Schur parameters, Toeplitz matrices, and Kre\u{\i}n shorted operators

TL;DR

This paper explores the mathematical relationships between Schur parameters, Toeplitz matrices, and Kren shorted operators, providing conditions for system controllability, observability, and solution uniqueness in operator theory.

Contribution

It introduces new connections and criteria linking Schur parameters, Toeplitz matrices, and Kren operators, advancing understanding of system properties and the Schur problem.

Findings

Criteria for controllability and observability of discrete-time systems.

Conditions for non-unitary contractions to be non-isometric or non-co-isometric.

Characterization and uniqueness criteria for the Schur problem solutions.

Abstract

We establish connections between Schur parameters of the Schur class operator-valued functions, the corresponding simple conservative realizations, lower triangular Toeplitz matrices, and Kre\u\i n shorted operators. By means of Schur parameters or shorted operators for defect operators of Toeplitz matrices necessary and sufficient conditions for a simple conservative discrete-time system to be controllable/observable and for a completely non-unitary contraction to be completely non-isometric/completely non-co-isometric are obtained. For the Schur problem a characterization of central solution and uniqueness criteria to the solution are given in terms of shorted operators for defect operators of contractive Toeplitz matrices, corresponding to data.