Orthogonal Laurent polynomials on the unit circle and snake-shaped matrix factorizations

TL;DR

This paper explores orthogonal Laurent polynomials on the unit circle, revealing a snake-shaped matrix factorization of the multiplication operator, with the shape dictated by monomial ordering and Schur parameters.

Contribution

It introduces a novel snake-shaped matrix factorization for the multiplication operator in orthogonal Laurent polynomial bases, linking monomial orderings and Schur parameters.

Findings

Matrix representation is a snake-shaped unitary or isometric matrix.

Shape of the snake is determined by monomial orthogonalization order.

Special cases include Hessenberg and CMV matrices.

Abstract

Let there be given a probability measure $\mu$ on the unit circle $\TT$ of the complex plane and consider the inner product induced by $\mu$. In this paper we consider the problem of orthogonalizing a sequence of monomials $\{z^{r_k}\}_k$, for a certain order of the $r_k\in\mathbb{Z}$, by means of the Gram-Schmidt orthogonalization process. This leads to a basis of orthonormal Laurent polynomials $\{\psi_k\}_k$. We show that the matrix representation with respect to the basis $\{\psi_k\}_k$ of the operator of multiplication by $z$ is an infinite unitary or isometric matrix allowing a 'snake-shaped' matrix factorization. Here the 'snake shape' of the factorization is to be understood in terms of its graphical representation via sequences of little line segments, following an earlier work of Delvaux and Van Barel. We show that the shape of the snake is determined by the order in which the monomials $\{z^{r_k}\}_k$ are orthogonalized, while the 'segments' of the snake are canonically determined in terms of the Schur parameters for $\mu$. Isometric Hessenberg matrices and unitary five-diagonal matrices (CMV matrices) follow as a special case of the presented formalism.