Commutativity, majorization, and reduction in Fan-Theobald-von Neumann systems

TL;DR

This paper explores the structure of Fan-Theobald-von Neumann systems, focusing on commutativity, automorphisms, majorization, and reduction, to deepen understanding of their algebraic and geometric properties.

Contribution

It introduces and studies automorphisms, majorization, and reduction within Fan-Theobald-von Neumann systems, expanding the theoretical framework established previously.

Findings

Characterization of automorphisms in Fan-Theobald-von Neumann systems

Development of majorization theory for these systems

Reduction techniques for simplifying system analysis

Abstract

A Fan-Theobald-von Neumann system is a triple $(V,W,\lambda)$, where $V$ and $W$ are real inner product spaces and $\lambda:V \to W$ is a norm-preserving map satisfying a Fan-Theobald-von Neumann type inequality together with a condition for equality. Examples include Euclidean Jordan algebras, systems induced by certain hyperbolic polynomials, and normal decompositions systems (Eaton triples). In the previous paper (arXiv:1902.06640) we presented some basic properties of such systems and described results on optimization problems dealing with certain combinations of linear/distance and spectral functions. We also introduced the concept of commutativity via the equality in the Fan-Theobald-von Neumann type inequality. In the present paper, we elaborate on the concept of commutativity and introduce/study automorphisms, majorization, and reduction in Fan-Theobald-von Neumann systems.