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

2023/02/12 by Gabriel Larotonda, Larotonda, Gabriel, Iván Rey +1 · 1 citation
Mathematics · Medicine · #Advanced Topics in Algebra #Ophthalmology and Eye Disorders #Advanced Operator Algebra Research

paper · pdf · doi:10.48550/arxiv.2302.05957

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 \mathfraku_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↦ V+[X,V] of \mathfraku_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.

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• Lie groups with a bi-invariant distance

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