Let $\mathbb{P}$ be the complete metric space consisting of positive invertible operators on an infinite-dimensional Hilbert space with the Thompson metric. We introduce the notion of operator means of probability measures on $\mathbb{P}$, in parallel with Kubo and Ando's definition of two-variable operator means, and show that every operator mean is contractive for the $\infty$-Wasserstein distance. By means of a fixed point method we consider deformation of such operator means, and show that the deformation of any operator mean becomes again an operator mean in our sense. Based on this deformation procedure we prove a number of properties and inequalities for operator means of probability measures.

中文翻译：

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概率度量的算子手段

令 $\mathbb{P}$ 是由具有 Thompson 度量的无限维希尔伯特空间上的正可逆算子组成的完整度量空间。我们在 $\mathbb{P}$ 上引入了概率测度的算子均值的概念，与 Kubo 和 Ando 对二变量算子均值的定义并行，并表明每个算子均值对于 $\infty$-Wasserstein 距离是收缩的. 通过不动点方法，我们考虑这种算子均值的变形，并表明任何算子均值的变形再次成为我们意义上的算子均值。基于这个变形过程，我们证明了概率度量的算子手段的许多性质和不等式。