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.

Abstract. The paper is concerned with the completeness property of root functions of general boundary value problems for n × n first order systems of ordinary differential equations on a finite interval. In comparison with the recent paper [45] we substantially relax the assumptions on boundary conditions guarantying the completeness of root vectors, allowing them to be nonweakly regular and even degenerate. Emphasize that in this case the completeness property substantially depends on the values of a potential matrix at the endpoints of the interval.It is also shown that the system of root vectors of the general n × n Dirac type system subject to certain boundary conditions forms a Riesz basis with parentheses. We also show that arbitrary complete dissipative boundary value problem for Dirac type operator with a summable potential matrix admits the spectral synthesis in L 2 ([0, 1]; C n ). Finally, we apply our results to investigate completeness and the Riesz basis property of the dynamic generator of spatially non-homogenous damped Timoshenko beam model.

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