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(l) for the reproducing kernel Hilbert space H(k(d)) on the unit ball B-d subset of C-d, where k(d) is the positive kernel k(d) (lambda, zeta) = 1/(1 - <lambda, zeta >) on B-d. The reproducing kernel space H(K-S) associated with the positive kernel K-S (lambda, zeta) = (I - S(lambda)S(zeta)*).k(d)(lambda, zeta) 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-lambda j* of the multiplication operators M-lambda j : f (lambda) bar right arrow lambda(j) f(lambda) on h(kd). We show that invariance of H(K-S) under M-lambda j* for each j = l ,..., d is equivalent to the existence of a realization for S(lambda) of the form S(lambda) = D+ C(I - lambda(1)A(1)-...-lambda(d)A(d))(-1) (lambda B-1(1) +...+ lambda B-d(d)) such that connecting operator [GRAPHICS] has adjoint U* which is isometric on a certain natural subspace (U is "weakly coisoAd B, metric") and has the additional property that the state operators A(1),...,A(d) pairwise commute; in this case one can take the state space to be the functional-model space H(K-S) and the state operators A(1),...,A(d) to be given by A(j) = M-lambda j*|H(K-S) (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 A I, Ad satisfy an additional stability property. (C) 2007 Elsevier Inc. All rights reserved.