A Gleason solution model for row contractions

TL;DR

This paper extends the deBranges-Rovnyak functional model to a broad class of completely non-coisometric row contractions in several variables, characterizing their unitary equivalence via a multivariable characteristic function.

Contribution

It generalizes the classical one-variable model to multivariable settings, including all commuting CNC row contractions, and characterizes their unitary invariants.

Findings

Characterization of CNC row contractions as Gleason solutions in multivariable spaces

Identification of the characteristic function as a complete unitary invariant in a subclass

Extension of the model to non-commuting and commuting row contractions

Abstract

In the deBranges-Rovnyak functional model for contractions on Hilbert space, any completely non-coisometric (CNC) contraction is represented as the adjoint of the restriction of the backward shift to a deBranges-Rovnyak space, $\mathscr{H} (b)$, associated to a contractive analytic operator-valued function, $b$, on the open unit disk. We extend this model to a large class of CNC row contractions of several copies of a Hilbert space into itself (including all CNC row contractions with commuting component operators). Namely, we completely characterize the set of all CNC row contractions, $T$, which are unitarily equivalent to an extremal Gleason solution for a deBranges-Rovnyak space, $\mathscr{H} (b_T)$, contractively contained in a vector-valued Drury-Arveson space of analytic functions on the open unit ball in several complex dimensions. Here, a Gleason solution is the appropriate several-variable analogue of the adjoint of the restricted backward shift and the characteristic function, $b_T$, belongs to the several-variable Schur class of contractive multipliers between vector-valued Drury-Arveson spaces. The characteristic function, $b_T$, is a unitary invariant, and we further characterize a natural sub-class of CNC row contractions for which it is a complete unitary invariant.