Criterion of Bari basis property for 2 × 2 Dirac‐type operators with strictly regular boundary conditions

Abstract

AbstractThe paper is concerned with the Bari basis property of a boundary value problem associated in with the following 2 × 2 Dirac‐type equation for : with a potential matrix and subject to the strictly regular boundary conditions . If , this equation is equivalent to one‐dimensional Dirac equation. We show that the normalized system of root vectors of the operator is a Bari basis in if and only if the unperturbed operator 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 , , boundary conditions be strictly regular, and let be the sequence biorthogonal to the normalized system of root vectors of the operator . Then, These abstract results are applied to noncanonical initial‐boundary value problem for a damped string equation.