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On generalized rezolvents and characteristic matrices of first-order symmetric systems


Abstract

We study general (not necessarily Hamiltonian) first-ordersymmetric system $J y'-B(t)y=\Delta(t) f(t)$ on an interval $\mathcal I=[a,b)$ with the regular endpoint $a$ and singular endpoint $b$. It isassumed that the deficiency indices $n_\pm(T_{\min})$ of thecorresponding minimal relation $T_{\min}$ in $L_\Delta^2(\mathcal I)$ satisfy$n_-(T_{\min})\leq n_+(T_{\min})$. We describe all generalized resolvents$y=R(\lambda)f, \; f\in L_\Delta^2(\mathcal I),$ of $T_{\min}$ in terms of boundary problemswith $\lambda$-depending boundary conditions imposed on regular andsingular boundary values of a function $y$ at the endpoints $a$and $b$ respectively. We also parametrize all characteristicmatrices $\Omega(\lambda)$ of the system immediately in terms of boundaryconditions. Such a parametrization is given both by the blockrepresentation of $\Omega(\lambda)$ and by the formula similar to thewell-known Krein formula for resolvents. These results develop the Straus' results on generalized resolvents and characteristicmatrices of differential operators.

Key words: First-order symmetric system, boundary problem with a spectral parameter, generalized resolvent, characteristic matrix.


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Article Information

TitleOn generalized rezolvents and characteristic matrices of first-order symmetric systems
SourceMethods Funct. Anal. Topology, Vol. 20 (2014), no. 4, 328-348
MathSciNet MR3309671
zbMATH 1324.47081
MilestonesReceived 13/01/2014
CopyrightThe Author(s) 2014 (CC BY-SA)

Authors Information

Vadim Mogilevskii
Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, 74 R. Luxemburg, Donets'k, 83050, Ukraine


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Citation Example

Tim Mogilevskii, On generalized rezolvents and characteristic matrices of first-order symmetric systems, Methods Funct. Anal. Topology 20 (2014), no. 4, 328-348.


BibTex

@article {MFAT733,
    AUTHOR = {Mogilevskii, Tim},
     TITLE = {On generalized rezolvents and characteristic matrices of first-order symmetric systems},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {20},
      YEAR = {2014},
    NUMBER = {4},
     PAGES = {328-348},
      ISSN = {1029-3531},
  MRNUMBER = {MR3309671},
 ZBLNUMBER = {1324.47081},
       URL = {http://mfat.imath.kiev.ua/article/?id=733},
}


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