Spectral sets and functions on Euclidean Jordan algebras

Spectral sets (functions) in Euclidean Jordan algebras are generalizations of permutation invariant sets (respectively, functions) in Rn. In this article, we study properties of such sets and functions and show how they are related to algebra automorphisms and majorization. We show that spectral sets/functions are indeed invariant under automorphisms, but the converse may not hold unless the algebra is Rn or simple. We study Schur-convex spectral functions and provide some applications. We also discuss the transfer principle and a related metaformula.