Original Article
Kyungpook Mathematical Journal 2009; 49(1): 167-181
Published online March 23, 2009
Copyright © Kyungpook Mathematical Journal.
Geometric Means of Positive Operators
Noboru Nakamura
Fujikoshi-kogyo Senior Highschool, Higashi-ishigane 7-5, Toyama, 930-0964, Japan
Based on Ricatti equation $XA^{-1}X = B$ for two (positive invertible) operators $A$ and $B$ which has the geometric mean $A sharp B$ as its solution, we consider a cubic equation $$ X(A sharp B)^{-1}X(A sharp B)^{-1}X=C $$ for $A, B$ and $C.$ The solution $X = (A sharp B) sharp_{frac{1}{3}}C$ 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 define reasonable geometric means of $k$ operators for all integers $k geq 2$ by induction. For three positive operators, in particular, we define the weighted geometric mean as an extension of that of two operators.
Keywords: positive operator, geometric mean, arithmetic-geometric mean inequality, reverse inequality