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 sy...

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Veröffentlicht in:Mathematische Nachrichten Jg. 281; H. 10; S. 1380 - 1399
Hauptverfasser: Arlinskiĭ, Yury, Belyi, Sergey, Derkach, Vladimir, Tsekanovskii, Eduard
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Berlin WILEY-VCH Verlag 01.10.2008
WILEY‐VCH Verlag
Schlagworte:
conservative system
Herglotz−Nevanlinna function
indefinite metric
Operator colligation
Pontryagin space
transfer function
ISSN:0025-584X, 1522-2616
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Zusammenfassung: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)
Bibliographie:istex:C70819F7830D4A6AB41DFA95C4C66E4578817E72
ArticleID:MANA200510686
ark:/67375/WNG-SVLVGXGK-L
ISSN:0025-584X
1522-2616
DOI:10.1002/mana.200510686