Weighted geometric means of positive operators

Saichi Izumino, Noboru Nakamura

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)213-228
Number of pages16
JournalKyungpook Mathematical Journal
Issue number2
Publication statusPublished - Jul 2010
Externally publishedYes


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

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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