On the boundary of the numerical range of some Jacobi operators

TL;DR

This paper investigates the boundary characteristics of the numerical range of Jacobi operators, revealing that non-round boundary points occur only at intersections with the essential spectrum and are not eigenvalues.

Contribution

It establishes conditions under which the boundary of the numerical range is non-round and clarifies the relationship with the essential spectrum for Jacobi operators.

Findings

Non-round boundary points occur only at the essential spectrum.

Such boundary points are not eigenvalues of the Jacobi operators.

The boundary behavior is characterized under specific conditions.

Abstract

In this paper, we study the numerical range of Jacobi operators and it is shown that under certain conditions, the boundary of the numerical range of these operators can be non-round only at the points where it touches the essential spectrum. It is further shown that these points cannot be the eigenvalue of the Jacobi operators.