Geometric commutation principle for weakly spectral sets in Euclidean Jordan algebras

TL;DR

This paper extends a geometric commutation principle from spectral sets to weakly spectral sets in Euclidean Jordan algebras, with implications for optimization and operator commutativity.

Contribution

It introduces an analog of the commutation principle for weakly spectral sets and explores its consequences and applications.

Findings

Established a geometric commutation principle for weakly spectral sets.

Demonstrated applications in optimization problems.

Connected weakly spectral sets with operator commutativity.

Abstract

A geometric commutation principle in Euclidean Jordan algebra, recently proved by Gowda, says that, for any spectral set $E$ in a Euclidean Jordan algebra $V$ and $a \in E$, $a$ strongly operator commutes with every element in the normal cone $N_E(a)$. Further, it can be used to establish strong operator commutativity relations in certain optimization problems. Knowing that every spectral sets are special cases of broader class of weakly spectral sets, we prove an analog of a geometric commutation principle for weakly spectral sets and study its consequences and applications.