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# On nonsymmetric rank one singular perturbations of selfadjoint operators

### Abstract

We consider nonsymmetric rank one singular perturbations of a selfadjoint operator, i.e., an expression of the form $\tilde A=A+\alpha\left\langle\cdot,\omega_1\right\rangle\omega_2$, $\omega_1\not=\omega_2$, $\alpha\in{\mathbb C}$, in a general case $\omega_1,\omega_2\in{\mathcal H}_{-2}$. Using a constructive description of the perturbed operator $\tilde A$, we investigate some spectral and approximations properties of $\tilde A$. The wave operators corresponding to the couple $A$, $\tilde A$ and a series of examples are also presented.

Key words: Singular perturbation, nonsymmetric perturbations, eigenvalue problem, M. Krein's formula.

### Article Information

 Title On nonsymmetric rank one singular perturbations of selfadjoint operators Source Methods Funct. Anal. Topology, Vol. 22 (2016), no. 2, 137-151 MathSciNet MR3522856 zbMATH 06665384 Milestones Received 22/09/2015; Revised 23/02/2016 Copyright The Author(s) 2016 (CC BY-SA)

### Authors Information

Mykola Dudkin
National Technical University of Ukraine "Kyiv Polytechnic Institute", 37 Prospect Peremogy, Kyiv, 03056, Ukraine

Tetiana Vdovenko
National Technical University of Ukraine "Kyiv Polytechnic Institute", 37 Prospect Peremogy, Kyiv, 03056, Ukraine

### Citation Example

Mykola Dudkin and Tetiana Vdovenko, On nonsymmetric rank one singular perturbations of selfadjoint operators, Methods Funct. Anal. Topology 22 (2016), no. 2, 137-151.

### BibTex

@article {MFAT841,
AUTHOR = {Dudkin, Mykola and Vdovenko, Tetiana},
TITLE = {On nonsymmetric rank one singular perturbations of selfadjoint operators},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {22},
YEAR = {2016},
NUMBER = {2},
PAGES = {137-151},
ISSN = {1029-3531},
MRNUMBER = {MR3522856},
ZBLNUMBER = {06665384},
URL = {http://mfat.imath.kiev.ua/article/?id=841},
}

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