Operator monotone functions of several variables

TL;DR

This paper introduces a new concept of operator monotonicity for multivariable functions, extending the classical one-variable case and preserving key relationships with operator convexity.

Contribution

It defines a multivariable operator monotonicity notion that generalizes the classical concept and maintains the link with operator convexity.

Findings

Established a new framework for multivariable operator monotonicity

Extended the relationship between operator convexity and monotonicity to multiple variables

Provided foundational definitions for future research in multivariable operator functions

Abstract

We propose a notion of operator monotonicity for functions of several variables, which extends the well known notion of operator monotonicity for functions of only one variable. The notion is chosen such that a fundamental relationship between operator convexity and operator monotonicity for functions of one variable is extended also to functions of several variables.