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

This article is cited in 23 scientific papers (total in 23 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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    This publication is cited in the following articles:
    1. A. M. Savchuk, A. A. Shkalikov, “Spectral Properties of the Complex Airy Operator on the Half-Line”, Funct. Anal. Appl., 51:1 (2017), 66–79  mathnet  crossref  crossref  mathscinet  isi  elib
    2. T. Sh. Kal'menov, B. T. Torebek, “On an ill-posed problem for the Laplace operator with nonlocal boundary condition”, Eurasian Math. J., 8:1 (2017), 50–57  mathnet
    3. E. A. Shiryaev, “Basis Property of Eigen- and Associated Functions of an Operator with Nondense Domain of Definition in the Example of the Orr–Sommerfeld problem”, Math. Notes, 102:6 (2017), 867–872  mathnet  crossref  crossref  isi  elib
    4. S. N. Tumanov, A. A. Shkalikov, “Eigenvalue dynamics of a $\mathcal{PT}$-symmetric Sturm–Liouville operator and criteria for similarity to a self-adjoint or a normal operator”, Dokl. Math., 96:3 (2017), 607–611  crossref  crossref  mathscinet  zmath  isi  elib
    5. M. Mahinzaeim, “Spectral properties and stability of a nonselfadjoint Euler-Bernoulli beam”, Methods Funct. Anal. Topol., 23:4 (2017), 346–366  mathscinet  zmath  isi
    6. 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  mathnet
    7. A. M. Savchuk, “Operator tipa Kalderona—Zigmunda i ego svyaz s asimptoticheskimi otsenkami dlya obyknovennykh differentsialnykh operatorov”, Differentsialnye i funktsionalno-differentsialnye uravneniya, SMFN, 63, no. 4, Rossiiskii universitet druzhby narodov, M., 2017, 689–702  mathnet  crossref
    8. I. N. Braeutigam, D. M. Polyakov, “On the asymptotics of eigenvalues of a fourth-order differential operator with matrix coefficients”, Differ. Equ., 54:4 (2018), 450–467  crossref  crossref  zmath  isi  elib  scopus
    9. Kh. K. Ishkin, “Conditions of spectrum localization for operators not close to self-adjoint operators”, Dokl. Math., 97:2 (2018), 170–173  mathnet  crossref  crossref  zmath  isi  elib  scopus
    10. D. M. Polyakov, “A one-dimensional Schrödinger operator with square-integrable potential”, Siberian Math. J., 59:3 (2018), 470–485  mathnet  crossref  crossref  isi  elib
    11. Ya. Sh. Il'yasov, N. F. Valeev, “On inverse spectral problem and generalized Sturm nodal theorem for nonlinear boundary value problems”, Ufimsk. matem. zhurn., 10:4 (2018), 123–129  mathnet  crossref  mathscinet  isi
    12. N. F. Valeev, E. A. Nazirova, “Pryamaya i obratnaya spektralnye zadachi v teorii kolebanii uprugoi plastiny s dopolnitelnymi tochechnymi vzaimodeistviyami”, Matematicheskaya fizika, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 152, VINITI RAN, M., 2018, 25–33  mathnet  mathscinet
    13. S. S. Mirzoyev, A. T. Gazilova, “On the Completeness of a Part of Root Vectors for a Class of Third-Order Quasi-Elliptic Operator Pencils”, Math. Notes, 105:5 (2019), 798–801  mathnet  crossref  crossref  isi  elib
    14. Shkalikov A.A., “Basis Properties of Root Functions of Differential Operators With Spectral Parameter in the Boundary Conditions”, Differ. Equ., 55:5 (2019), 631–643  crossref  isi
    15. Vladykina V.E., Shkalikov A.A., “Spectral Properties of Ordinary Differential Operators With Involution”, Dokl. Math., 99:1 (2019), 5–10  crossref  isi
    16. 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  elib
    17. V. E. Vladykina, A. A. Shkalikov, “Regular Ordinary Differential Operators with Involution”, Math. Notes, 106:5 (2019), 674–687  mathnet  crossref  crossref  isi  elib
    18. Agibalova A.V. Lunyov A.A. Malamud M.M. Oridoroga L.L., “Completeness Property of One-Dimensional Perturbations of Normal and Spectral Operators Generated By First Order Systems”, Integr. Equ. Oper. Theory, 91:4 (2019), UNSP 37  crossref  isi
    19. Kukushkin V M., “Asymptotics of Eigenvalues For Differential Operators of Fractional Order”, Fract. Calc. Appl. Anal., 22:3 (2019), 658–680  crossref  isi
    20. Kostin A.B., Sherstyukov V.B., “Basis Property of the System of Root Functions of the Oblique Derivative Problem For the Laplace Operator in a Disk”, Differ. Equ., 55:10 (2019), 1349–1361  crossref  isi
    21. Valeev N.F., Nazirova E.A., Azizova R.G., “Multiparameter Inverse Spectral Problems in the Oscillation Model of An Orthotropic Plate”, Azerbaijan J. Math., 9:2 (2019), 88–99  isi
    22. Boulton L., “Perturbations of Gibbs Semigroups and the Non-Selfadjoint Harmonic Oscillator”, J. Funct. Anal., 278:7 (2020), UNSP 108415  crossref  isi
    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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