A new systems theory perspective on canonical Wiener-Hopf factorization on the unit circle

TL;DR

This paper introduces a new systems theory approach to canonical Wiener-Hopf factorization for operator-valued functions on the unit circle, using transfer functions of infinite-dimensional systems and Krein space theory.

Contribution

It provides explicit formulas for canonical factorizations of operator-valued functions via a novel systems theory perspective.

Findings

Explicit formulas for factorizations derived

Application of strict bounded real lemma to operator functions

Use of Krein space theory in factorization process

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 and the origin 0, and that have the form G(z)=I+F(z) with F(z) taking strictly contractive values on the unit circle. 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 systems, together with associated Krein space theory, to derive explicit formulas for the left and right canonical factorizations.