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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Spectral theory of rank one perturbations of normal compact operators

Author: A. D. Baranov
Original publication: Algebra i Analiz, tom 30 (2018), nomer 5.
Journal: St. Petersburg Math. J. 30 (2019), 761-802
MSC (2010): Primary 47B15; Secondary 47A55
Published electronically: July 26, 2019
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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.

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Additional Information

A. D. Baranov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia; National Research University, Higher School of Economics, St. Petersburg, Russia

Keywords: Spectral synthesis, nonvanishing moments, domination, completeness, spectrum, invariant subspace, functional model
Received by editor(s): March 15, 2018
Published electronically: July 26, 2019
Additional Notes: Theorems 2.1–2.6 and the results of §§3–6 were obtained with the support of Russian Science Foundation project no. 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 no. 17-51-150005-NCNI-a) and CNRS, France (project PRC CNRS/RFBR 2017–2019 “Noyaux reproduisants dans des espaces de Hilbert de fonctions analytiques”).
Article copyright: © Copyright 2019 American Mathematical Society