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 B d ⊂ C, where kd is the positive kernel kd(λ,ζ) = 1/(1 − 〈λ, ζ〉) on B . 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 M∗ λj of the multiplication operators Mλj : f(λ) 7→ λjf(λ) on H(kd). We show that invariance of H(KS) under M ∗ λj for each j = 1, . . . , d is equivalent to the existence of a realization for S(λ) of the form S(λ) = D + C(I − λ1A1 − · · · − λdAd) (λ1B1 + · · ·+ λdBd) such that connecting operator U =   A1 B1 .. .. Ad Bd C D   has adjointU∗ 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 Aj = Mλ∗ j H(KS) (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.