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

Abstract

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 $$2 \times 2$$ system 0.1 $$\begin{aligned}&L y = -i B^{-1} y' + Q(x) y = \lambda y , \quad B = \begin{pmatrix} b_1 &{}\quad 0 \\ 0 &{}\quad b_2 \end{pmatrix}, \quad y = \begin{pmatrix} y_1 \\ y_2 \end{pmatrix}, \end{aligned}$$ on a finite interval assuming that the potential matrix Q is summable, and $$b_1 b_2^{-1} \notin \mathbb {R}$$ (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 $$n \times n$$ system in terms of the coefficients of linear boundary forms.