Free holomorphic functions and operator model theory

TL;DR

This paper explores classes of noncommutative multivariable functions to develop operator model and dilation theories beyond the unit ball, extending free analogue of Nagy-Foias theory to new domains.

Contribution

It introduces a framework for developing operator models and dilation theories for a broad class of noncommutative domains defined by free holomorphic functions.

Findings

Identifies large classes of noncommutative functions suitable for model theory

Extends free Nagy-Foias theory to new noncommutative domains

Lays groundwork for future operator theory in noncommutative multivariable analysis

Abstract

This paper is an attempt to find large classes of noncommutative multivariable functions $g:\Omega\subset [B(\cH)^n]_1^-\to B(\cH)^n$ for which a reasonable operator model theory and dilation theory can be developed for the noncommutative domain $g(\Omega)$. In other words, we want to transfer the free analogue of Nagy-Foias theory from the unit ball $[B(\cH)^n]_1$ to other noncommutative domains, using appropriate maps.