Commutation principles in Euclidean Jordan algebras and normal decomposition systems

TL;DR

This paper extends the commutation principle for spectral functions in Euclidean Jordan algebras to more general invariant sets and functions, and explores similar principles within normal decomposition systems.

Contribution

It generalizes the commutation principle to invariant sets and functions beyond spectral sets, and applies the concept to normal decomposition systems.

Findings

Extended the commutation principle to invariant sets and functions.

Established similar principles in normal decomposition systems.

Broadened the applicability of spectral function minimization results.

Abstract

The commutation principle of Ramirez, Seeger, and Sossa \cite{ramirez-seeger-sossa} proved in the setting of Euclidean Jordan algebras says that when the sum of a Fr\'{e}chet differentiable function $\Theta(x)$ and a spectral function $F(x)$ is minimized over a spectral set $\Omega$, any local minimizer $a$ operator commutes with the Fr\'{e}chet derivative $\Theta^{\prime}(a)$. In this paper, we extend this result to sets and functions which are (just) invariant under algebra automorphisms. We also consider a similar principle in the setting of normal decomposition systems.