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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
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.
spectral synthesis, nonvanishing moments, domination, completeness, spectrum, invariant subspace, functional model.
|Russian Science Foundation
|Russian Foundation for Basic Research
||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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St. Petersburg Mathematical Journal, 2019, 30:5, 761–802
MSC: Primary 47B15; Secondary 47A55
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
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\paper Spectral theory of rank one perturbations of normal compact operators
\jour Algebra i Analiz
\jour St. Petersburg Math. J.
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Abakumov E., Baranov A., Belov Yu., “Krein-Type Theorems and Ordered Structure For Cauchy-de Branges Spaces”, J. Funct. Anal., 277:1 (2019), 200–226
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