2012/12/24 by Anton Baranov, Baranov, Anton D., Dmitry V. Yakubovich +1
Engineering · Mathematics · #34L10 #47A55 #47B25 #47B32 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Material Science and Thermodynamics #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1212.5965
openalex publication_date 2012/12/24 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
We study spectral properties of nonselfadjoint rank one perturbations of\ncompact selfadjoint operators. The problems under consideration include\ncompleteness of eigenvectors, relations between completeness of the perturbed\noperator and its adjoint, and the spectral synthesis problem. We obtain new\ncriteria for completeness and spectral synthesis in this class as well as a\nseries of counterexamples which show that the spectral structure of rank one\nperturbations is, in general, unexpectedly rich and complicated.\n A parallel spectral theory is developed for one-dimensional singular\nperturbations of unbounded selfadjoint operators. Our approach is based on a\nfunctional model for this class which translates the properties of operators to\ncompleteness problems for systems of reproducing kernels and their\nbiorthogonals in some spaces of analytic (entire) functions.\n