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A Panorama of Modern Operator Theory and Related TopicsCanonical Transfer-function Realization for Schur-Agler-class Functions of the Polydisk

A Panorama of Modern Operator Theory and Related Topics: Canonical Transfer-function Realization... [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.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

A Panorama of Modern Operator Theory and Related TopicsCanonical Transfer-function Realization for Schur-Agler-class Functions of the Polydisk

Part of the Operator Theory: Advances and Applications Book Series (volume 218)
Editors: Dym, Harry; Kaashoek, Marinus A.; Lancaster, Peter; Langer, Heinz; Lerer, Leonid
Ball, Joseph A.; Bolotnikov, Vladimir
A Panorama of Modern Operator Theory and Related Topics — Jan 3, 2012
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Publisher
Springer Basel
Copyright
© Springer Basel AG 2012
ISBN
978-3-0348-0220-8
Pages
75 –122
DOI
10.1007/978-3-0348-0221-5_4
Publisher site
See Chapter on Publisher Site

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.]

Published: Jan 3, 2012

Keywords: Operator-valued Schur-Agler functions; Agler decomposition; unitary realization.

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