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 \(\Omega\).