On solvability of some boundary value problems for elliptic type operator-differential equations

Conditions on the coefficients of a class of higher order operator-differential equations that provide solvability of some boundary value problems well-posed for these equations, are obtained in the paper. These boundary value problems are generalizations of some mixed problems for a semiharmonic equation. Relation of solvability of boundary value problems with the sharp value of the norms of intermediate derivative operators in some subspaces are found. The obtained results are applied to the proof of multiple completeness of some system of derivatives of chains of eigen and adjoint vectors corresponding to eigenvalues of the appropriate polynomial operator bundle on the half-plane, and to the completeness of decreasing elementary solutions of a homogeneous equation.