A New Systems Theory Perspective on Canonical Wiener-Hopf Factorization on the Unit Circle

S. ter Horst; M. Kurula; A. C.M. Ran

doi:10.1007/s00020-025-02813-x

A New Systems Theory Perspective on Canonical Wiener-Hopf Factorization on the Unit Circle

S. ter Horst
, M. Kurula
, A. C.M. Ran^*

^*Corresponding author for this work

Mathematics

Research output: Contribution to Journal › Article › Academic › peer-review

Abstract

We establish left and right canonical factorizations of Hilbert-space operator-valued functions G(z) that are analytic on neighborhoods of the complex unit circle T and the origin 0 and that have the form G(z)=I+F(z) with F(z) taking strictly contractive values on T. Such functions can be realized as transfer functions of infinite dimensional dichotomous discrete-time linear systems, and we employ the strict bounded real lemma for this class of operators, together with associated Kreĭn space theory, to derive explicit formulas for the left and right canonical factorizations.

Original language	English
Article number	27
Pages (from-to)	1-21
Number of pages	21
Journal	Integral Equations and Operator Theory
Volume	97
Issue number	4
Early online date	15 Oct 2025
DOIs	https://doi.org/10.1007/s00020-025-02813-x
Publication status	Published - Dec 2025

Bibliographical note

Publisher Copyright:
© The Author(s) 2025.

Funding

This work is based on research supported in part by the National Research Foundation of South Africa (NRF) and the DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS). Any opinion, finding and conclusion or recommendation expressed in this material is that of the authors and the NRF and CoE-MaSS do not accept any liability in this regard. Part of the research was conducted during a visit of the second author to North-West University in March and April of 2024, supported by a scholarship of the Magnus Ehrnrooth Foundation. This work is based on the research supported in part by the National Research Foundation of South Africa (Grant Numbers 118513, 127364 and 145688), the DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (Grant Number 2024-030-OPA) and the Magnus Ehrnrooth Foundation

Funders	Funder number
Magnus Ehrnroothin Säätiö
National Research Foundation	118513, 145688, 127364
DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences	2024-030-OPA

Keywords

Canonical Wiener-Hopf factorization
dichotomous systems
Kreĭn space
KYP-inequality

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N2 - We establish left and right canonical factorizations of Hilbert-space operator-valued functions G(z) that are analytic on neighborhoods of the complex unit circle T and the origin 0 and that have the form G(z)=I+F(z) with F(z) taking strictly contractive values on T. Such functions can be realized as transfer functions of infinite dimensional dichotomous discrete-time linear systems, and we employ the strict bounded real lemma for this class of operators, together with associated Kreĭn space theory, to derive explicit formulas for the left and right canonical factorizations.

AB - We establish left and right canonical factorizations of Hilbert-space operator-valued functions G(z) that are analytic on neighborhoods of the complex unit circle T and the origin 0 and that have the form G(z)=I+F(z) with F(z) taking strictly contractive values on T. Such functions can be realized as transfer functions of infinite dimensional dichotomous discrete-time linear systems, and we employ the strict bounded real lemma for this class of operators, together with associated Kreĭn space theory, to derive explicit formulas for the left and right canonical factorizations.

KW - Canonical Wiener-Hopf factorization

KW - dichotomous systems

KW - Kreĭn space

KW - KYP-inequality

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A New Systems Theory Perspective on Canonical Wiener-Hopf Factorization on the Unit Circle

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