A non-commutative de Branges-Rovnyak model for row contractions
Robert T. W. Martin, Jeet Sampat
TL;DR
The paper develops a non-commutative, multivariable extension of the de Branges–Rovnyak model for completely non-coisometric contractions by embedding CNC row contractions into NC de Branges–Rovnyak spaces built from contractive left multipliers between vector-valued free Hardy spaces. It introduces graded NC model maps, the NC characteristic function $B_T$, and shows that every CNC row contraction is unitarily equivalent to the extremal Gleason solution on $\\mathscr{H}(B_T)$, with $B_T$ CE and unique up to weak coincidence; a Crofoot-type transform relates $B_T$ to the partial-isometric part’s characteristic function $B_V$. The framework also connects to Frostman shifts, and provides a clear link to commutative multivariable models via CCNC theory, yielding a complete invariant theory for NC row contractions and a practical model-construction method via NC resolvents and graded kernels. This work unifies NC RKHS methods with transfer-function realizations, enabling explicit characterizations and unitary classifications of CNC row contractions in the NC, multivariate setting, with potential implications for NC dilation theory and operator-valued function theory.
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.