Operator Connections and Borel Measures on the Unit Interval

TL;DR

This paper establishes a one-to-one correspondence between operator connections and finite Borel measures on the unit interval, providing integral representations and exploring decompositions of these mathematical objects.

Contribution

It introduces a novel integral representation linking connections and Borel measures, and analyzes their decompositions, advancing the understanding of operator means.

Findings

Connections correspond uniquely to finite Borel measures.

Every mean is an average of weighted harmonic means.

Decomposition results for connections and means are provided.

Abstract

A connection is a binary operation for positive operators satisfying the monotonicity, the transformer inequality and the joint-continuity from above. A mean is a normalized connection. In this paper, we show that there is a one-to-one correspondence between connections and finite Borel measures on the unit interval via a suitable integral representation. Every mean can be regarded as an average of weighted harmonic means. Moreover, we investigate decompositions of connections, means, symmetric connections and symmetric means.