On the Properties of a Dirac Operator with Irregular BoundaryConditions

We study the basis properties of root functions of a $$2\times 2 $$ Dirac operator with integrable complex-valued potential and irregular boundary conditions. Under certain conditions on the spectrum of the operator, we prove that the root function system of this operator is incomplete in the space $$L_2(0,\pi )\oplus L_2(0,\pi )$$ but forms an unconditional basis in the closed linear span of itself.