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

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 $\frac{n}{2}+1$.