Spectral sets, extremal functions and exceptional matrices

TL;DR

This paper investigates the properties of matrices that maximize the operator norm of holomorphic functions applied to them, revealing special orthogonality characteristics of these extremal matrices.

Contribution

It introduces and analyzes a class of exceptional matrices characterized by orthogonality properties of their singular vectors in the context of extremal operator norm problems.

Findings

Exceptional matrices have mutually orthogonal principal singular vectors.

Orthogonality characterizes a special class of extremal matrices.

The study links spectral properties with extremal function behavior.

Abstract

Let $A$ be a square matrix and let $\Omega$ be an open set in the plane containing the spectrum of $A$. We consider the problem of maximizing the operator norm $\|f(A)\|$ amongst all holomorphic functions $f$ from $\Omega$ into the closed unit disk. If $f_0$ is extremal for this problem and if $\|f_0(A)\|>1$, then it turns out that the matrix $f_0(A)$ has special properties, among them the fact that its principal left and right singular vectors are mutually orthogonal. We study this class of exceptional matrices $f_0(A)$. In particular, we are interested in the extent to which they are characterized by the aforementioned orthogonality property.