Positivity, Betweenness and Strictness of Operator Means

TL;DR

This paper explores the properties of operator means, providing new characterizations of positivity, betweenness, and strictness using operator monotone functions, measures, and operator equations.

Contribution

It introduces novel characterizations of key properties of operator means through various mathematical frameworks, enhancing understanding of their structure.

Findings

Characterizations of positivity, betweenness, and strictness in operator means.

Connections between operator means, monotone functions, and measures.

Operator equations describing properties of operator means.

Abstract

An operator mean is a binary operation assigned to each pair of positive operators satisfying monotonicity, continuity from above, the transformer inequality and the fixed-point property. It is well known that there are one-to-one correspondences between operator means, operator monotone functions and Borel measures. In this paper, we provide various characterizations for the concepts of positivity, betweenness and strictness of operator means in terms of operator monotone functions, Borel measures and certain operator equations.