Schur-class multipliers on the Arveson space: De Branges–Rovnyak reproducing kernel spaces and commutative transfer-function realizations

An interesting and recently much studied generalization of the classical Schur class is the class of contractive operator-valued multipliers S(λ) for the reproducing kernel Hilbert space H(kd) on the unit ball Bd⊂Cd, where kd is the positive kernel kd(λ,ζ)=1/(1−〈λ,ζ〉) on Bd. The reproducing kernel space H(KS) associated with the positive kernel KS(λ,ζ)=(I−S(λ)S∗(ζ))⋅kd(λ,ζ) is a natural multivariable generalization of the classical de Branges–Rovnyak canonical model space. A special feature appearing in the multivariable case is that the space H(KS) in general may not be invariant under the adjoints of the multiplication operators on H(kd). We show that invariance of H(KS) under for each j=1,…,d is equivalent to the existence of a realization for S(λ) of the form S(λ)=D+C−1(I−λ1A1−⋯−λdAd)(λ1B1+⋯+λdBd) such that connecting operator has adjoint U∗ which is isometric on a certain natural subspace (U is “weakly coisometric”) and has the additional property that the state operators A1,…,Ad pairwise commute; in this case one can take the state space to be the functional-model space H(KS) and the state operators A1,…,Ad to be given by (a de Branges–Rovnyak functional-model realization). We show that this special situation always occurs for the case of inner functions S (where the associated multiplication operator MS is a partial isometry), and that inner multipliers are characterized by the existence of such a realization such that the state operators A1,…,Ad satisfy an additional stability property.