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

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 + zC(I - zA)B-1, 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.