Completeness Property of One-Dimensional Perturbations of Normal and Spectral Operators Generated by First Order Systems

The paper is concerned with the completeness property of rank one perturbations of the unperturbed operators generated by special boundary value problems (BVP) for the following 2 × 2 system Ly=-iB-1y′+Q(x)y=λy,B=(b100b2),y=(y1y2),on a finite interval assuming that the potential matrix Q is summable, and b1b2-1∉R (essentially non-Dirac type case). We assume that the unperturbed operator generated by a BVP belongs to one of the following three subclasses of the class of spectral operators: (a) normal operators; (b) operators similar either to a normal or almost normal; (c) operators that meet Riesz basis property with parentheses; We show that in each of the three cases there exists (non-unique) operator generated by a quasi-periodic BVP and its certain rank-one perturbations (in the resolvent sense) generated by special BVPs which are complete while their adjoint are not. In connection with the case (b) we investigate Riesz basis property of quasi-periodic BVP under certain assumptions on the potential matrix Q. We also find a simple formula for the rank of the resolvent difference for operators corresponding to two BVPs for n× n system in terms of the coefficients of linear boundary forms. © 2019, Springer Nature Switzerland AG.

Авторы
Agibalova A.V.1 , Lunyov A.A.2 , Malamud M.M. 3 , Oridoroga L.L.1
Издательство
Springer Basel
Номер выпуска
4
Язык
Английский
Статус
Опубликовано
Номер
37
Том
91
Год
2019
Организации
  • 1 Donetsk National University, Universitetskaya Str. 24, Donetsk, 83055, Ukraine
  • 2 Facebook, Inc., 1 Hacker Way, Menlo Park, CA 94025, United States
  • 3 Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation
Ключевые слова
Completeness of root vectors; Normal operator; Rank one perturbation; Resolvent operator; Riesz basis property; Systems of ordinary differential equations
Дата создания
24.12.2019
Дата изменения
24.12.2019
Постоянная ссылка
https://repository.rudn.ru/ru/records/article/record/55123/