The least squares mean of positive Hilbert–Schmidt operators

We show that, the least squares mean on the Riemannian manifold ΣΣ of positive operators in the extended Hilbert–Schmidt algebra of linear operators on a Hilbert space equipped with the canonical trace metric is the unique solution of the corresponding Karcher equation. This allows us to conclude that, the least squares mean is the restriction of the Karcher mean on the open cone of all bounded positive definite operators, and hence inherits the basic properties of that mean. Conversely, the Karcher mean on the positive definite operators is shown to be the unique monotonically strongly continuous extension of the least squares mean on ΣΣ.