Geometric Means of Positive Operators

• Journal title : Kyungpook mathematical journal
• Volume 49, Issue 1,  2009, pp.167-181
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2009.49.1.167
Title & Authors
Geometric Means of Positive Operators
Nakamura, Noboru;

Abstract
Based on Ricatti equation $\small{XA^{-1}X=B}$ for two (positive invertible) operators A and B which has the geometric mean $\small{A{\sharp}B}$ as its solution, we consider a cubic equation $\small{X(A{\sharp}B)^{-1}X(A{\sharp}B)^{-1}X=C}$ for A, B and C. The solution X = $\small{(A{\sharp}B){\sharp}_{\frac{1}{3}}C}$ 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 $\small{k{\geq}2}$ 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;
Language
English
Cited by
1.
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