An octagon containing the numerical range of a bounded linear operator

TL;DR

This paper introduces an octagon-shaped polygon that encloses the numerical range of a bounded linear operator on a complex Hilbert space, constructed solely from the operator's norms, providing a geometric containment tool.

Contribution

The paper presents a novel geometric construction of an octagon that contains the numerical range, using only norms, and establishes its properties in relation to the spectral norm.

Findings

The octagon is symmetric with respect to the origin.

It is tangent to the numerical range at at least four points.

The construction applies to the spectral norm case.

Abstract

A polygon is derived that contains the numerical range of a bounded linear operator on a complex Hilbert space, using only norms. In its most general form, the polygon is an octagon, symmetric with respect to the origin, and tangent to the closure of the numerical range in at least four points when the spectral norm is used.