Weighted geometric means of positive operators

A weighted version of the geometric mean of k (≥̧ 3) positive invertible operators is given. For operators A₁,... , 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 A₁^α A^α_k^k if A₁,... , 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.