Verblunsky-type coefficients for Dirac and canonical systems generated by Toeplitz and Hankel matrices, respectively

TL;DR

This paper introduces Verblunsky-type coefficients for Toeplitz and Hankel matrices, establishing a one-to-one correspondence with positive-definite matrices and exploring their spectral theory connections.

Contribution

It extends Verblunsky's theorem to Toeplitz and Hankel matrices, linking them to discrete Dirac and canonical systems with new coefficient characterizations.

Findings

Established one-to-one correspondence between positive-definite matrices and Verblunsky-type coefficients.

Connected the coefficients to spectral theory of Dirac and canonical systems.

Provided theoretical framework for analyzing Toeplitz and Hankel matrices via these coefficients.

Abstract

We introduce Verblunsky-type coefficients of Toeplitz and Hankel matrices, which correspond to the discrete Dirac and canonical systems generated by Toeplitz and Hankel matrices, respectively. We prove one to one correspondences between positive-definite Toeplitz (Hankel) matrices and their Verblunsky-type coefficients as analogs of the well-known Verblunsky's theorem. Several interconnections with the spectral theory are described as well.