Weighted Geometric Means of Positive Operators

• Journal title : Kyungpook mathematical journal
• Volume 50, Issue 2,  2010, pp.213-228
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2010.50.2.213
Title & Authors
Weighted Geometric Means of Positive Operators
Izumino, Saichi; Nakamura, Noboru;

Abstract
A weighted version of the geometric mean of k ($\small{\geq\;3}$) positive invertible operators is given. For operators $\small{A_1,{\ldots},A_k}$ and for nonnegative numbers $\small{{\alpha}_1,\ldots,{\alpha}_k}$ such that $\small{\sum_\limits_{i=1}^k\;\alpha_i=1}$, 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 $\small{A_1^{\alpha_1}{\cdots}A_k^{{\alpha}_k}}$ if $\small{A_1,{\ldots},A_k}$ 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.
Keywords
positive operator;weighted geometric mean;arithmetic-geometric mean inequality;reverse inequality;
Language
English
Cited by
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