Completeness property of one-dimensional perturbations of normal and spectral operators generated by first order systems

TL;DR

This paper investigates the completeness of rank-one perturbations of operators generated by boundary value problems for a specific 2x2 system, focusing on spectral properties and basis conditions in various subclasses of spectral operators.

Contribution

It demonstrates the existence of complete rank-one perturbations for operators in different subclasses, including normal and almost normal, and explores Riesz basis properties under certain conditions.

Findings

Existence of complete perturbations in all subclasses

Explicit formula for resolvent difference rank

Analysis of Riesz basis property for quasi-periodic BVPs

Abstract

The paper is concerned with completeness property of rank one perturbations of unperturbed operators generated by special boundary value problems (BVP) for the following $2 \times 2$ system \begin{equation} L y = -i B^{-1} y' + Q(x) y = \lambda y , \quad B = \begin{pmatrix} b_1 & 0 \\ 0 & b_2 \end{pmatrix}, \quad y = \begin{pmatrix} y_1 \\ y_2 \end{pmatrix}, \end{equation} on a finite interval assuming that a potential matrix $Q$ is summable, and $b_1 b_2^{-1} \notin \mathbb{R}$ (essentially non-Dirac type case). We assume that 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 (in general, 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 a potential matrix $Q$. We also find a simple formula for the rank of the resolvent difference for operators corresponding to two BVPs for $n \times n$ system in terms of the coefficients of boundary linear forms.