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Eurasian Math. J., 2017, Volume 8, Number 1, Pages 119–127 (Mi emj251)  

This article is cited in 2 scientific papers (total in 2 papers)

Unconditional bases of subspaces related to non-self-adjoint perturbations of self-adjoint operators

A. K. Motovilovab, A. A. Shkalikovc

a Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 6 Joliot-Curie St, 141980 Dubna, Moscow Region, Russia
b Faculty of Natural and Engineering Sciences, Dubna State University, 19 Universitetskaya St, 141980 Dubna, Moscow Region, Russia
c Faculty of Mathematics and Mechanics, M.V. Lomonosov Moscow State University, 1 Leninskiye Gory St, 119991 Moscow GSP-1, Russia

Abstract: Assume that $T$ is a self-adjoint operator on a Hilbert space $\mathcal{H}$ and that the spectrum of $T$ is contained in the union $\bigcup_{j\in J}\Delta_j$, $J\subseteq\mathbb{Z}$, of the segments $\Delta_j = [\alpha_j,\beta_j]\subset \mathbb{R}$ such that $\alpha_{j+1}>\beta_j$ and
$$ \inf_j(\alpha_{j+1}-\beta_j)=d>0. $$
If $B$ is a bounded (in general non-self-adjoint) perturbation of $T$ with $||B||=:b<d/2$, then the spectrum of the perturbed operator $A=T+B$ lies in the union $\bigcup_{j\in J}U_b(\Delta_j)$ of the mutually disjoint closed $b$-neighborhoods $U_b(\Delta_j)$ of the segments $\Delta_j$ in $\mathbb{C}$. Let $Q_j$ be the Riesz projection onto the invariant subspace of $A$ corresponding to the part of the spectrum of $A$ lying in $U_b(\Delta_j)$, $j\in J$. Our main result is as follows: The subspaces $\mathcal{L}_j=Q_j(\mathcal{H})$, $j \in J$ form an unconditional basis in the whole space $\mathcal{H}$.

Keywords and phrases: Riesz basis, unconditional basis of subspaces, non-self-adjoint perturbations.

Funding Agency Grant Number
Russian Foundation for Basic Research
Deutsche Forschungsgemeinschaft
This work was supported by the Deutsche Forschungsgemeinschaft (DFG) and by the Russian Foundation for Basic Research (RFBR).


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Bibliographic databases:
MSC: 47A55, 47A15
Received: 23.01.2017
Language:

Citation: A. K. Motovilov, A. A. Shkalikov, “Unconditional bases of subspaces related to non-self-adjoint perturbations of self-adjoint operators”, Eurasian Math. J., 8:1 (2017), 119–127

Citation in format AMSBIB
\Bibitem{MotShk17}
\by A.~K.~Motovilov, A.~A.~Shkalikov
\paper Unconditional bases of subspaces related to non-self-adjoint perturbations of self-adjoint operators
\jour Eurasian Math. J.
\yr 2017
\vol 8
\issue 1
\pages 119--127
\mathnet{http://mi.mathnet.ru/emj251}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000411744800008}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. K. Motovilov, A. A. Shkalikov, “Sokhranenie svoistva bezuslovnoi bazisnosti pri nesamosopryazhennykh vozmuscheniyakh samosopryazhennykh operatorov”, Funkts. analiz i ego pril., 53:3 (2019), 45–60  mathnet  crossref  mathscinet  elib
    2. B. D. Djordjevic, “On a singular sylvester equation with unbounded self-adjoint a and B”, Complex Anal. Oper. Theory, 14:4 (2020), 43  crossref  mathscinet  zmath  isi  scopus
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