Cascade Connection of Multiparametric Linear Systems and a Conservative Realization of Decomposable Inner Operator Functions in Bidisk

Abstract

We introduce the notion of cascade connection of multiparametric discrete time-invariant linear dynamical systems with unit delay. This allows us to construct the explicit example of conservative realization of a decomposable operator-valued function $θ(z_1, z_2)=θ_2(z_2)θ_1(z_1)$, where $θ_1$ and $θ_2$ are inner operator-valued functions on the unit disk, vanishing at zero, and discuss some properties of such realization.