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.

Full Text

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
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},
       URL = {http://mfat.imath.kiev.ua/article/?id=1055},
}

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