Operator means of probability measures and generalized Karcher equations

Abstract

Abstract. In this article we consider means of positive bounded linear operators on a Hilbert space. We present a complete theory that provides a framework which extends the theory of the Karcher mean, its approximating matrix power means, and a large part of Kubo-Ando theory to arbitrary many variables, in fact, to the case of probability measures with bounded support on the cone of positive definite operators. This framework characterizes each operator mean extrinsically as unique solutions of generalized Karcher equations which are obtained by exchanging the matrix logarithm function in the Karcher equation to arbitrary operator monotone functions over the positive real half-line. If the underlying Hilbert space is finite dimensional, then these generalized Karcher equations are Riemannian gradients of convex combinations of strictly geodesically convex log-determinant divergence functions, hence these new means are the global minimizers of them, in analogue to the case of the Karcher mean as pointed out. Our framework is based on fundamental contraction results with respect to the Thompson metric, which provides us nonlinear contraction semigroups in the cone of positive definite operators that form a decreasing net approximating these operator means in the strong topology from above.

Operator means of probability measures

“…for given µ ∈ P ∞ (P), which is the original case of the generalized Karcher equation in [40]. So the properties (i)-(viii) for G, except (iii) and (vii), are known in [40]. Below we will prove (iii) and (vii) for G based on the convergence P r → G as r → 0.…”

“…If we write Aτ σ B for the solution X 0 , then τ σ becomes a two-variable operator mean again and the representing function of τ σ is exactly determined by those of τ and σ, see [17]. The restriction to two-variable operator means was also discussed in [40,49] for the generalized Karcher equation mentioned in Remark 3.2.…”

“…But it follows from Theorem 2.8 that the power means are d W ∞ -contractive on P ∞ (P) bigger than P cp (P). for given µ ∈ P ∞ (P), which is the original case of the generalized Karcher equation in [40]. So the properties (i)-(viii) for G, except (iii) and (vii), are known in [40].…”

“…For instance, in view of definition of the power means P r in (5.3), note that P r for r ∈ (0, 1] satisfies (6.6) without the normality of Φ. But it is unknown to us whether this is also the case for P r for r ∈ [−1, 0), in particular, for H. In a different approach in [40,Theorem 6.4] it was shown that (6.6) holds for a certain wide class of operator means on P ∞ (P) under unital positive linear maps Φ, but the normality of Φ seems necessary there. 6.3.…”

Existence and uniqueness of the Karcher mean on unital C⁎-algebras

“…H. Karcher [3] gave an equational characterization of this mean on Riemannian manifolds, and in [7] Y. Lim and the author showed that this equational characterization could be extended to the cone of positive invertible operators in the C * -algebra of bounded linear maps on a Hilbert space. Later M. Palfia studied more general classes of means and barycenters and more general versions of the Karcher equation [11].…”