The numerical range of periodic banded Toeplitz operators

Abstract

We prove that the closure of the numerical range of a \((n+1)\)-periodic and \((2m+1)\)-banded Toeplitz operator can be expressed as the closure of the convex hull of the uncountable union of numerical ranges of certain symbol matrices. In contrast to the periodic 3-banded (or tridiagonal) case, we show an example of a 2-periodic and 5-banded Toeplitz operator such that the closure of its numerical range is not equal to the numerical range of a single finite matrix.