Orthogonal Polynomials and Operator Theory. Open Problems and Applications

In this contribution we review first the connections between orthogonal polynomials on the real line, Jacobi matrices, their LU and UL factorizations, spectral measures, spectral transformations and Stieltjes functions as well as their application in the study of some integrable systems of the Toda hierarchy. In a second step, we analyze symmetric operators associated with a Sobolev type inner product. The connection between the Cholesky factorization of five diagonal matrices associated with such operators and the UL factorization of the square of a shifted Jacobi matrix associated with a Christoffel perturbation of the measure is pointed out. Finally, we deal with orthogonal polynomials associated with probability measures supported on the unit circle, Hessenberg and CMV matrices, spectral measures and their spectral transformations, Carathéodory functions as well as their application in the analysis of Schur flows both from an algebraic and analytic way.