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

Abstract

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 × 2 Dirac-type equation for y = col(y 1 , y 2 ):) and subject to the strictly regular boundary conditionsWe show that the system of root vectors {f n } n∈Z of the operator L U (Q) forms 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:boundary conditions be strictly regular, and let {g n } n∈Z be the sequence biorthogonal to the system of root vectors {f n } n∈Z of the operator L U (Q). ThenThese abstract results are applied to non-canonical initial-boundary value problem for a damped string equation.