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

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 $${\mathbb {T}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> and the origin 0 and that have the form $$G(z)=I+F(z)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>I</mml:mi> <mml:mo>+</mml:mo> <mml:mi>F</mml:mi> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> with F ( z ) taking strictly contractive values on $${\mathbb {T}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> . 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.