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

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A Panorama of Modern Operator Theory and Related Topics

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 218))

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.

Mathematics Subject Classification (2000). 47A57.

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Authors and Affiliations

  1. Department of Mathematics, Virginia Tech, Blacksburg, VA, 24061-0123, USA

    Joseph A. Ball

  2. Department of Mathematics, The College of William and Mary, Williamsburg, VA, 23187-8795, USA

    Vladimir Bolotnikov

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  1. Joseph A. Ball
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Correspondence to Joseph A. Ball .

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Editors and Affiliations

  1. Dept. Mathematics, Weizmann Institute of Science, Rehovot, Israel

    Harry Dym

  2. Dept. Mathematics &, Computer Sciences, Free University Amsterdam, De Boelelaan 1081 A, Amsterdam, 1081 HV, Netherlands

    Marinus A. Kaashoek

  3. Dept. Mathematics & Statistics, University of Calgary, University Drive NW 2500, Calgary, T2N 1N4, Alberta, Canada

    Peter Lancaster

  4. Inst. Analysis und Scientific Computing, TU Wien, Wiedner Hauptstr. 8-10, Wien, 1040, Austria

    Heinz Langer

  5. Dept. Mathematics, Technion Israel Institute of Technology, Haifa, 32000, Israel

    Leonid Lerer

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Ball, J.A., Bolotnikov, V. (2012). Canonical Transfer-function Realization for Schur-Agler-class Functions of the Polydisk. In: Dym, H., Kaashoek, M., Lancaster, P., Langer, H., Lerer, L. (eds) A Panorama of Modern Operator Theory and Related Topics. Operator Theory: Advances and Applications(), vol 218. Springer, Basel. https://doi.org/10.1007/978-3-0348-0221-5_4

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