Spectral functions on Jordan algebras: differentiability and convexity properties

A spectral function on a formally real Jordan algebra is a real-valued function which depends only on the eigenvalues of its argument. One convenient way to create them is to start from a function f : Rexp.r [arrow] R which is symmetric in the components of its argument, and to define the function F(u) := f([delta](u)) where [delta](u) is the vector of eigenvalues of u. In this paper, we show that this construction preserves a number of properties which are frequently used in the framework of convex optimization: differentiability, convexity properties and Lipschitz continuity of the gradient for the Euclidean norm with the same constant as for f.