Crouzeix's conjecture for classes of matrices

TL;DR

Crouzeix's conjecture posits a universal bound $\|p(A)\| \le 2 \sup_{z \in W(A)} |p(z)|$ for all matrices $A$ and polynomials $p$, a claim supported by extensive numerical evidence but lacking a general proof. The paper surveys existing results, constructs new matrix classes satisfying CC, and reveals a fundamental link between cyclicity and CC that reduces full verification to a differentiation operator on a function space associated with $A$. It further develops a framework connecting CC to extremal functions, symmetric matrices, and truncated Toeplitz operators, including a 3×3 equivalence among CC for TTOs, symmetric matrices, and TTOs on model spaces. These insights provide structural reductions and new avenues toward a complete proof, with potential practical impact on estimating operator norms from numerical ranges in applications.

Abstract

For a matrix $A$ which satisfies Crouzeix's conjecture, we construct several classes of matrices from $A$ for which the conjecture will also hold. We discover a new link between cyclicity and Crouzeix's conjecture, which shows that Crouzeix's Conjecture holds in full generality if and only if it holds for the differentiation operator on a class of analytic functions. We pose several open questions, which if proved, will prove Crouzeix's conjecture. We also begin an investigation into Crouzeix's conjecture for symmetric matrices and in the case of $3 \times 3$ matrices, we show Crouzeix's conjecture holds for symmetric matrices if and only if it holds for analytic truncated Toeplitz operators.

Theorems & Definitions (32)

• Conjecture 1.1
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Proposition 2.6