A non-commutative de Branges-Rovnyak model for row contractions
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, , which is the inverse of the monic, affine linear pencil of in a certain NC unit row-ball of the NC universe of all row tuples of square matrices of all finite sizes.
Keywords
Cite
@article{arxiv.2601.05211,
title = {A non-commutative de Branges-Rovnyak model for row contractions},
author = {Robert T. W. Martin and Jeet Sampat},
journal= {arXiv preprint arXiv:2601.05211},
year = {2026}
}