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Characteristic matrices and spectral functions of first order symmetric systems with maximal deficiency index of the minimal relation

Vadim Mogilevskii
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Abstract

Let $H$ be a finite dimensional Hilbert space and let $[H]$ be the set of all li ear operators in $H$. We consider first-order symmetric system $J y'-B(t)y=\Lambda(t) f(t)$ with $[H]$-valued coefficients defined on an interval $[a,b) $ with the regular endpoint $a$. It is assumed that the corresponding minimal relation $T_{\rm min}$ has maximally possible deficiency index $n_+(T_{\rm min})=\dim H$. The main result is a parametrization of all characteristic matrices and pseudospectral (spectral) functions of a given system by means of a Nevanlinna type boundary parameter $\tau$. Similar parametrization for regular systems has earlier been obtained by Langer and Textorius. We also show that the coefficients of the parametrization form the matrix $W(\lambda)$ with the properties similar to those of the resolvent matrix in the extension theory of symmetric operators.

Key words: First-order symmetric system, characteristic matrix, spectral function, pseudospectral function, Fourier transform.


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TitleCharacteristic matrices and spectral functions of first order symmetric systems with maximal deficiency index of the minimal relation
SourceMethods Funct. Anal. Topology, Vol. 21 (2015), no. 1, 76-98
MathSciNet   MR3407922
zbMATH 06533469
Milestones  Received 08/08/2014; Revised 26/11/2014
CopyrightThe Author(s) 2015 (CC BY-SA)

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Vadim Mogilevskii, Characteristic matrices and spectral functions of first order symmetric systems with maximal deficiency index of the minimal relation, Methods Funct. Anal. Topology 21 (2015), no. 1, 76-98.


BibTex

@article {MFAT749,
    AUTHOR = {Mogilevskii, Vadim},
     TITLE = {Characteristic matrices and spectral functions of first order symmetric systems with maximal deficiency index of the minimal relation},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {21},
      YEAR = {2015},
    NUMBER = {1},
     PAGES = {76-98},
      ISSN = {1029-3531},
  MRNUMBER = {MR3407922},
 ZBLNUMBER = {06533469},
       URL = {http://mfat.imath.kiev.ua/article/?id=749},
}


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