Open Access

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

### Abstract

$J$-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.

### Article Information

 Title On $J$-self-adjoint extensions of the Phillips symmetric operator Source Methods Funct. Anal. Topology, Vol. 16 (2010), no. 4, 333-348 MathSciNet MR2777192 Copyright The 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

### Citation Example

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.

### BibTex

@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},
URL = {http://mfat.imath.kiev.ua/article/?id=546},
}