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# Symmetric extensions of symmetric linear relations (operators) preserving the multivalued part

### Abstract

Let $\mathfrak H$ be a Hilbert space and let $A$ be a symmetric linear relation (in particular, a nondensely defined operator) in $\mathfrak H$. By using the concept of a boundary triplet for $A^*$ we characterize symmetric extensions $\widetilde A\supset A$ preserving the multivalued part of $A$. Such a characterization is given in terms of an abstract boundary parameter and the Weyl function of the boundary triplet. Application of these results to the Hamiltonian system $Jy'-B(t)y=\lambda\Delta(t) y$ enabled us to describe its matrix solutions generating the generalized Fourier transform with the nonempty set of respective spectral functions.

Key words: Symmetric linear relation, symmetric extension, boundary triplet, Hamiltonian system, spectral function.

### Article Information

 Title Symmetric extensions of symmetric linear relations (operators) preserving the multivalued part Source Methods Funct. Anal. Topology, Vol. 24 (2018), no. 2, 152-177 MathSciNet MR3827126 Milestones Received 26/09/2017; Revised 05/11/2017 Copyright The Author(s) 2018 (CC BY-SA)

### Authors Information

V. I. Mogilevskii
Department of Mathematical Analysis and Informatics, Poltava National V. G. Korolenko Pedagogical University, 2 Ostrogradski str., Poltava, 36000, Ukraine

### Citation Example

V. I. Mogilevskii, Symmetric extensions of symmetric linear relations (operators) preserving the multivalued part, Methods Funct. Anal. Topology 24 (2018), no. 2, 152-177.

### BibTex

@article {MFAT1055,
AUTHOR = {Mogilevskii, V. I.},
TITLE = {Symmetric extensions of symmetric linear relations
(operators) preserving the multivalued part},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {24},
YEAR = {2018},
NUMBER = {2},
PAGES = {152-177},
ISSN = {1029-3531},
MRNUMBER = {MR3827126},
URL = {http://mfat.imath.kiev.ua/article/?id=1055},
}

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