### Abstract

A weighted version of the geometric mean of k (≥̧ 3) positive invertible operators is given. For operators A_{1},... , A_{k} and for nonnegative numbers α1,... , αk such that we define weighted geometric means of two types, the first type by a direct construction through symmetrization procedure, and the second type by an indirect construction through the non-weighted (or uniformly weighted) geometric mean. Both of them reduce to A_{1}^{α} A^{α}_{k}^{k} if A_{1},... , A_{k} commute with each other. The first type does not have the property of permutation invariance, but satisfies a weaker one with respect to permutation invariance. The second type has the property of permutation invariance. We also show a reverse inequality for the arithmetic-geometric mean inequality of the weighted version.

Original language | English |
---|---|

Pages (from-to) | 213-228 |

Number of pages | 16 |

Journal | Kyungpook Mathematical Journal |

Volume | 50 |

Issue number | 2 |

Publication status | Published - Jul 2010 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Kyungpook Mathematical Journal*,

*50*(2), 213-228.

**Weighted geometric means of positive operators.** / Izumino, Saichi; Nakamura, Noboru.

Research output: Contribution to journal › Article

*Kyungpook Mathematical Journal*, vol. 50, no. 2, pp. 213-228.

}

TY - JOUR

T1 - Weighted geometric means of positive operators

AU - Izumino, Saichi

AU - Nakamura, Noboru

PY - 2010/7

Y1 - 2010/7

N2 - A weighted version of the geometric mean of k (≥̧ 3) positive invertible operators is given. For operators A1,... , Ak and for nonnegative numbers α1,... , αk such that we define weighted geometric means of two types, the first type by a direct construction through symmetrization procedure, and the second type by an indirect construction through the non-weighted (or uniformly weighted) geometric mean. Both of them reduce to A1α Aαkk if A1,... , Ak commute with each other. The first type does not have the property of permutation invariance, but satisfies a weaker one with respect to permutation invariance. The second type has the property of permutation invariance. We also show a reverse inequality for the arithmetic-geometric mean inequality of the weighted version.

AB - A weighted version of the geometric mean of k (≥̧ 3) positive invertible operators is given. For operators A1,... , Ak and for nonnegative numbers α1,... , αk such that we define weighted geometric means of two types, the first type by a direct construction through symmetrization procedure, and the second type by an indirect construction through the non-weighted (or uniformly weighted) geometric mean. Both of them reduce to A1α Aαkk if A1,... , Ak commute with each other. The first type does not have the property of permutation invariance, but satisfies a weaker one with respect to permutation invariance. The second type has the property of permutation invariance. We also show a reverse inequality for the arithmetic-geometric mean inequality of the weighted version.

KW - Arithmetic-geometric mean inequality

KW - Positive operator

KW - Reverse inequality

KW - Weighted geometric mean

UR - http://www.scopus.com/inward/record.url?scp=77955976669&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77955976669&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:77955976669

VL - 50

SP - 213

EP - 228

JO - Kyungpook Mathematical Journal

JF - Kyungpook Mathematical Journal

SN - 1225-6951

IS - 2

ER -