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;
A weighted version of the geometric mean of k () positive invertible operators is given. For operators and for nonnegative numbers 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 if 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.
positive operator;weighted geometric mean;arithmetic-geometric mean inequality;reverse inequality;
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