Darboux Transformations for Orthogonal Polynomials on the Real Line and on the Unit Circle

Abstract

Jacobi matrices constitute a communicating vessel between orthogonal polynomials on the real line (OPRL) and the theory of self-adjoint operators. In this context, commutation methods in operator theory lead to Darboux transformations of Jacobi matrices, which are equivalent to Christoffel transformations of OPRL. Jacobi matrices have unitary analogues known as CMV matrices, which provide a similar link between orthogonal polynomials on the unit circle (OPUC) and the theory of unitary operators. It has been recently shown that Darboux transformations also make sense in this setting, although some differences arise with respect to the Jacobi case. This paper is a survey on Darboux transformations for Jacobi and CMV matrices which present them in a unified way, highlighting their similarities and differences and connecting them to OPRL and OPUC.