Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Geometric Means of Positive Operators

  • Received : 2008.01.30
  • Accepted : 2008.05.16
  • Published : 2009.03.31
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Abstract

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.

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References

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