On some uses of factorizations in linear system theory

We present a short, and far from exhaustive, survey of some of the uses of factorization theory in the study of linear systems. The whole development is based on model theory, be it in the context of vectorial polynomial, rational or Hardy spaces. Invariant subspaces of operators are related to factorizations of matrix functions. The connection to system theory is via realizations based on various coprime factorizations. Among topics touched upon are the study of factorizations of rational functions and in particular the important cases of inner and all-pass functions. We survey the subject of coprime factorizations and their connection, via spectral factorizations, to various Riccati equations. In this connection we describe some geometric aspects of Wiener-Hopf factorizations. We proceed to describe connections with geometric control theory and give a Hilbert space characterization of stabilizability and detectability subspaces. Finally we describe a complete parametrization of all minimal spectral factors in the case of a coercive spectral density function.