Conservative discrete time-invariant systems and block operator CMV matrices

TL;DR

This paper constructs new conservative realizations of operator-valued Schur functions using Schur parameters, representing systems with block operator CMV matrices, and applies these models to contractions and dilations.

Contribution

It introduces a novel approach to system realizations using Schur parameters and block operator CMV matrices, extending scalar polynomial techniques to operator theory.

Findings

Realizations use only Schur parameters, simplifying the system construction.

Block operator CMV matrices serve as models for contractions and dilations.

New models for minimal dilations are developed using these matrices.

Abstract

It is well known that an operator-valued function $\Theta$ from the Schur class ${\bf S}(\mathfrak M,\mathfrak N)$, where $\mathfrak M$ and $\mathfrak N$ are separable Hilbert spaces, can be realized as the transfer function of a simple conservative discrete time-invariant linear system. The known realizations involve the function $\Theta$ itself, the Hardy spaces or the reproducing kernel Hilbert spaces. On the other hand, as in the classical scalar case, the Schur class operator-valued function is uniquely determined by its so called "Schur parameters". In this paper we construct simple conservative realizations using the Schur parameters only. It turns out that the unitary operators corresponding to the systems take the form of five diagonal block operator matrices, which are the analogs of Cantero--Moral--Vel\'azquez (CMV) matrices appeared recently in the theory of scalar orthogonal polynomials on the unit circle. We obtain new models given by truncated block operator CMV matrices for an arbitrary completely non-unitary contraction. It is shown that the minimal unitary dilations of a contraction in a Hilbert space and the minimal Naimark dilations of a semi-spectral operator measure on the unit circle can also be expressed by means of block operator CMV matrices.