Abstract interpolation problem in generalized Nevanlinna classes

Abstract

The abstract interpolation problem (AIP) in the Schur class was posed by V. Katznelson, A. Kheifets, and P. Yuditskii in 1987. In the present paper we consider an analog of AIP for the generalized Nevanlinna class $N_κ(L)$ in the nondegenerate case. We associate with the data set of the AIP a symmetric linear relation $\hat A$ acting in a Pontryagin space. The description of all solutions of the AIP is reduced to the problem of description of all $L$-resolvents of this symmetric linear relation $\hat A$. The latter set is parametrized by application of the indefinite version of Kreın’s representation theory for symmetric linear relations in Pontryagin spaces developed by M. G. Kreın and H. Langer in [22] and a formula for the $L$-resolvent matrix obtained by V. Derkach and M. Malamud in [11].