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

### 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.

### Article Information

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

### Citation Example

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,
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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