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# On exit space extensions of symmetric operators with applications to first order symmetric systems

### Abstract

Let $A$ be a symmetric linear relation with arbitrary deficiency indices. By using the conceptof the boundary triplet we describe exit space self-adjointextensions $\widetilde A^\tau$ of $A$ in terms of a boundary parameter $\tau$. We characterize certain geometrical properties of $\widetilde A^\tau$ and describe all $\widetilde A^\tau$ with ${\rm mul}\, \widetilde A^\tau=\{0\}$. Applying these results to general (possibly non-Hamiltonian) symmetric systems $Jy'- B(t)y=\Delta(t)y, \; t \in [a,b\rangle,$ we describe all matrix spectral functions of theminimally possible dimension such that the Parseval equality holdsfor any function $f\in L_\Delta^2([a,b \rangle)$.

Key words: Symmetric relation, exit space extension, boundary triplet, first order symmetric system, spectral function.

### Article Information

 Title On exit space extensions of symmetric operators with applications to first order symmetric systems Source Methods Funct. Anal. Topology, Vol. 19 (2013), no. 3, 268-292 MathSciNet MR3136731 zbMATH 1289.47046 Milestones Received 21/03/2013; Revised 02/04/2013 Copyright The Author(s) 2013 (CC BY-SA)

### Authors Information

V. I. Mogilevskii
Department of Mathematical Analysis, Lugans'k Taras Shevchenko National University, 2 Oboronna, Lugans'k, 91011, Ukraine

### Citation Example

V. I. Mogilevskii, On exit space extensions of symmetric operators with applications to first order symmetric systems, Methods Funct. Anal. Topology 19 (2013), no. 3, 268-292.

### BibTex

@article {MFAT686,
AUTHOR = {Mogilevskii, V. I.},
TITLE = {On exit space  extensions of symmetric operators with applications to first order symmetric systems},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {19},
YEAR = {2013},
NUMBER = {3},
PAGES = {268-292},
ISSN = {1029-3531},
MRNUMBER = {MR3136731},
ZBLNUMBER = {1289.47046},
URL = {http://mfat.imath.kiev.ua/article/?id=686},
}