Gowda, Muddappa SJeong, Juyoung2019-10-112019-10-112017-01-0111649http://hdl.handle.net/11603/15685This 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.This item may be protected under Title 17 of the U.S. Copyright Law. It is made available by UMBC for non-commercial research and education. For permission to publish or reproduce, please see http://aok.lib.umbc.edu/specoll/repro.php or contact Special Collections at speccoll(at)umbc.eduAlgebra automorphismEuclidean Jordan algebraMajorizationOptimizationSpectral functionSpectral setText