On the convexity of numerical range over certain fields

TL;DR

This paper investigates the convexity properties of the numerical range of matrices over certain fields, especially focusing on cases involving Galois extensions and ordered fields, under specific assumptions.

Contribution

It provides new results on the convexity of the numerical range for matrices over fields with Galois extensions, extending classical concepts to more general algebraic settings.

Findings

Convexity of numerical range established under certain field assumptions.

Results extend classical numerical range properties to Galois extension fields.

Many results applicable to ordered fields.

Abstract

Let $L$ be a degree $2$ Galois extension of the field $K$ and $M$ an $n\times n$ matrix with coefficients in $L$. Let $\langle \ ,\ \rangle : L^n\times L^n\to L$ be the sesquilinear form associated to the involution $\sigma: L\to L$ fixing $K$. This sesquilinear form defines the numerical range $\mathrm{Num}(M)$ of any $n\times n$ matrix over $L$. In this paper we study the convexity of $\mathrm{Num}(M)$ (under certain assumptions on $K$ and/or $M$). Many of the results are for ordered fields.