The numerical range of a periodic tridiagonal operator reduces to the numerical range of a finite matrix

TL;DR

This paper proves that the numerical range of an (n+1)-periodic tridiagonal operator can be fully characterized by the numerical range of a specific finite matrix, simplifying analysis of such operators.

Contribution

It establishes a precise reduction of the numerical range of periodic tridiagonal operators to finite matrices, providing a new tool for their analysis.

Findings

Numerical range of (n+1)-periodic tridiagonal operator equals that of a 2(n+1)×2(n+1) matrix.

Closure of the numerical range is exactly the numerical range of the finite matrix.

Simplifies the study of spectral properties of periodic tridiagonal operators.

Abstract

In this paper we show that the closure of the numerical range of an $n+1$-periodic tridiagonal operator is equal to the numerical range of a $2(n+1)\times 2(n+1)$ complex matrix.