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Brune sections in the non-stationary case. (English) Zbl 0998.93019
The classical inverse scattering problem consists in finding all representations of a given Schur-class function \(s\) in the form \(s=T_{\Theta}(\sigma)=(\theta_{11}\sigma + \theta_{12})(\theta_{21}\sigma +\theta_{22})^{-1}\) where \(\sigma\) is also a Schur-class function, and \(\Theta=\left( \begin{smallmatrix} \theta_{11} & \theta_{12} \cr \theta_{21} & \theta_{22}\end{smallmatrix}\right)\) is a meromorphic function in \({\mathbf D}\) that is \(J\)-inner, i.e., \(\Theta(z)J\Theta(z)^*\leq J\) at all points of holomorphy in \({\mathbf D}\), and \(\Theta(z)J\Theta(z)^*= J\) a.e. on \({\mathbf T}\), with \(J=\left(\begin{smallmatrix} 1 & 0 \cr 0 & -1 \end{smallmatrix}\right)\). The elementary such \(\Theta\)’s are of the form \(\Theta(z)=\left(\begin{smallmatrix} 1 & 0 \cr 0 & 1\end{smallmatrix}\right)-(1-z)/[\varepsilon(1-za^*)(1-a)]\left(\begin{smallmatrix} 1 & -k \cr k^* & -1\end{smallmatrix}\right)\). For the Schur sections (called also Blaschke factors) one has \(a\in {\mathbf D}, k\in {\mathbf D}, \varepsilon=(1-|k|^2)/(1-|a|^2)\), while in the case of Brune sections, \(a\) and \(k\) are on \({\mathbf T}\) (with \(a\neq 1\)), and \(\varepsilon\) is a strictly positive number. The inverse scattering problem is closely related to the theory of linear time-invariant dissipative systems, and it has numerous generalizations and ramifications. In the time-varying setting, Schur-class functions are replaced by upper triangular doubly infinite contractive matrices. The analogue of the Blaschke factor is known [see, e.g., D. Alpay, P. Dewilde and H. Dym, Oper. Theory, Adv. Appl. 47, 61-135 (1990; Zbl 0727.47005)]. Here, the authors introduce the time-varying counterpart of Brune sections. However, on the way some unbounded operators appear, and to remedy this dificulty, they use the so-called Zadeh extension. The theory results in a factorization theorem that generalizes the factorization of \(J\)-inner functions to the time-varying setting in the case of finitely specified systems.
MSC:
93C05 Linear systems in control theory
30G30 Other generalizations of analytic functions (including abstract-valued functions)
15A06 Linear equations (linear algebraic aspects)
30E05 Moment problems and interpolation problems in the complex plane
47A57 Linear operator methods in interpolation, moment and extension problems
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
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