On a generalized Hilbert problem

The problem analyzed is to find functions f±, meromorphic in C±, respectively, with values that are linear operators on a Banach space, and such that their boundary values on R satisfy the equation f−=ω f+, where the operator-valued function ω as well as the positions of the poles of f± and the ranges of their residues are given. Uniqueness results are obtained, under certain conditions an index is proved to exist, and the determination of f± is reduced to the solution of a generalization of Marchenko’s fundamental equation. The results are applied to inverse scattering and inverse spectral problems.