Canonical Transfer-function Realization for Schur-Agler-class Functions of the Polydisk

Abstract

Associated with any Schur-class function S((z)) (i.e., a contractive operator-valued holomorphic function on the unit disk) is the de Branges- Rovnyak kernel Ks((z,C)) = [=([I-S(z))S(C)) * ]/(1–) and the reproducing kernel Hilbert space H(KS) which serves as the canonical functional-model statespace for a coisometric transfer-function realization s((z)) = D+z(A)1B of S. To obtain a canonical functional-model unitary transfer-function realization, it is now well understood that one must work with a certain (2 × 2)- block matrix kernel and associated two-component reproducing kernel Hilbert space. In this paper we indicate how these ideas extend to the multivariable setting where the unit disk is replaced by the unit polydisk in d complex variables. For the case d> 2, one must replace the Schur class by the more restrictive Schur-Agler class (defined in terms of the validity of a certain von Neumann inequality) in order to get a good realization theory paralleling the single-variable case. This work represents one contribution to the recent extension of the state-space method to multivariable settings, an area of research where Israel Gohberg was a prominent and leading practitioner.