GEOMETRIC MEANS OF POSITIVE OPERATORS II

Based on Ricatti equation XA 1 X = B for two (positive invertible) operators A and B which has the geometric mean A)B as its solution, we consider a cubic equation X(A)B) 1 X(A)B) 1 X = C for A;B and C: The solution X = (A)B)) 1 C 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 dene reasonable geometric means of k operators for all integers k 2 by induction. For three positive operators, in particular, we dene the weighted geometric mean as an extension of that of two operators.

基于Ricatti方程xa 1 x = b，对于两个（正逆转）操作员A和B，其几何平均值a）b作为解决方案，我们考虑一个立方方程x（a）b）1 x（a）1 x（a）1 x = c的a; b和c：解决方案x =（a）b）1 c是三个算法者的候选者。但是，与两个运算符的几何平均值不同，该解决方案并不是在排列下不变的。为了提供缺乏财产，我们通过Ando-Li-Mathias进行了限制过程。我们通过诱导使所有整数k 2的k运算符的合理几何方法取消了几何方法。特别是对于三个积极的运算符，我们将加权几何平均值作为两个操作员的扩展。