The numerical range of a class of periodic tridiagonal operators

TL;DR

This paper characterizes the numerical range of specific periodic tridiagonal operators by expressing it as the convex hull of unions of numerical ranges of associated matrices, providing explicit descriptions and conjectures for the general case.

Contribution

It introduces a method to compute the numerical range of periodic tridiagonal operators using unions of matrix numerical ranges and improves this for special cases.

Findings

Numerical range is the convex hull of unions of matrix numerical ranges.

Explicit descriptions are provided for certain special cases.

A conjecture is proposed for the general case.

Abstract

In this paper we compute the closure of the numerical range of certain periodic tridiagonal operators. This is achieved by showing that the closure of the numerical range of such operators can be expressed as the closure of the convex hull of the uncountable union of numerical ranges of certain symbol matrices. For a special case, this result can be improved so that it is the convex hull of the union of the numerical ranges of only two matrices. A conjecture is stated for the general case.