Canonical transfer-function realization for Schur-Agler-class functions on domains with matrix polynomial defining function in ℂn

Abstract

It is well known that a Schur-class function s(z), i.e., a holomorphic function on the unit disk whose values are contraction operators between two Hilbert spacesU(the input space) and y(the output space), can be written as the characteristic function S(z)=D+zC(I−zA)−1 of the unitary colligation U = (Formula presented.) (or as the transfer function of the associated conservative linear system) where U defines a unitary operator from X⊕U to X⊕Y where the Hilbert space X is an appropriately chosen state space. Moreover, this transfer function is essentially uniquely determined if U is also required to satisfy a certain minimality condition (U should be “closely-connected”). In addition, by choosing the state space X to be the two-component de Branges-Rovnyak reproducing kernel Hilbert space ℋ(K), one can arrive at a unique canonical functional-model form for a U meeting the minimality requirement. Recent work of the authors and others has extended the notion of Schur class and transfer-function representation for Schur-class functions to several-variable complex domains with matrix-polynomial defining function. In this setting the term “Schur-Agler class” is used since one also imposes that a certain von Neumann inequality be satisfied. In this article we develop an analogue of the two-component de Branges-Rovnyak reproducing kernel Hilbert space for this more general setting and thereby arrive at a two-component canonical functional model colligation for the analogue of closely-connected unitary transfer-function realization for this Schur-Agler class. A number of new technical issues appear in this setting which are not present in the classical case.