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

TL;DR

This paper explores the iterative process of Schur class operator-valued functions and provides a method to construct conservative realizations of their iterates based on the original function's realization.

Contribution

It introduces a novel construction method for conservative realizations of Schur iterates using the original function's realization, advancing operator theory and system realization techniques.

Findings

Constructed conservative realizations of Schur iterates from the original function.

Extended classical Schur algorithm to operator-valued functions.

Provided explicit formulas for realizations of iterates.

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 $\Theta$.