Journal of Operator Theory

Volume 47, Issue 2, Spring 2002 pp. 245-286.

System theory, operator models and scattering: the time-varying case

Authors: Daniel Alpay (1), Joseph A. Ball (2), and Yossi Peretz (3)
Author institution: (1) Dept. of Math. and Computer Science, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva 84105, Israel
(2) Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061, USA
(3) Dept. of Math. and Computer Science, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva 84105, Israel

Summary: It is well known that linear system theory, Lax-Phillips scattering theory, and operator model theory for a contraction operator are all intimately related. A common thread in all three theories is a contractive, analytic, operator-valued function on the unit disk $W(z)$ having a representation of the form $W(z) = D + z C (I - zA)^{-1}B$, known, depending on the context, as the transfer function, the scattering function, or the characteristic function. We present the time-varing analogue of this framework. Also included is a time-varying analogue of the Abstract Interpolation Problem of Katsnelson-Kheifets-Yuditskii.

Keywords: Time-varying system, realization, frequency response function, transfer function, isometric/coisometric/unitary system, Lax-Phillips scattering

Contents Full-Text PDF