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

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 un(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↦V+[X,V] of un(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.