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 $\Theta (\lambda )$ be a function from the Schur class $\mathbf{S}(\mathfrak{M},\mathfrak{N})$ of contractive functions holomorphic on the unit disk. The operator generalization of the classical Schur algorithm associates with $\Theta$ the sequence of contractions (the Schur parameters of $\Theta$) $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 ${\Theta }_0=\Theta$, $Θ_n\in {\bf S}(\sD_{Γ_n},\sD_{Γ^*_n})$ $n=1,...$ (the Schur iterares of $\Theta$) 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 $\tau =\left[\begin{array}{cc}D& C\\ B& A\end{array}\right];\mathfrak{M},\mathfrak{N},\mathfrak{H}$ 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 ${\Theta }_n$ by means of the conservative and simple realization of $\Theta$.