De Branges-Rovnyak functional-model reproducing kernel spaces: multivariable generalizations
by
Joseph Ball
Department of Mathematics, Virginia Tech
Coauthors: Dmitry Kaliuzhnyi-Verbovetskyi, Cora Sadosky, and Victor Vinnikov

It is well known that a holomorphic function S on the unit disk with values equal to contraction operators between two Hilbert spaces U and Y (i.e., a Schur-class function S) can be realized in the form of the characteristic function S(z) = D + z C (I - zA)^-1 B of a unitary colligation

U = [ A B C D ]

(so U is a unitary from H ⊕U to H ⊕Y for an appropriate Hilbert space H). One convenient way for producing such a realization A, B, C, D is through the associated de Branges-Rovnyak functional model space H(S) (together with a closely related extended version D(S)) with reproducing kernel function K_S given by K_S(z, w) = [I - S(z) S(w)^*]/(1 - z[`w]). The de Branges-Rovnyak model space can also be used to give a functional-model for the Sz.-Nagy dilation of a contraction operator and for the associated Lax-Phillips discrete-time scattering system. Recently there has appeared a variety of work generalizing various aspects of these ideas to multivariable settings (e.g., where the unit disk is replaced by (1) the unit ball in C^d (Drury-Arveson space in place of the Hardy space), (2) d-tuples of operators T₁, ..., T_d on a Hilbert space such that the block row [ T₁  ... T_d] is a contraction (row-contraction), (3) the unit polydisk in C^d, or (4) d-tuples T₁, ..., T_d of operators on a Hilbert space such that each T_k is a contraction (the noncommutative polydisk). We focus here on recent work of the speaker (joint with Dmitry Kaliuzhnyi-Verbovetskyi, Cora Sadosky and Victor Vinnikov) on the (commutative) polydisk setting. According to a seminal result of Jim Agler, a holomorphic function S on the polydisk with values equal to operators between two Hilbert spaces U and Y can be realized in the form S(z) = D + C (I - (z₁P₁ + ...+ z_d P_d)A)^-1 (z₁P₁ + ...+ z_d P_d) B for a unitary colligation U as above mapping H ⊕U to H ⊕Y, where P₁, ..., P_d is a spanning family of pairwise-orthogonal projection operators on H. We discuss how polydisk versions of de Branges-Rovnyak functional model spaces can be used to provide a concrete functional-model version of this realization theorem. We also discuss connections with Lax-Phillips scattering and uniqueness issues which are here much more intricate than in the single-variable case.

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Date received: February 11, 2008