Hilbert's metric on symmetric cones

Abstract

Let $Ω$ be a symmetric cone. In this note, we introduce the Hilbert projective metric on $Ω$ in terms of Jordan algebras and we apply it to prove that given a linear transformation $g$ such that $g(Ω)\subset Ω$ and a real number $p$, $|p|>1$, then there exists a unique element $x\inΩ$ satisfying $g(x)=x^p$.