2005/09/08 by Yuri Arlinskiĭ, Yuri Arlinskii, Sergey Belyi +7
Computer Science · Mathematics · #47A10 (Primary) #47N50 #81Q10 (Secondary) #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.FA #math.SP #msc:47A10 #msc:47N50 #msc:81Q10
paper · pdf · doi:10.48550/arxiv.math/0509185
21 pages
arxiv created 2005/09/08 · openalex publication_date 2005/09/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper the realization problems for the Krein-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 \Pk 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 \Pk. The case of scalar functions of the class Nκ(R) is considered in details and some examples are presented.