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

Abstract

In this paper we prove a conjecture stated by the first two authors in [12] 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) × (n + 1) matrices. Furthermore, when n + 1 is odd, we show that the size of such matrices simplifies to n 2 + 1.