On the Crouzeix ratio for $N\times N$ matrices
Bartosz Malman, Javad Mashreghi, Ryan O’Loughlin, Thomas Ransford · arXiv (Cornell University) · 2024
The Crouzeix ratio $ψ(A)$ 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 $ψ(A)\le 1+\sqrt{2}$, and it is conjectured that $ψ(A)\le 2$. In this note, we show that $ψ(A)\le C_N$, 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 $A\mapsto ψ(A)$.