Entire Symmetric Operators in de Branges–Pontryagin Spaces and a Truncated Matrix Moment Problem

The role of de Branges–Pontryagin spaces as functional models for entire symmetric operators with finite equal deficiency indices and proper gauges in Pontryagin spaces is reviewed and then extended to symmetric operators that are not entire. These results are used to derive an operator representation for generalized Carathéodory functions. Enroute, boundary mappings and the characteristic function of S are defined. Generalized resolvents of symmetric operators S with non dense domains corresponding to single-valued representing extensions \({{\widetilde{S}} }\) $\tilde{S}$ are characterized in terms of the characteristic function of S. These results are applied to obtain a description of the set of solutions of an indefinite truncated matrix moment problem.