Iterates of the Schur class operator-valued function and their conservative realizations

Abstract

Let $\mathfrak M$ and $\mathfrak N$ be separable Hilbert spaces and let $Θ(λ)$ be a function from the Schur class ${\bf S}(\mathfrak M,\mathfrak N)$ of contractive functions holomorphic on the unit disk. The operator generalization of the classical Schur algorithm associates with $Θ$ the sequence of contractions (the Schur parameters of $Θ$) $Γ_0=Θ(0)\in \bL(\sM,\sN), Γ_n\in\bL(\sD_{Γ_{n-1}}, \sD_{Γ^*_{n-1}}) $ and the sequence of functions $Θ_0 = Θ$, $Θ_n\in {\bf S}(\sD_{Γ_n},\sD_{Γ^*_n})$ $ n=1,...$ (the Schur iterares of $Θ$) connected by the relations \[ Γ_n=Θ_n(0), Θ_n(λ) = Γ_n+λD_{Γ^*_n} Θ_{n+1}(λ) (I + λΓ^*_nΘ_{n+1} (λ))^{-1}D_{Γ_n}, |λ|<1. \] The function $Θ(λ)\in {\bf S}(\sM,\sN)$ can be realized as the transfer function \[ Θ(λ)=D+λC(I-λA)^{-1}B \] of a linear conservative and simple discrete-time system $τ= {\begin{bmatrix}D & C \cr B & A\end{bmatrix}; \mathfrak M, \mathfrak N,\mathfrak H}$ with the state space $\mathfrak H$ and the input and output spaces $\mathfrak M$ and $\mathfrak N $, respectively. In this paper we give a construction of conservative and simple realizations of the Schur iterates $Θ_n$ by means of the conservative and simple realization of $Θ$.