particular, we use the geometry of the numerical range to prove the LSC conjecture for compressions of the shift corresponding to unicritical Blaschke products of degree $4$.