Hilbert's metric on symmetric cones

Abstract

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