Operator monotone functions and L\"owner functions of several variables

TL;DR

This paper extends classical results on operator monotone functions to multiple variables, providing characterizations of such functions and their properties in the context of matrix analysis and complex function theory.

Contribution

It generalizes L"owner's and Nevanlinna's theorems to several variables, including a complete characterization of rational operator monotone functions of two variables.

Findings

Characterization of multivariable matrix monotone functions

Generalization of Nevanlinna's theorem to multiple variables

Complete description of rational operator monotone functions of two variables

Abstract

We prove generalizations of L\"owner's results on matrix monotone functions to several variables. We give a characterization of when a function of $d$ variables is locally monotone on $d$-tuples of commuting self-adjoint $n$-by-$n$ matrices. We prove a generalization to several variables of Nevanlinna's theorem describing analytic functions that map the upper half-plane to itself and satisfy a growth condition. We use this to characterize all rational functions of two variables that are operator monotone.