[Paper Review] Operator Monotone Functions: Characterizations and Integral Representations

Pattrawut Chansangiam|arXiv (Cornell University)|May 11, 2013

Mathematical Inequalities and Applications10 references3 citations

TL;DR

This paper provides a comprehensive survey of operator monotone functions, establishing their characterizations via matrix of divided differences and Hansen-Pedersen conditions. The key contribution is a complete integral representation for operator monotone functions on $[0,ty)$ using Borel measures, showing they are convex combinations of elementary functions $t \mapsto \frac{t(1+\xi)}{t+\xi}$, with uniqueness of the representing measure.

ABSTRACT

Operator monotone functions, introduced by Lowner in 1934, are an important class of real-valued functions. They arise naturally in matrix and operator theory and have various applications in other branches of mathematics and related fields. This concept is closely related to operator convex/concave functions. In this paper, we provide their important examples and characterizations in terms of matrix of divided differences. Various characterizations and the relationship between operator monotonicity and operator convexity are given by Hansen-Pedersen characterizations. Moreover, operator monotone functions on the nonnegative reals have special properties, namely, they admit integral representations with respect to suitable Borel measures.

Motivation & Objective

• To provide a unified survey of operator monotone functions and their role in matrix and operator theory.
• To establish characterizations of operator monotonicity using the matrix of divided differences.
• To clarify the relationship between operator monotonicity and operator convexity via Hansen-Pedersen conditions.
• To derive and prove the integral representation of operator monotone functions on $[0,\infty)$ using Borel measures.
• To demonstrate the uniqueness and construction of the representing measure for such functions.

Proposed method

• Characterize operator monotone functions using the positivity of the matrix of divided differences for all $n \in \mathbb{N}$.
• Apply the Hansen-Pedersen characterization to link operator monotonicity and operator convexity through functional calculus and spectral resolution.
• Transform functions on $[0,\infty)$ to the interval $(-1,1)$ via the Cayley transform $t = \frac{1+x}{1-x}$ to leverage known integral representations.
• Use the Krein-Milman theorem and weak-* compactness of probability measures on $[-1,1]$ to represent functions as integrals over extreme points $\phi_\lambda(x) = \frac{x}{1 - \lambda x}$.
• Apply the Riesz-Markov-Kakutani representation theorem to ensure the uniqueness of the representing measure via moment equality on polynomials.
• Transform the resulting integral back to the original domain $[0,\infty)$, yielding the representation $f(t) = \int_{[0,\infty]} \frac{t(1+\xi)}{t+\xi} \, dm(\xi)$.

Experimental results

Research questions

• RQ1 How can operator monotone functions be characterized using the matrix of divided differences for all $n$?
• RQ2 What is the precise relationship between operator monotonicity and operator convexity as captured by the Hansen-Pedersen conditions?
• RQ3 Can every operator monotone function on $[0,\infty)$ be represented as an integral of elementary operator monotone functions?
• RQ4 What is the uniqueness condition for the representing Borel measure in such integral representations?
• RQ5 How does the transformation via the Cayley map facilitate the derivation of the integral representation?

Key findings

• Every operator monotone function on $[0,\infty)$ admits a unique integral representation of the form $f(t) = \int_{[0,\infty]} \frac{t(1+\xi)}{t+\xi} \, dm(\xi)$ for some Borel probability measure $m$ on $[0,\infty]$.
• The representing measure $m$ is unique, as established by the equality of all moments $\int \xi^k \, dm(\xi)$ for $k \in \mathbb{N}_0$ and the Stone-Weierstrass theorem.
• The elementary functions $t \mapsto \frac{t(1+\xi)}{t+\xi}$ serve as the building blocks for all operator monotone functions on $[0,\infty)$, forming the extreme points of the convex set of such functions.
• The function $t \mapsto t^p$ for $p \in (0,1)$ has a known integral representation with density $\frac{\sin p\pi}{\pi} \cdot \frac{\xi^{p-1}}{1+\xi}$, confirming consistency with the general framework.
• The transformation $x = \frac{1-t}{1+t}$ maps $[0,\infty)$ to $(-1,1)$, enabling the use of known results on Pick functions and their integral representations.
• The proof relies on weak-* convergence of measures and the Riesz representation theorem to ensure that pointwise convergence of functions implies convergence of their representing measures.