# Operator means of probability measures

## TL;DR

This paper introduces a new framework for operator means of probability measures on positive invertible operators, establishing their contractive properties and deformation stability, with implications for inequalities in infinite-dimensional Hilbert spaces.

## Contribution

It extends the concept of operator means to probability measures, demonstrating contractivity and deformation invariance, and deriving new properties and inequalities.

## Key findings

- Operator means are contractive under the $
abla$-Wasserstein distance.
- Deformations of operator means remain within the class of operator means.
- New inequalities for operator means of probability measures are established.

## Abstract

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.