The concept of bispectrality (in short, a function in two variables that is an eigenfunction for an operator in each variable) is especially interesting for orthogonal polynomials. Indeed, depending on the type of operators (differential, difference, q-difference, etc.) and their orders, the bispectrality characterizes the most important
families of orthogonal polynomials, from the classical, classical discrete or q -classical
polynomials, to the Krall and exceptional polynomials. In my opinion, one of the
most interesting (and difficult) problems in relation to orthogonality and bispectrality
is the characterization of the two algebras associated with each family of bispectral
polynomials. In this talk I will review the state of the art about this problem.
In this talk we expose a survey of Darboux transformations for Jacobi and CMV
matrices, relating them with orthogonal polynomials on the real line, orthogonal
polynomials on the unit circle and integrable systems, and also highlighting their
similarities and di_x001B_erences. We will show the interest of the above mentioned spurious
solutions of inverse Darboux for CMV matrices: they are related to Sobolev type inner
products.