标题： 矩阵单位圆上的正交劳伦特多项式与托达型可积系统 显示英文标题

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

摘要： 矩阵单位圆上的正交Laurent多项式和Toda型可积系统理论通过两个左和右的Cantero-Morales-Velazquez块矩 moment 矩阵的Gauss-Borel分解相连接，这些矩阵是使用准确定矩阵测度构建的。这些矩 moment 矩阵的块Gauss-Borel分解问题导致了两组双正交矩阵正交Laurent多项式和矩阵Szegő多项式，这些多项式可以用块矩 moment 矩阵的边界截断的Schur补来表示。推导了Christoffel-Darboux理论的相应块扩展。研究了导致Toda型可积系统的准确定矩阵测度的变形。在此矩阵场景中给出了可积理论；波函数和伴随波函数，Lax和Zakharov-Shabat方程，双线性方程以及与Darboux变换相关的离散流。我们推广了Cafasso的Toeplitz格子矩阵扩展中关于Szegő多项式的Verblunsky系数的可积流。对Miwa位移的分析使得能够找到Christoffel-Darboux核与矩阵正交Laurent多项式的Miwa位移之间的有趣联系。 显示英文摘要

摘要： 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.