Completeness and spectral synthesis of nonselfadjoint one-dimensional perturbations of selfadjoint operators

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• 作者：Anton D. Baranov^a ; ^{a.d.baranov@spbu.ru" class="auth_mail" title="E-mail the corresponding author} ; Dmitry V. Yakubovich^b ; ^c ; ^{dmitry.yakubovich@uam.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 34L10 ; 47B32 ; 47A55 ; secondary ; 47A45
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：22 October 2016
• 年：2016
• 卷：302
• 期：Complete
• 页码：740-798
• 全文大小：989 K

文摘

We study spectral properties of nonselfadjoint rank one perturbations of compact selfadjoint operators. The problems under consideration include completeness of eigenvectors, relations between completeness of the perturbed operator and its adjoint, and the spectral synthesis problem. We obtain new criteria for completeness and spectral synthesis in this class as well as a series of counterexamples which show that the spectral structure of rank one perturbations is, in general, unexpectedly rich and complicated.

A parallel spectral theory is developed for one-dimensional singular perturbations of unbounded selfadjoint operators. Our approach is based on a functional model for this class which translates the properties of operators to completeness problems for systems of reproducing kernels and their biorthogonals in some spaces of analytic (entire) functions.