Complex Jacobi matrices generated by Darboux transformations

In this paper, we study complex Jacobi matrices obtained by the Christoffel and Geronimus transformations at a nonreal complex number. In particular, we show that a Nevai class is invariant under such transformations and the ratio asymptotic still holds outside the spectrum of the corresponding Jacobi matrix. In general, these transformations can be iterated and, for example, we demonstrate how Geronimus transformations can lead to $R_{II}$-recurrence relations, which in turn are related to pencils of Jacobi matrices.