Some constants related to numerical ranges

TL;DR

This paper investigates constants related to the numerical range of matrices to advance understanding of the conjecture that the numerical range is a 2-spectral set, including theoretical review and numerical experiments.

Contribution

It introduces a study of constants associated with the numerical range and presents initial results and numerical tests to explore the conjecture.

Findings

Partial results on constants related to numerical ranges

Identification of open problems in the field

Numerical experiments supporting theoretical insights

Abstract

In an attempt to progress towards proving the conjecture the numerical range W (A) is a 2--spectral set for the matrix A, we propose a study of various constants. We review some partial results, many problems are still open. We describe our corresponding numerical tests.