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 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 matrices. Furthermore, when n + 1 is odd, we show that the size of such matrices simplifies to .