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Operator means of probability measures and generalized Karcher equations. (English) Zbl 1347.47013

Let \(\mathbb P\) denote the cone of positive definite matrices of order \(n\) with complex entries. The weighted multivariable geometric mean or Karcher mean of the \(k\)-tuple \(\mathbb A:=(A_1,A_2,\dots,A_k)\in\mathbb P^k\) relative to the positive probability vector \(\omega:=(w_1,w_2,\dots,w_k)\) is defined as the center of mass \(\Lambda(w_1,w_2,\dots,w_k;A_1,A_2,\dots, A_k):=\arg\min_X\sum_{i=1}^kw_id^2(X,A_i)\), where \(X\) runs over \(\mathbb P\). Here \(d\) denotes the Riemannian distance on the manifold \(\mathbb P\). It is known that the Karcher mean is the unique positive definite solution of the corresponding Karcher equation: \(\sum_{i=1}^kw_i \log(X^{-1}A_i)=0\).
Among other things, the author extends the theory of the Karcher mean by taking any operator monotone function instead of the logarithmic function in the Karcher equation. Another generalization that is considered is to replace the sums by integrals with respect to probability measures supported on \(\mathbb P\). In order to prove his results, the author, among other things, deduces a generalized form of the Karcher mean via strictly geodesically convex divergence functions and obtains certain new integral formulae. The framework for these results is based on contraction results relative to the Thompson metric, which yield nonlinear contraction semigroups in the cone of positive definite operators.

MSC:

47A64 Operator means involving linear operators, shorted linear operators, etc.
46L05 General theory of \(C^*\)-algebras
53C20 Global Riemannian geometry, including pinching
53C35 Differential geometry of symmetric spaces
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