On realization of the Krein-Langer class Nk of matrix-valued functions in Hilbert spaces with indefinite metric

In this paper the realization problems for the Krein-Langer class $N_{\kappa }$ of matrix-valued functions are being considered. We found the criterion when a given matrix-valued function from the class $N_{\kappa }$ 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 $\text{Pk}$ with indefinite metric. We specify three subclasses of the class $N_{\kappa }(R)$ 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 $\mathrm{\Pi }\kappa$. Alternatively we show that the class $N_{\kappa }(R)$ can be realized as transfer matrix-functions of some canonical impedance systems with self-adjoint main operators in rigged spaces $\text{Pk}$. The case of scalar functions of the class $N_{\kappa }(R)$ is considered in details and some examples are presented.