Crouzeix's conjecture holds for tridiagonal $3\times 3$ matrices with elliptic numerical range centered at an eigenvalue

TL;DR

This paper proves Crouzeix's conjecture in its strong form for all tridiagonal 3x3 matrices with elliptic numerical range centered at an eigenvalue, extending previous results and confirming the conjecture in this specific case.

Contribution

The authors establish the conjecture's validity in its strong form for all tridiagonal 3x3 matrices with elliptic numerical range centered at an eigenvalue, a significant special case.

Findings

Crouzeix's conjecture holds for all such matrices.

Extension of Choi's main result.

Validation of the conjecture in a new matrix class.

Abstract

M. Crouzeix formulated the following conjecture in (Integral Equations Operator Theory 48, 2004, 461--477): For every square matrix $A$ and every polynomial $p$, $$ \|p(A)\| \le 2 \max_{z\in W(A)}|p(z)|, $$ where $W(A)$ is the numerical range of $A$. We show that the conjecture holds in its strong, completely bounded form, i.e., where $p$ above is allowed to be any matrix-valued polynomial, for all tridiagonal $3\times 3$ matrices with constant main diagonal: $$ \left[\begin{matrix}a&b_1&0\\c_1&a&b_2\\0&c_2&a\end{matrix}\right],\qquad a,b_k,c_k\in\mathbb C, $$ or equivalently, for all complex $3\times 3$ matrices with elliptic numerical range and one eigenvalue at the center of the ellipse. We also extend the main result of D. Choi in (Linear Algebra Appl. 438, 3247--3257) slightly.