Orthogonal polynomials and perturbations on measures supported on the real line and on the unit circle. A matrix perspective

The connection between measures supported on the real line (resp. on the unit circle), Hankel (resp. Toeplitz) matrices, Jacobi (resp. Hessenberg and CMV) matrices, Stieltjes (resp. Carathéodory) functions constitutes a key element in the analysis of orthogonal polynomials on the real line (resp. on the unit circle). In the present contribution, we focus our attention on perturbations of the measures supported either on the real line or the unit circle and their consequences on the behavior of the corresponding sequences of orthogonal polynomials and the matrices associated with the multiplication operator in terms on those polynomial bases. The matrix perspective related to such perturbations from the point of view of factorizations (LULU and QRQR) is emphasized. Finally, we show the role of spectral transformations in the analysis of some integrable systems.