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Algebra i Analiz, 2018, Volume 30, Issue 5, Pages 1–56 (Mi aa1613)  

This article is cited in 5 scientific papers (total in 5 papers)

Research Papers

Spectral theory of rank one perturbations of normal compact operators

A. D. Baranovab

a Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia
b National Research University Higher School of Economics, St. Petersburg, Russia

Abstract: A functional model is constructed for rank one perturbations of compact normal operators that act in a certain Hilbert spaces of entire functions generalizing the de Branges spaces. By using this model, completeness and spectral synthesis problems are studied for such perturbations. Previously, the spectral theory of rank one perturbations was developed in the selfadjoint case by D. Yakubovich and the author. In the present paper, most of known results in the area are extended and simplified significantly. Also, an ordering theorem for invariant subspaces with common spectral part is proved. This result is new even for rank one perturbations of compact selfadjoint operators.

Keywords: spectral synthesis, nonvanishing moments, domination, completeness, spectrum, invariant subspace, functional model.

Funding Agency Grant Number
Russian Science Foundation 14-21-00035
Russian Foundation for Basic Research 17-51-150005-NCNI-a
CNRS, France PRC CNRS/RFBR 2017-2019
Theorems 2.1–2.6 and the results of §§ 3–6 were obtained with the support of Russian Science Foundation project № 14-21-00035. Theorems 2.7 and 2.8 and the results of §§ 7,8 were obtained as a part of joint grant of Russian Foundation for Basic Research (project № 17-51-150005-NCNI-a) and CNRS, France (project PRC CNRS/RFBR 2017-2019 “Noyaux reproduisants dans des espaces de Hilbert de fonctions analytiques”).

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English version:
St. Petersburg Mathematical Journal, 2019, 30:5, 761–802

Bibliographic databases:

MSC: Primary 47B15; Secondary 47A55
Received: 15.03.2018

Citation: A. D. Baranov, “Spectral theory of rank one perturbations of normal compact operators”, Algebra i Analiz, 30:5 (2018), 1–56; St. Petersburg Math. J., 30:5 (2019), 761–802

Citation in format AMSBIB
\by A.~D.~Baranov
\paper Spectral theory of rank one perturbations of normal compact operators
\jour Algebra i Analiz
\yr 2018
\vol 30
\issue 5
\pages 1--56
\jour St. Petersburg Math. J.
\yr 2019
\vol 30
\issue 5
\pages 761--802

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    This publication is cited in the following articles:
    1. E. Abakumov, A. Baranov, Yu. Belov, “Krein-type theorems and ordered structure for Cauchy-de branges spaces”, J. Funct. Anal., 277:1 (2019), 200–226  crossref  isi
    2. A. V. Agibalova, A. A. Lunyov, M. M. Malamud, L. L. Oridoroga, “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  mathscinet  isi  scopus
    3. A. Yu. Trynin, “On the uniform approximation of functions of bounded variation by Lagrange interpolation polynomials with a matrix ${\mathcal L}_n^{(\alpha_n,\beta_n)}$ of Jacobi nodes”, Izv. Math., 84:6 (2020), 1224–1249  mathnet  crossref  crossref  mathscinet  isi  elib
    4. Putinar M., Yakubovich D., “Spectral Dissection of Finite Rank Perturbations of Normal Operators”, J. Operat. Theor., 85:1 (2021), 45–78  crossref  mathscinet  isi  scopus
    5. A. Yu. Trynin, “O skhodimosti obobschenii sink-approksimatsii na klasse Privalova–Chanturiya”, Sib. zhurn. industr. matem., 24:3 (2021), 122–137  mathnet  crossref
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