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 ${\bf S}(\mathfrak M,\mathfrak N)$ 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 $\Theta$) $\Gamma_0=\Theta(0)\in \bL(\sM,\sN), \Gamma_n\in\bL(\sD_{\Gamma_{n-1}}, \sD_{\Gamma^*_{n-1}}) $ and the sequence of functions $\Theta_0 = \Theta$, $\Theta_n\in {\bf S}(\sD_{\Gamma_n},\sD_{\Gamma^*_n})$ $ n=1,...$ (the Schur iterares of $\Theta$) connected by the relations \[ \Gamma_n=\Theta_n(0), \Theta_n(\lambda) = \Gamma_n+\lambda D_{\Gamma^*_n} \Theta_{n+1}(\lambda) (I + \lambda\Gamma^*_n\Theta_{n+1} (\lambda))^{-1}D_{\Gamma_n}, |\lambda|<1. \] The function $\Theta(\lambda)\in {\bf S}(\sM,\sN)$ can be realized as the transfer function \[ \Theta(\lambda)=D+\lambda C(I-\lambda A)^{-1}B \] of a linear conservative and simple discrete-time system $\tau = {\begin{bmatrix}D & C \cr B & A\end{bmatrix}; \mathfrak M, \mathfrak N,\mathfrak H}$ 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 $\Theta_n$ by means of the conservative and simple realization of $\Theta$.