Free holomorphic functions and operator model theory

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