Convexity and differentiability properties of spectral functions and spectral mappings on Euclidean Jordan algebras (Q875019)

The author of this paper discusses properties of the so called eigenvalue function of an Euclidean Jordan algebra. Let \({\mathcal J}\) be an Euclidean Jordan algebra of rank \(r\) and let \(f:{\mathbb R}^r\to {\mathbb R}\cup \{+\infty\}\) be a function invariant with respect to component permutations of its arguments. Define as \(F:{\mathcal J} \to {\mathbb R} \cup \{+\infty\}\) the spectral function generated by \(f\), \(F(u)=f(\lambda(u))\). The main purpose of the paper is then to study properties that \(f\) transfers to \(F\). The author focuses on convexity and differentiability properties, extending to the framework of Jordan algebras various well-known facts in the case of symmetric matrices. Two remarkable features of the paper can be highlighted. The author obtains a concise formula for the directional derivative of a single eigenvalue (Section 5). Spectral mappings are also considered, in particular he gives a formula (Section 6) for the Jacobian of a spectral function.