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<p>We establish left and right canonical factorizations of Hilbert-space operator-valued functions <i>G</i>(<i>z</i>) that are analytic on neighborhoods of the complex unit circle <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathbb {T}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">T</mi> </math></EquationSource> </InlineEquation> and the origin 0 and that have the form <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(G(z)=I+F(z)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>I</mi> <mo>+</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with <i>F</i>(<i>z</i>) taking strictly contractive values on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\({\mathbb {T}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">T</mi> </math></EquationSource> </InlineEquation>. 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.</p>

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

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  • 发表日期：

    2025年10月15日

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  • S. ter Horst,
  • M. Kurula,
  • A. C. M. Ran

摘要

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 \({\mathbb {T}}\) T and the origin 0 and that have the form \(G(z)=I+F(z)\) G ( z ) = I + F ( z ) with F(z) taking strictly contractive values on \({\mathbb {T}}\) 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.