In the de Branges-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 de Branges-Rovnyak space, 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 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 de Branges-Rovnyak space, 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. 1 2 R.T.W. MARTIN AND A. RAMANANTOANINARecall that the shift, S : H 2 (D) → H 2 (D) is the canonical isometry of multiplication by z and its adjoint, the backward shift, S * , acts as the difference quotient:The shift plays a central role in the classical theory of Hardy spaces [29,42,25]. If T is any CNC contraction, there is a (essentially unique) contractive, operator-valued analytic function, bT , on the unit disk, bT (z) ∈ L (H, K) (i.e. a member of the operatorvalued Schur class), so that T is unitarily equivalent to X where X * := S * | H (b T ) is the restriction of the backward shift of the vector-valued Hardy space H 2 (D) ⊗ K to the de Branges-Rovnyak space H (bT ) [7,11]. (Any de Branges-Rovnyak space H (b) associated to a contractive, operator-valued analytic function, b, on the disk is always contractively contained in vector-valued Hardy space and is always co-invariant for the shift [48].) This provides a natural model for CNC contractions as adjoints of restrictions of the backward shift to de Branges-Rovnyak reproducing kernel Hilbert spaces, and this is the model we extend to several variables in this paper. A canonical several-variable extension of the Hardy space of the disk is the Drury-Arveson space, H 2 d , the unique RKHS of analytic functions on the open unit ball B d := (C d )1 corresponding to the several-variable Szegö kernel. (If Y is a Banach space, let (Y )1 denote the open unit ball and let [Y ]1 denote the closed unit ball.) The Schur classes of contractive, operator-valued functions on the disk are promoted to the multi-variable Schur classes, S d (J, K), of contractive, operator-valued multipliers between vector-valued Drury-Arveson spaces H 2 d ⊗ J and H 2 d ⊗ K (see Subsection 2.1), and the appropriate analogue of the adjoint o...

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