Canonical conservative state/signal shift realizations of passive discrete time behaviors

Abstract

A passive linear discrete time invariant s/s (state/signal) system Σ = ( V ; X , W ) consists of a Hilbert (state) space X , a Kreĭn (signal) space W , a maximal nonnegative (generating) subspace V of the Kreĭn space K : = − X [ ∔ ] X [ ∔ ] W . The sets of trajectories ( x ( ⋅ ) ; w ( ⋅ ) ) generated by V on the discrete time intervals I ⊂ Z are defined by ( x ( n + 1 ) ; x ( n ) ; w ( n ) ) ∈ V , n ∈ I . This system is forward conservative, or backward conservative, or conservative if V ⊂ V [ ⊥ ] , V [ ⊥ ] ⊂ V , or V [ ⊥ ] = V , respectively. The set W + Σ of all signal components w ( ⋅ ) of trajectories ( x ( ⋅ ) ; w ( ⋅ ) ) of Σ on I = Z + with x ( 0 ) = 0 and w ( ⋅ ) ∈ ℓ 2 ( Z + ; W ) is called the future time domain behavior of Σ. The Fourier transform W ˆ + Σ of W + Σ is called the future frequency domain behavior of Σ. This set is a maximal nonnegative right-shift invariant subspace in the Kreĭn space K 2 ( D ; W ) that as a topological vector space coincides with the usual Hardy space H 2 ( D ; W ) , but has the indefinite Kreĭn space inner product inherited from W . A subspace of K 2 ( D ; W ) with the above properties is called a passive future frequency domain behavior on W . It has been shown earlier by the present authors that every passive future frequency domain behavior W ˆ + on W may be realized as the future frequency domain behavior of some passive s/s system Σ, and that it is possible to require, in addition, that Σ is (a) controllable and forward conservative, (b) observable and backward conservative, or (c) simple and conservative. These three types of realizations are determined by W ˆ + up to unitary similarity. Canonical functional shift realizations of the types (a) and (b) have been obtained earlier by the present authors, and their connection to the classical de Branges–Rovnyak models have been discussed. Here we present analogous results for a realization of the type (c).