Operator means of probability measures and generalized Karcher equations
Operator means of probability measures and generalized Karcher equations
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概率测量的算子平均值和广义 Karcher 方程
DOI：
10.1016/j.aim.2015.11.019
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发表时间：
2016
影响因子：
1.7
通讯作者：
Miklos Palfia
中科院分区：
文献类型：
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作者：
C. Perfetti;G. Rius;X. Borrise and K. Abe.;Fumiharu Kobayashi;畠中省伍・高島義徳・橋爪章人・原田明;畠中省伍・高島義徳・原田明;Fumiharu Kobayashi;Miklos Palfia
本文研究Hilbert空间上正有界线性算子的均值。我们提出了一个完整的理论，它提供了一个框架，将Karcher均值理论，它的近似矩阵幂均值理论，以及Kubo-Ando理论的大部分扩展到任意多变量，实际上，扩展到正定算子锥上有界支持的概率测度的情况。该框架将每个算子均值外在表征为广义Karcher方程的唯一解，该解通过将Karcher方程中的矩阵对数函数交换为正实半线上的任意算子单调函数而得到。如果潜在的希尔伯特空间是有限维的，那么这些广义Karcher方程是严格测地凸对数行列式散度函数的凸组合的黎曼梯度，因此这些新均值是它们的全局最小值，类似于前面指出的Karcher均值的情况。我们的框架是基于关于Thompson度规的基本收缩结果，它为我们提供了正定算子锥上的非线性收缩半群，这些正定算子在强拓扑中形成一个逼近这些算子均值的递减网。
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 their global minimizers, in analogy 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.