A note on real operator monotone functions

TL;DR

This paper introduces and characterizes real operator monotone functions for tuples of operators, revealing their structure and properties within free function calculus, and establishing conditions for their linearity.

Contribution

It provides a complete characterization of real operator monotone functions on open convex free domains and shows that free holomorphic functions with this property are affine linear.

Findings

Characterization of real operator monotone functions on open convex free domains

Proof that free holomorphic functions with real operator monotonicity are affine linear

Connection between real operator monotonicity and completely positive parts

Abstract

In this paper we initiate the study of real operator monotonicity for functions of tuples of operators, which are multivariate structured maps with a functional calculus called free functions that preserve the order between real parts (or Hermitian parts) of bounded linear Hilbert space operators. We completely characterize such functions on open convex free domains in terms of ordinary operator monotone free functions on self-adjoint domains. Further assuming the more stringent free holomorphicity, we prove that all such functions are affine linear with completely positive nonconstant part. This problem has been proposed by David Blecher at the biannual OTOA conference held in Bangalore in December 2016.