On the Riesz basis property of root vectors system for 2 × 2 Dirac type operators

Abstract

The paper is concerned with the Riesz basis property of a boundary value problem associated in L 2 [0, 1] ⊗ C 2 with the following 2 × 2 Dirac type equationIt is proved that the system of root functions of a linear boundary value problem constitutes a Riesz basis in L 2 [0, 1] ⊗ C 2 provided that the boundary conditions are strictly regular. By analogy with the case of ordinary differential equations, boundary conditions are called strictly regular if the eigenvalues of the corresponding unperturbed (Q = 0) operator are asymptotically simple and separated. As distinguished from the Dirac case there is no simple algebraic criterion of the strict regularity whenever b1 + b2 = 0. However under certain restrictions on coefficients of the boundary linear forms we present certain algebraic criteria of the strict regularity in the latter case. In particular, it is shown that regular separated boundary conditions are always strictly regular while periodic (antiperiodic) boundary conditions are strictly regular if and only if b1 + b2 = 0.The proof of the main result is based on existence of triangular transformation operators for system (0.1). Their existence is also established here in the case of a summable Q. In the case of regular (but not strictly regular) boundary conditions we prove the Riesz basis property with parentheses. The main results are applied to establish the Riesz basis property of the dynamic generator of spatially non-homogenous damped Timoshenko beam model.2010 Mathematics Subject Classification. 47E05, 34L40, 34L10, 35L35.

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

“…In the sequel we follow the scheme proposed in [27] for investigating the Riesz basis property of BVP for Dirac type system (B = B * ) with a summable potential matrix Q. The following result is similar to that of Proposition 3.1 from [27].…”

“…Next we obtain a formula for the characteristic determinant ∆(·) of the BVP (3.1)-(3.3) similar to that used in [27,Lemma 4.1].…”

“…(i) The Riesz basis property for 2 × 2 Dirac operators L U 1 ,U 2 has been investigated in numerous papers (see [41,8,6,9,11,12,18,24,27,39] and references therein). The most complete result was recently obtained independently and by different methods in [24,27] and [39]. Namely, assuming that B = B * and Q(·) ∈ L 1 ([0, 1]; C 2×2 ) it is proved in [24,27] (the general case of b 1 b 2 < 0) and [39] (the Dirac case, b 1 = −b 2 ) that the system of root vectors of equation (1.3) subject to regular boundary conditions constitutes a Riesz basis with parentheses in L 2 ([0, 1]; C 2 ) and ordinary Riesz basis provided that BC are strictly regular.…”

“…(ii) Note also that an important role in proving Riesz basis property in [24,27] and [39] is playing the following asymptotic formula λ n = λ 0 n + o(1), as n → ∞, n ∈ Z, (3.36) for the eigenvalues {λ n } n∈Z of the operator L C,D (B, Q) with regular BC (and summable potential matrix Q), where {λ 0 n } n∈Z is the sequence of eigenvalues of the unperturbed operator L C,D (B, 0). Note also that formula (3.36) has recently been applied to investigation of spectral properties of Dirac systems on star graphs [1].…”

“…Besides, P. Djakov and B. Mityagin [10] imposing certain smoothness condition on Q(·) proved equiconvergence of the spectral decompositions for 2 × 2 Dirac equations subject to general regular boundary conditions. Moreover, the Riesz basis property for 2 × 2 Dirac operators L U 1 ,U 2 has been investigated in numerous papers (see [41,8,6,9,11,12,18,24,27,39] and references therein, and discussion in Remark 3.14).…”