### Abstract

Based on Ricatti equation XA^{-1}X = 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 |
---|---|

Pages (from-to) | 167-181 |

Number of pages | 15 |

Journal | Kyungpook Mathematical Journal |

Volume | 49 |

Issue number | 1 |

Publication status | Published - Jul 2009 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Kyungpook Mathematical Journal*,

*49*(1), 167-181.

**Geometric means of positive operators.** / Nakamura, Noboru.

Research output: Contribution to journal › Article

*Kyungpook Mathematical Journal*, vol. 49, no. 1, pp. 167-181.

}

TY - JOUR

T1 - Geometric means of positive operators

AU - Nakamura, Noboru

PY - 2009/7

Y1 - 2009/7

N2 - 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.

AB - 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.

KW - Arithmetic-geometric mean inequality

KW - Geometric mean

KW - Positive operator

KW - Reverse inequality

UR - http://www.scopus.com/inward/record.url?scp=72949092768&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=72949092768&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:72949092768

VL - 49

SP - 167

EP - 181

JO - Kyungpook Mathematical Journal

JF - Kyungpook Mathematical Journal

SN - 1225-6951

IS - 1

ER -