The Normed Ordered Cone of Operator Connections

TL;DR

This paper explores the structure of operator connections, showing they form a normed ordered cone isometrically linked to monotone functions and measures, and characterizes when a connection is a mean.

Contribution

It establishes an isometric order-isomorphism between the cone of operator connections, monotone functions, and Borel measures, and characterizes means as connections with norm 1.

Findings

Operator connections form a normed ordered cone.

The cone of connections is isometrically order-isomorphic to monotone functions.

Convergence of connections, functions, and measures are equivalent.

Abstract

A connection in Kubo-Ando sense is a binary operation for positive operators on a Hilbert space satisfying the monotonicity, the transformer inequality and the continuity from above. A mean is a connection $\sigma$ such that $A \sigma A =A$ for all positive operators $A$. In this paper, we consider the interplay between the cone of connections, the cone of operator monotone functions on $\R^+$ and the cone of finite Borel measures on $[0,\infty]$. %We define a norm for a connection in such a way that the set of operator connections becomes %a normed ordered cone. %On the other hand, the cone of operator monotone functions on $\R^+$ %and the cone of finite Borel measures on $[0,\infty]$ are equipped with suitable norms. The set of operator connections is shown to be isometrically order-isomorphic, as normed ordered cones, to the set of operator monotone functions on $\R^+$. This set is isometrically isomorphic, as normed cones, to the set of finite Borel measures on $[0,\infty]$. It follows that the convergences of the sequence of connections, the sequence of their representing functions and the sequence of their representing measures are equivalent. In addition, we obtain characterizations for a connection to be a mean. In fact, a connection is a mean if and only if it has norm 1.