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

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 × 2 system 1991 Mathematics Subject Classification. Primary 47E05; Secondary 34L10, 47B15. Key words and phrases. Systems of ordinary differential equations, normal operator, completeness of root vectors, resolvent operator, rank one perturbation, Riesz basis property.

Criterion of Bari basis property for $2 \times 2$ Dirac-type operators with strictly regular boundary conditions

“…Theorem 2.4. Let boundary conditions (1.6) be strictly regular and let Q ∈ L p ([0, 1]; C 2×2 ) for some p ∈ [1,2]. Then some normalized system of root vectors of the operator L U (Q) is a Bari (ℓ p ) *sequence in L 2 ([0, 1]; C 2 ) if and only if the operator L U (0) is self-adjoint, i.e.…”

“…Let T be an operator with compact resolvent in a separable Hilbert space H and let {λ n } n∈Z be a sequence of its eigenvalues counting multiplicities. Let also p ∈ [1,2]. Assume that for some N ∈ N eigenvalues λ n , |n| N, are algebraically simple.…”

“…Marchenko in [35,Chapter 1.3]. As a development of [34], in [1,24,26,2] completeness conditions for nonregular and even degenerate boundary conditions were found with applications to dissipative and normal operators. In the joint paper [26] the author and M.M.…”

“…Theorem 1.1 (Theorem 7.15 in [28]). Let K ∈ L p ([0, 1]; C 2×2 ) be a compact set for some p ∈ [1,2], let Q, Q ∈ K and boundary conditions (1.3) be strictly regular. Then for some normalized systems of root vectors {f n } n∈Z and { f n } n∈Z of the operators L U (Q) and L U ( Q) the following uniform relations hold for Q, Q ∈ K:…”

Criterion of Bari basis property for 2 × 2 Dirac‐type operators with strictly regular boundary conditions

“…Marchenko (see [37,Chapter 1.3]). As a development of [35,36], in [1,2,26,27], completeness conditions for nonregular and even degenerate boundary conditions were found with applications to dissipative and normal operators. In the joint paper [27], the author and M.M.…”