Defect Functions of Holomorphic Contractive Operator Functions and the Scattering Suboperator through the Internal Channels of a System. Part I

Abstract

Let \(\theta (\zeta )\) be a Schur operator function, i.e., it is defined on the unit disk \({\mathbb D}\,{:=}\,\{\zeta \in {\mathbb C}: |\zeta | < 1\}\) and its values are contractive operators acting from one Hilbert space into another one. In the first part of the paper the outer and \(*\)-outer Schur operator functions \(\varphi (\zeta )\) and \(\psi (\zeta )\) which describe respectively the deviations of the function \(\theta (\zeta )\) from inner and \(*\)-inner operator functions are studied. If \(\varphi (\zeta )\ne 0\), then it means that in the scattering system for which \(\theta (\zeta )\) is the transfer function a portion of “information” comes inward the system and does not go outward, i.e., it is left in the internal channels of the system ([11, Sect. 6]). The function \(\psi (\zeta )\) has the analogous property. For this reason these functions are called defect ones of the function \(\theta (\zeta )\). The explicit form of the defect functions \(\varphi (\zeta )\) and \(\psi (\zeta )\) is obtained and the analytic connection of these functions with the function \(\theta (\zeta )\) is described ([11, Sect. 3 and Sect. 5]). The operator functions \(\left( \begin{matrix} \varphi (\zeta ) \\ \theta (\zeta ) \end{matrix}\right) \) and \((\psi (\zeta ), \theta (\zeta ))\) are Schur functions as well ([11, Sect. 3]). It is important that there exists the unique contractive operator function \(\chi (t),t\in \partial {\mathbb D}\), such that the operator function \(\left( \begin{matrix} \chi (t) &{} \varphi (t) \\ \psi (t) &{} \theta (t) \end{matrix}\right) ,t\in \partial {\mathbb D},\) is also contractive (Sect. 6). The second part of the paper is devoted to introducing and studying the properties of the function \(\chi (t)\). Specifically, it is shown that the function \(\chi (t)\) is the scattering suboperator through the internal channels of the scattering system for which \(\theta (\zeta )\) is the transfer function (Sect. 6).