Commutation Principles in Euclidean Jordan Algebras and Normal Decomposition Systems

Abstract

The commutation principle of Ramírez, Seeger, and Sossa [SIAM J. Optim., 23 (2013), pp. 687--694] proved in the setting of Euclidean Jordan algebras says that when the sum of a Fréchet differentiable function $\Theta$ and a spectral function $F$ is minimized (maximized) over a spectral set $\Omega$, any local minimizer (respectively, maximizer) $a$ operator commutes with the Fréchet derivative $\Theta'(a)$. 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.