Weakly invariant norms: geometry of spheres in the space of skew-Hermitian matrices

TL;DR

This paper explores the geometric structure of spheres in the space of skew-Hermitian matrices under weakly unitarily invariant norms, linking eigenvalue majorization to geometric and functional properties.

Contribution

It provides a detailed characterization of norming functionals, criteria for commutator support hyperplanes, and analyzes the action of the Lie algebra on the sphere, revealing conditions for norm preservation.

Findings

Eigenvalue majorization encodes geometric properties of the sphere.

Characterization of norming functionals for weakly invariant norms.

Conditions under which the adjoint action preserves the norm.

Abstract

Let $N$ be a weakly unitarily invariant norm (i.e. invariant for the coadjoint action of the unitary group) in the space of skew-Hermitian matrices $\mathfrak{u}_n(\mathbb C)$. In this paper we study the geometry of the unit sphere of such a norm, and we show how its geometric properties are encoded by the majorization properties of the eigenvalues of the matrices. We give a detailed characterization of norming functionals of elements for a given norm, and we then prove a sharp criterion for the commutator $[X,[X,V]]$ to be in the hyperplane that supports $V$ in the unit sphere. We show that the adjoint action $V\mapsto V+[X,V]$ of $\mathfrak{u}_n(\mathbb C)$ on itself pushes vectors away from the unit sphere. As an application of the previous results, for a strictly convex norm, we prove that the norm is preserved by this last action if and only if $X$ commutes with $V$. We give a more detailed description in the case of any weakly $Ad$-invariant norm.