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

Let there be given a probability measure μμ on the unit circle TT of the complex plane and consider the inner product induced by μμ. In this paper we consider the problem of orthogonalizing a sequence of monomials {zrk}k{zrk}k, for a certain order of the rk∈Zrk∈Z, by means of the Gram–Schmidt orthogonalization process. This leads to a sequence of orthonormal Laurent polynomials {ψk}k{ψk}k. We show that the matrix representation with respect to {ψk}k{ψk}k of the operator of multiplication by zz 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 S. Delvaux and M. Van Barel. We show that the shape of the snake is determined by the order in which the monomials {zrk}k{zrk}k are orthogonalized, while the ‘segments’ of the snake are canonically determined in terms of the Schur parameters for μμ. Isometric Hessenberg matrices and unitary five-diagonal matrices (CMV matrices) follow as a special case of the presented formalism.