Integral Representations and Decompositions of Operator Monotone Functions on the Nonnegative Reals

TL;DR

This paper establishes a one-to-one correspondence between operator monotone functions on nonnegative reals and finite Borel measures on [0,1], providing integral representations and decompositions for these functions.

Contribution

It introduces a novel integral representation linking operator monotone functions to measures, enabling their decomposition and analysis.

Findings

Established a bijection between operator monotone functions and measures on [0,1]

Provided integral representations for operator monotone functions

Enabled decomposition of operator monotone functions into basic components

Abstract

In this paper, we show that there is a one-to-one correspondence between operator monotone functions on the nonnegative reals and finite Borel measures on the unit interval. This correspondence appears as an integral representation of special operator monotone functions $x \mapsto 1\,!_t\,x$ for $t \in [0,1]$ with respect to a finite Borel measure on $[0,1]$, here $!_t$ denotes the $t$-weighted harmonic mean. Hence such functions form building blocks for arbitrary operator monotone functions on the nonnegative reals. Moreover, we use this integral representation to decompose operator monotone functions.