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 Szeg\H{o} 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 Szeg\H{o} 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.