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On realization of the Kreĭn–Langer class Nκ of matrix‐valued functions in Pontryagin spaces

In this paper the realization problems for the Kreĭn–Langer class Nκ of matrix‐valued functions are being considered. We found the criterion when a given matrix‐valued function from the class Nκ can be realized as linear‐fractional transformation of the transfer function of canonical conservative system of the M. Livsic type (Brodskii–Livsic rigged operator colligation) with the main operator acting on a rigged Pontryagin space Πκ with indefinite metric. We specify three subclasses of the class Nκ (R) of all realizable matrix‐valued functions that correspond to different properties of a realizing system, in particular, when the domains of the main operator of a system and its conjugate coincide, when the domain of the hermitian part of a main operator is dense in Πκ . Alternatively we show that the class Nκ (R) can be realized as transfer matrix‐functions of some canonical impedance systems with self‐adjoint main operators in rigged spaces Πκ . The case of scalar functions of the class Nκ (R) is considered in details and some examples are presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)