Weighted geometric means of positive operators

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