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 M ∗ λj of the multiplication operators Mλj :f (λ) →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 ) −1(λ1B1+· · ·+λdBd ) such that connecting operator U = ⎡ ⎢⎣ A1 B1 ... ... Ad Bd C D ⎤ ⎥⎦ 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 Aj =M ∗ λj |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. © 2007 Elsevier Inc. All rights reserved.