The maximal factorable minorant (Q1808892)

The transfer function \(\theta(z)\) is an important characteristic of a linear dynamical system. Let \(\theta(z): U\to V\) be a contractive operator function analytic on the unit disk \(D= \{z\in \mathbb{C}: |z|< 1\}\). A theorem of \textit{B. Sz.-Nagy} and \textit{C. Foias} [Harmonic analysis of operators on Hilbert spaces, North Holland (1970; Zbl 0201.45003)] asserts that there exists an outer function \(\varphi(z)\) on \(D\), whose values are operators from \(U\) to an auxiliary space \(E\) such that \(\varphi^* \varphi\leq I-\theta^* \theta\) a.e. on \(\partial D\) and if \(\phi(z)\) is an analytic contractive operator function such that \(\phi^* \phi\leq I-\theta^* \theta\) a.e., then \(\phi^* \phi\leq \varphi^* \varphi\) a.e. The function \(\varphi(z)\) is unique upto a constant unitary factor on the left and is called the maximal factorable minorant (MFM) of \(I- \theta^* \theta\). Some important qualitative properties of unitary systems like observability, controllability etc. are characterized by the MFM \(\varphi(z)\) and these properties are often not conserved through the cascade coupling of the two systems. It is desired to use the MFM as a tool to consider the conditions for the conservation of qualitative properties for a cascade coupling. If the system \(\alpha\) is a cascade coupling of two systems \(\alpha_1\) and \(\alpha_2\), then the transfer function of \(\alpha\) is a product of two transfer functions of \(\alpha_1\) and \(\alpha_2\). In this paper, the authors have investigated the building of the MFM of \(\theta(z)= \theta_1(z) \theta_2(z)\) from the MFM of \(\theta_1(z)\) and \(\theta_2(z)\) and conditions in which the MFM of \(\theta(z)\) has simplest form.