The numerical range of some periodic tridiagonal operators is the convex hull of the numerical ranges of two finite matrices

TL;DR

This paper proves a conjecture that the numerical range of certain periodic tridiagonal operators equals the convex hull of the numerical ranges of two finite matrices, with simplifications for odd cases.

Contribution

It establishes the closure of the numerical range for a class of periodic tridiagonal operators as the convex hull of two finite matrices' numerical ranges, confirming a conjecture.

Findings

Numerical range equals convex hull of two matrices' ranges

Simplified matrix size when n+1 is odd

Confirmed conjecture for a class of operators

Abstract

In this paper we prove a conjecture stated by the first two authors establishing the closure of the numerical range of a certain class of $n+1$-periodic tridiagonal operators as the convex hull of the numerical ranges of two tridiagonal $(n+1) \times (n+1)$ matrices. Furthermore, when $n+1$ is odd, we show that the size of such matrices simplifies to $\frac{n}{2}+1$.