Several authors, e.g. T. Ando [Topics on operator inequalities, Hokkaido Univ., Sapporo (1978; Zbl 0388.47024)], M. Fiedler and V. Ptak [Linear Algebra Appl. 251, 1-20 (1997; Zbl 0872.15014)], have considered the concept of geometric mean \(A\# B\) of positive (semi-) definite linear operators \(A\) and \(B\).
If \(\Omega\) denotes the cone of all square elements of a Euclidean Jordan algebra then the quadratic representation \(x\mapsto P(x)\) transforms \(\Omega\) injectively into the cone of positive definite linear operators of \(V\) which leave \(\Omega\) invariant. The author introduces a geometric mean \(a\# b\) for any two \(a, b\in\Omega\). On one hand this geometric mean can be characterized in an intrinsic way as midpoint of the minimal geodesic passing through \(a\) and \(b\) with respect to a natural Riemannian metric on \(\Omega\). On the other hand it is shown that \(P(a\# b)=P(a)\# P(b)\) so that the geometric mean on \(\Omega\) allows an immediate transfer to the geometric mean of positive definite linear operators.