Abstract
Based on Ricatti equation for two (positive invertible) operators A and B which has the geometric mean as its solution, we consider a cubic equation for A, B and C. The solution X = is a candidate of the geometric mean of the three operators. However, this solution is 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 by induction. For three positive operators, in particular, we define the weighted geometric mean as an extension of that of two operators.
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