On the Crouzeix ratio for $N\times N$ matrices

TL;DR

This paper investigates the Crouzeix ratio for complex matrices, establishing an upper bound that depends only on the matrix size, advancing understanding of this ratio's behavior.

Contribution

It proves that the Crouzeix ratio is bounded above by a constant depending solely on the matrix size, improving previous universal bounds.

Findings

Crouzeix ratio is bounded by a size-dependent constant

The bound is strictly less than 1+√2 for all finite N

Continuity properties of the ratio map are analyzed

Abstract

The Crouzeix ratio $\psi(A)$ of an $N\times N$ complex matrix $A$ is the supremum of $\|p(A)\|$ taken over all polynomials $p$ such that $|p|\le 1$ on the numerical range of $A$. It is known that $\psi(A)\le 1+\sqrt{2}$, and it is conjectured that $\psi(A)\le 2$. In this note, we show that $\psi(A)\le C_N$, where $C_N$ is a constant depending only on $N$ and satisfying $C_N<1+\sqrt{2}$. The proof is based on a study of the continuity properties of the map $A\mapsto \psi(A)$.