Geometric means on symmetric cones

Abstract.

Let $\Omega$ be the interior of the cone of all square elements of a Euclidean Jordan algebra V, and let $x\rightarrow P(x)$ be the quadratic representation of V. Then P transforms injectively $\Omega$ into the cone of positive definite operators on V, which make $\Omega$ invariant.¶Given $a,b\in \Omega$ there is a natural definition of the geometric mean $P(a)\#P(b)$ of the positive definite operators P(a) and P(b). But it is not clear whether there is $x\in \Omega$ such that $P(x)=P(a)\#P(b)$. In this paper we prove its existence, denoted by $a\#b$ and called the geometric mean of a and b. The arithmetic-geometric-harmonic means inequalities are established. Also $a\#b$ is characterized as the midpoint of the minimal geodesic passing a and b with respect to a natural Riemannian metric on $\Omega$.