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On $J$-self-adjoint extensions of the Phillips symmetric operator


$J$-self-adjoint extensions of the Phillips symmetric operator $S$ are %\break studied. The concepts of stable and unstable $C$-symmetry are introduced in the extension theory framework. The main results are the following: if ${A}$ is a $J$-self-adjoint extension of $S$, then either $\sigma({A})=\mathbb{R}$ or $\sigma({A})=\mathbb{C}$; if ${A}$ has a real spectrum, then ${A}$ has a stable $C$-symmetry and ${A}$ is similar to a self-adjoint operator; there are no $J$-self-adjoint extensions of the Phillips operator with unstable $C$-symmetry.

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TitleOn $J$-self-adjoint extensions of the Phillips symmetric operator
SourceMethods Funct. Anal. Topology, Vol. 16 (2010), no. 4, 333-348
MathSciNet MR2777192
CopyrightThe Author(s) 2010 (CC BY-SA)

Authors Information

S. Kuzhel
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine

O. Shapovalova
National Pedagogical Dragomanov University, Kyiv, Ukraine

L. Vavrykovych
Nizhin State University, 2 Kropyv'yanskogo Str., Nizhin, 16602, Ukraine 

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S. Kuzhel, O. Shapovalova, and L. Vavrykovych, On $J$-self-adjoint extensions of the Phillips symmetric operator, Methods Funct. Anal. Topology 16 (2010), no. 4, 333-348.


@article {MFAT546,
    AUTHOR = {Kuzhel, S. and Shapovalova, O. and Vavrykovych, L.},
     TITLE = {On $J$-self-adjoint extensions of the Phillips symmetric operator},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {16},
      YEAR = {2010},
    NUMBER = {4},
     PAGES = {333-348},
      ISSN = {1029-3531},
  MRNUMBER = {MR2777192},
       URL = {},

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