Corners of multidimensional numerical ranges

TL;DR

This paper investigates the geometric structure of the $n$-dimensional numerical range of linear operators on Hilbert spaces, revealing that corner points' components are eigenvalues of the operator.

Contribution

It establishes that each component of a corner point in the $n$-dimensional numerical range corresponds to an eigenvalue of the operator, a novel geometric insight.

Findings

Corner points' components are eigenvalues.

Provides geometric characterization of numerical range.

Extends understanding of operator spectra.

Abstract

The $n$-dimensional numerical range of a densely defined linear operator $T$ on a complex Hilbert space $\H$ is the set of vectors in $\C^n$ of the form $(< Te_1,e_1>,...,< Te_n,e_n>)$, where $e_1,...,e_n$ is an orthonormal system in $\H$, consisting of vectors from the domain of $T$. We prove that the components of every corner point of the $n$-dimensional numerical range are eigenvalues of $T$.