An observation concerning boundary points of the numerical range

TL;DR

This paper extends Hübner's theorem by showing that boundary points with infinite upper curvature of a linear operator's numerical range are contained in its spectrum, using classical ideas of Berberian.

Contribution

It proves that Hübner's result holds under weaker conditions, specifically infinite upper curvature, simplifying the understanding of boundary points in the numerical range.

Findings

Boundary points with infinite upper curvature are in the spectrum.

The proof is short, simple, and relies on classical ideas.

The result generalizes Hübner's theorem.

Abstract

A theorem of H\"ubner states that non-round boundary points of the numerical range of a linear operator, i.e. points where the boundary has infinite curvature, are contained in the spectrum of the operator. In this note, answering a question of Salinas and Velasco, we will show that H\"ubner's result remains true under the weaker assumption that the boundary has infinite upper curvature. Our short and simple proof relies on some classical ideas of Berberian.