Darboux transformations for CMV matrices

Abstract

We develop a theory of Darboux transformations for CMV matrices, canonical representations of the unitary operators. In perfect analogy with their self-adjoint version -the Darboux transformations of Jacobi matrices -they are equivalent to Laurent polynomial modifications of the underlying measures. We address other questions which emphasize the similarities between Darboux transformations for Jacobi and CMV matrices, like their (almost) isospectrality or the relation that they establish between the corresponding orthogonal polynomials, showing also that both transformations are connected by the Szegő mapping.Nevertheless, we uncover some features of the Darboux transformations for CMV matrices which are in striking contrast with those of the Jacobi case. In particular, when applied to CMV matrices, the matrix realization of the inverse Darboux transformations -what we call 'Darboux transformations with parameters' -leads to spurious solutions whose interpretation deserves future research. Such spurious solutions are neither unitary nor band matrices, so Darboux transformations for CMV matrices are much more subject to the subtleties of the algebra of infinite matrices than their Jacobi counterparts.A key role in our theory is played by the Cholesky factorizations of infinite matrices. Actually, the Darboux transformations introduced in this paper are based on the Cholesky factorizations of degree one Hermitian Laurent polynomials evaluated on CMV matrices. These transformations are also generalized to higher degree Laurent polynomials, as well as to the extension of CMV matrices to quasi-definite functionals -what we call 'quasi-CMV' matrices.Furthermore, we show that this CMV version of Darboux transformations plays a role in integrable systems like the Schur flows or the Ablowitz-Ladik model which parallels that of Darboux for Jacobi matrices in the Toda lattice.

Double Affine Hecke Algebra of Rank 1 and Orthogonal Polynomials on the Unit Circle

“…Remarkably, the orthonormal polynomialsŜ (3) n (x) are the eigenvectors of the symmetric Jacobi matrix J that can be constructed as the sum of two involution operators [7] (see also [4]):…”

Christoffel Transformation for a Matrix of Bi-variate Measures

“…Recently, Cantero, Marcellán, Moral and Velázquez [14] presented an approach to the Darboux transformations for CMV matrices. In particular, for the Christoffel transformation they show that given a Hermitian polynomial L(z), a linear functional µ (supported on the unit circle) and the perturbed oneμ = L(z)µ, if L(C) has Cholesky factorization L(C) = AA † , then L(Ĉ) = A † A, where C andĈ are the CMV matrices associated with µ andμ, respectively.…”

“…Besides, if AB is admissible, then (AB) † is also admissible and (AB) † = B † A † . However, the associative law can fail even if all the involved matrix products are admissible [14].…”

CMV Biorthogonal Laurent Polynomials: Perturbations and Christoffel Formulas

“…Generic orders in the basis used to span the space of orthogonal Laurent polynomials in the unit circle were discussed in [24]. The CMV (Cantero-Moral-Velázquez) matrices [22], which constitute the representation of the multiplication operator in terms of the basis of orthonormal Laurent polynomials, were discussed in [26] where the connection with Darboux transformations and their applications to integrable systems has been analyzed. Regarding the CMV ordering, orthogonal Laurent polynomials, and Toda systems, see section 4.4 of [27], where discrete Toda flows and their connection with Darboux transformations were discussed.…”