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

The paper is concerned with the Bari basis property of a boundary value problem associated in L-2([0, 1];C-2) with the following 2 x 2 Dirac-type equation for y = col(y(1),y(2)): [GRAPHICS] . with a potential matrix Q is an element of L-2([0, 1];C-2x2) and subject to the strictly regular boundary conditions Uy := {U-1,U-2}y = 0. If b(2) = -b(1) = 1, this equation is equivalent to one-dimensional Dirac equation. We show that the normalized system of root vectors {f(n)}n is an element of Z of the operatorL(u)(Q) is a Bari basis in L-2([0, 1];C-2) if and only if the unperturbed operator L-u(0) is self-adjoint. We also give explicit conditions for this in terms of coefficients in the boundary conditions. The Bari basis criterion is a consequence of our more general result: Let Q is an element of L-p([0, 1];C-2x2),p is an element of [1,2], boundary conditions be strictly regular, and let {gn}(n is an element of Z) be the sequence biorthogonal to the normalized system of root vectors {f(n)}(n is an element of Z) of the operator L-u(Q). Then, [GRAPHICS] . These abstract results are applied to noncanonical initial-boundary value problem for a damped string equation.