The numerical range of periodic banded Toeplitz operators

TL;DR

This paper characterizes the numerical range of certain periodic banded Toeplitz operators, showing it as a convex hull of unions of symbol matrix ranges, and provides a counterexample to previous assumptions.

Contribution

It extends the understanding of numerical ranges for periodic banded Toeplitz operators and presents a novel counterexample in the non-tridiagonal case.

Findings

Numerical range is the convex hull of unions of symbol matrix ranges.

For certain operators, the numerical range cannot be represented by a single finite matrix.

The paper provides a counterexample in the 2-periodic, 5-banded case.

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.