Convexity and differentiability properties of spectral functions and spectral mappings on Euclidean Jordan algebras

We study in this paper several properties of the eigenvalues function of a Euclidean Jordan algebra, extending several known results in the framework of symmetric matrices. In particular, we give a concise form for the directional differential of a single eigenvalue. We especially focus on spectral functions F on Euclidean Jordan algebras, which are the composition of a symmetric real-valued function f with the eigenvalues function. We explore several properties of f that are transferred to F, in particular convexity, strong convexity and differentiability. Spectral mappings are also considered, a special case of which is the gradient mapping of a spectral function. Answering a problem proposed by H. Sendov, we give a formula for the Jacobian of these functions.