Geometric means of positive operators

Based on Ricatti equation XA^-1X = B for two (positive invertible) operators A and B which has the geometric mean A#B as its solution, we consider a cubic equation for A, B and C. The solution X = (A#B)#1/3 C is a candidate of the geometric mean of the three operators. However, this solution i3s not invariant under permutation unlike the geometric mean of two operators. To supply the lack of the property, we adopt a limiting process due to Ando-Li-Mathias. We define reasonable geometric means of k operators for all integers k ≥ 2 by induction. For three positive operators, in particular, we define the weighted geometric mean as an extension of that of two operators.