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

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
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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
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
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
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
4. Putinar M., Yakubovich D., “Spectral Dissection of Finite Rank Perturbations of Normal Operators”, J. Operat. Theor., 85:1 (2021), 45–78
5. A. Yu. Trynin, “O skhodimosti obobschenii sink-approksimatsii na klasse Privalova–Chanturiya”, Sib. zhurn. industr. matem., 24:3 (2021), 122–137
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