Paper detail

On the Riesz basis property of root vectors system for $2 \times 2$ Dirac type operators

The paper is concerned with the Riesz basis property of a boundary value problem associated in $L^2[0,1] \otimes \mathbb{C}^2$ with the following $2 \times 2$ Dirac type equation $$ L y = -i B^{-1} y' + Q(x) y = λy, \quad B = \begin{pmatrix} b_1 & 0 \\ 0 & b_2 \end{pmatrix}, \quad y = \begin{pmatrix} y_1 \\ y_2 \end{pmatrix}, \quad (1) $$ with a summable potential matrix $Q \in L^1[0,1] \otimes \mathbb{C}^{2 \times 2}$ and $b_1 < 0 < b_2$. If $b_2 = -b_1 =1$ this equation is equivalent to one dimensional Dirac equation. It is proved that the system of root functions of a linear boundary value problem constitutes a Riesz basis in $L^2[0,1] \otimes \mathbb{C}^2$ 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 $b_1 + b_2 \not = 0$. 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 $b_1 + b_2 \not = 0.$ The proof of the main result is based on existence of triangular transformation operators for system (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.