# Canonical Transfer-function Realization for Schur-Agler-class Functions on Domains with Matrix Polynomial Defining Function in $$\mathbb{C}^n$$

Ontology type: schema:Chapter

### Chapter Info

DATE

2012

AUTHORS ABSTRACT

It is well known that a Schur-class function $$\mathbf{S}(z)$$, i.e.,a holomorphic function on the unit disk whose values are contraction operators between two Hilbert spaces u(the input space) and y (the output space), can be written as the characteristic function $$\mathbf{S}(z)=D+{_Z}C(I-{_Z}A)^{-1}B$$ of the unitary colligation $$\mathbf{U}=\begin{array}{llll}[A & B \\ C & D] \end{array}$$ (or as the transfer function of the associated conservative linear system) where U defines a unitary operator from $$\mathbf{X}\bigoplus\mathbf{U}\; to \; \mathbf{X}\bigoplus\mathbf{Y}$$ 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 $$\mathcal{H}({\hat{K}})$$, 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. More... »

PAGES

23-55

### Book

TITLE

Recent Progress in Operator Theory and Its Applications

ISBN

978-3-0348-0345-8
978-3-0348-0346-5

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-3-0348-0346-5_3

DOI

http://dx.doi.org/10.1007/978-3-0348-0346-5_3

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1033003030

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