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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.

### Article Information

 Title On generalized rezolvents and characteristic matrices of first-order symmetric systems Source Methods Funct. Anal. Topology, Vol. 20 (2014), no. 4, 328-348 MathSciNet MR3309671 zbMATH 1324.47081 Milestones Received 13/01/2014 Copyright The Author(s) 2014 (CC BY-SA)

### Authors Information

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

### 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},
}