Title: Spectral sets and functions on Euclidean Jordan algebras

Abstract
This thesis studies spectral and weakly spectral sets/functions on Euclidean Jordan algebras. These are generalizations of similar well-known concepts on  the algebras of  real symmetric and complex Hermitian matrices. A spectral set in a Euclidean Jordan algebra V is the inverse image of a permutation invariant set in R^n under the eigenvalue map (which takes an element x in V to its eigenvalue vector in R^n consisting of eigenvalues of x written in the decreasing order). A spectral function on V is the composition of a permutation invariant function on R^n and the eigenvalue map.

In this thesis, we study properties of such sets/functions and show how they are related to algebra automorphisms and majorization. We show they are indeed invariant under algebra automorphisms of V, hence weakly spectral with converse holding when V is essentially simple.

For a spectral set K, we discuss the transfer principle and a related metaformula.  When K is also a cone, we show that the dual of K is a spectral cone under certain conditions. We also discuss the dimension of K, and characterize the pointedness/solidness of K.  Specializing, we study permutation invariant (proper) polyhedral cones in R^n. We show that the Lyapunov rank of such a cone divides n.

Lastly, we study Schur-convexity of a spectral function and describe some applications.