A non-commutative de Branges-Rovnyak model for row contractions

TL;DR

This paper generalizes the de Branges-Rovnyak model to non-commutative multivariate settings, providing a unitary equivalence for CNC row contractions using NC Hardy spaces and characteristic functions.

Contribution

It introduces a non-commutative model for CNC row contractions, extending classical theory to multivariate NC operator settings with a complete unitary invariant.

Findings

CNC row contractions are unitarily equivalent to adjoints of restricted backward shifts.

The characteristic function serves as a complete unitary invariant.

Constructs a model reproducing kernel Hilbert space of NC functions.

Abstract

We extend the de Branges-Rovnyak model for completely non-coisometric (CNC) linear contractions on a Hilbert space to the non-commutative multivariate setting of CNC row contractions. Namely, we show that any CNC contraction from several copies of a Hilbert space into a single copy is unitarily equivalent to the adjoint of the restricted backward right shifts acting on the de Branges-Rovnyak space of a contractive left multiplier between vector-valued "free Hardy spaces" of square-summable power series in several non-commuting (NC) variables. This contractive, operator-valued left multiplier, the characteristic function of the CNC row contraction, is a complete unitary invariant and it is always column-extreme as a contractive left multiplier. Our construction builds a model reproducing kernel Hilbert space of NC functions using a "non-commutative resolvent" of the row contraction, $T$, which is the inverse of the monic, affine linear pencil of $T$ in a certain NC unit row-ball of the NC universe of all row tuples of square matrices of all finite sizes.