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

In this paper the realization problems for the Kre?n–Langer class N_κ of matrix‐valued functions are being considered. We found the criterion when a given matrix‐valued function from the class N_κ 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 Π_κ with indefinite metric. We specify three subclasses of the class N_κ (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 Π_κ . Alternatively we show that the class N_κ (R) can be realized as transfer matrix‐functions of some canonical impedance systems with self‐adjoint main operators in rigged spaces Π_κ . The case of scalar functions of the class N_κ (R) is considered in details and some examples are presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)