Completeness of elementary solutions to a class of second order operator-differential equations

We present conditions of solvability of a boundary value problem for a class of second order operator-differential equations on a finite segment, study the behavior of the resolvent of the corresponding operator pencil, prove the double completeness of a system of the derived chains of eigenvectors and associated vectors corresponding to a boundary value problem on a segment, and establish the completeness of elementary solutions to the homogeneous equation in the solution space.