# Geometric means of positive operators

Research output: Contribution to journalArticle

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 language English 167-181 15 Kyungpook Mathematical Journal 49 1 Published - Jul 2009 Yes

### Fingerprint

Geometric mean
Positive Operator
Operator
Cubic equation
Invertible
Proof by induction
Permutation
Limiting
Integer
Invariant

### Keywords

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

### ASJC Scopus subject areas

• Mathematics(all)
• Applied Mathematics

### Cite this

In: Kyungpook Mathematical Journal, Vol. 49, No. 1, 07.2009, p. 167-181.

Research output: Contribution to journalArticle

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