Matrix Orthogonal Laurent Polynomials on the Unit Circle and Toda Type Integrable Systems

Abstract

Matrix orthogonal Laurent polynomials in the unit circle and the theory of Toda-like integrable systems are connected using the Gauss--Borel factorization of two, left and a right, Cantero-Morales-Velazquez block moment matrices, which are constructed using a quasi-definite matrix measure. A block Gauss-Borel factorization problem of these moment matrices leads to two sets of biorthogonal matrix orthogonal Laurent polynomials and matrix Szegő polynomials, which can be expressed in terms of Schur complements of bordered truncations of the block moment matrix. The corresponding block extension of the Christoffel-Darboux theory is derived. Deformations of the quasi-definite matrix measure leading to integrable systems of Toda type are studied. The integrable theory is given in this matrix scenario; wave and adjoint wave functions, Lax and Zakharov-Shabat equations, bilinear equations and discrete flows --connected with Darboux transformations--. We generalize the integrable flows of the Cafasso's matrix extension of the Toeplitz lattice for the Verblunsky coefficients of Szegő polynomials. An analysis of the Miwa shifts allows for the finding of interesting connections between Christoffel--Darboux kernels and Miwa shifts of the matrix orthogonal Laurent polynomials.

arXiv comment40 pages, amsart

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Cited by in corpus (12)

• Multivariate orthogonal polynomial and integrable systems
• Christoffel transformations for matrix orthogonal polynomials in the real line and the non-Abelian 2D Toda lattice hierarchy
• Darboux transformations for multivariate orthogonal polynomials
• Transformation theory and Christoffel formulas for matrix biorthogonal polynomials on the real line
• Matrix-valued Gegenbauer polynomials
• Ladder relations for a class of matrix valued orthogonal polynomials
• Linear spectral transformations for multivariate orthogonal polynomials and multispectral Toda hierarchies
• The Toda and Painlevé Systems Associated with Semiclassical Matrix-Valued Orthogonal Polynomials of Laguerre Type
• CMV biorthogonal Laurent polynomials: Christoffel formulas for Christoffel and Geronimus perturbations
• Matrix biorthogonal polynomials in the unit circle: Riemann-Hilbert problem and matrix discrete Painleve II system
• CMV biorthogonal Laurent polynomials. II: Christoffel formulas for Geronimus-Uvarov perturbations
• Matrix-valued Laurent polynomials, parametric linear systems and integrable systems