Characteristic functions and functional models on noncommutative varieties

Abstract

In this paper we develop a Sz.-Nagy-Foiaş type theory of characteristic functions and functional models on noncommutative domains and varieties in \(B(\mathcal{H})^n\), where \(B(\mathcal{H})\) is the algebra of all bounded linear operators on a Hilbert space \(\mathcal{H}\), associated with admissible free holomorphic functions for operator model theory. This includes, in particular, a large class of commutative domains in \(B(\mathcal{H})^n\). In the commutative case, the characteristic functions are multipliers of reproducing kernel Hilbert spaces on domains in \(\mathbb{C}^n\). We also show that the universal operator model associated with any admissible noncommutative variety has the quasi-wandering property.