Spectral sets and functions on Euclidean Jordan algebras

Abstract Spectral sets (functions) in Euclidean Jordan algebras are generalizations of permutation invariant sets (respectively, functions) in R n . 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 R n or simple. We study Schur-convex spectral functions and provide some applications. We also discuss the transfer principle and a related metaformula.