Geometric means of positive operators

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11 Citations (Scopus)

Abstract

Based on Ricatti equation XA-1X = B for two (positive invertible) operators A and B which has the geometric mean A#B as its solution, we consider a cubic equation for A, B and C. The solution X = (A#B)#1/3 C is a candidate of the geometric mean of the three operators. However, this solution i3s 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 ≥ 2 by induction. For three positive operators, in particular, we define the weighted geometric mean as an extension of that of two operators.

Original languageEnglish
Pages (from-to)167-181
Number of pages15
JournalKyungpook Mathematical Journal
Volume49
Issue number1
DOIs
Publication statusPublished - Jul 2009
Externally publishedYes

Keywords

  • Arithmetic-geometric mean inequality
  • Geometric mean
  • Positive operator
  • Reverse inequality

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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