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Uspekhi Mat. Nauk, 2016, Volume 71, Issue 5(431), Pages 113–174 (Mi umn9740)  

This article is cited in 28 scientific papers (total in 28 papers)

Perturbations of self-adjoint and normal operators with discrete spectrum

A. A. Shkalikov

Moscow State University

Abstract: The spectral properties of operators of the form $A=T+B$ are analyzed, where $B$ is a non-symmetric operator subordinate to a self-adjoint or normal operator $T$. The different definitions of perturbations with respect to $T$ are considered: completely subordinated, subordinate with order $p<1$, locally subordinate. Analogues of these types of perturbations are considered also for operators defined in terms of quadratic forms. For perturbations of different types, series of statements on the completeness property of the root vectors of the operator and on the basis or unconditional basis property are proved. The spectra of the operators $T$ and $T+B$ are compared as well. A survey of research in this area is presented.
Bibliography: 89 titles.

Keywords: perturbations of linear operators, resolvent estimates, conditions for $p$-subordination, conditions for local subordination, sums of the quadratic forms of operators, unconditional bases, Riesz bases, Abel–Lidskii summability method.

Funding Agency Grant Number
Russian Science Foundation 14-11-00754
This work was carried out with the support of the Russian Science Foundation (project no. 14-11-00754).


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English version:
Russian Mathematical Surveys, 2016, 71:5, 907–964

Bibliographic databases:

UDC: 517.984
MSC: Primary 47A55; Secondary 47B15, 47B25
Received: 25.07.2016

Citation: A. A. Shkalikov, “Perturbations of self-adjoint and normal operators with discrete spectrum”, Uspekhi Mat. Nauk, 71:5(431) (2016), 113–174; Russian Math. Surveys, 71:5 (2016), 907–964

Citation in format AMSBIB
\by A.~A.~Shkalikov
\paper Perturbations of self-adjoint and normal operators with discrete spectrum
\jour Uspekhi Mat. Nauk
\yr 2016
\vol 71
\issue 5(431)
\pages 113--174
\jour Russian Math. Surveys
\yr 2016
\vol 71
\issue 5
\pages 907--964

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    23. Kh. K. Ishkin, R. I. Marvanov, “Equivalence criterion for two asymptotic formulae”, Ufa Math. J., 12:1 (2020), 30–42  mathnet  crossref  isi
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