Crouzeix's Conjecture and related problems

TL;DR

This paper investigates Crouzeix's conjecture, demonstrating its validity for certain contractions with well-separated eigenvalues, and explores extremal functions and shift compressions as key examples.

Contribution

It provides new results confirming the conjecture for specific classes of operators and analyzes extremal functions using the pseudohyperbolic metric.

Findings

Crouzeix's conjecture holds for contractions with sufficiently separated eigenvalues.

Extremal functions exhibit specific properties related to the pseudohyperbolic metric.

Shift operator compressions serve as illustrative examples for the theory.

Abstract

In this paper, we establish several results related to Crouzeix's conjecture. We show that the conjecture holds for contractions with eigenvalues that are sufficiently well-separated. This separation is measured by the so-called separation constant, which is defined in terms of the pseudohyperbolic metric. Moreover, we study general properties of related extremal functions and associated vectors. Throughout, compressions of the shift serve as illustrating examples which also allow for refined results.