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

Abstract

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 ( k d ) on the unit ball B d ⊂ C d , where k d is the positive kernel k d ( λ , ζ ) = 1 / ( 1 − 〈 λ , ζ 〉 ) on B d . The reproducing kernel space H ( K S ) associated with the positive kernel K S ( λ , ζ ) = ( I − S ( λ ) S ( ζ ) ∗ ) ⋅ k d ( 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 ( K S ) in general may not be invariant under the adjoints M λ j ∗ of the multiplication operators M λ j : f ( λ ) ↦ λ j f ( λ ) on H ( k d ) . We show that invariance of H ( K S ) under M λ j ∗ for each j = 1 , … , d is equivalent to the existence of a realization for S ( λ ) of the form S ( λ ) = D + C ( I − λ 1 A 1 − ⋯ − λ d A d ) −1 ( λ 1 B 1 + ⋯ + λ d B d ) such that connecting operator U = [ A 1 B 1 ⋮ ⋮ A d B d 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 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 λ 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 M S is a partial isometry), and that inner multipliers are characterized by the existence of such a realization such that the state operators A 1 , … , A d satisfy an additional stability property.