The uniqueness criterion for a solution to a boundary value problem for the operator $\frac{\partial ^{2p} }{\partial t^{2p}}-A$ with an elliptic operator $A$ of arbitrary order

We establish the uniqueness criterion for a solution to the operator $\frac{\partial ^{2p}}{\partial t^{2p}}-A(x,D)$ with the Dirichlet time-dependent boundary conditions and general boundary conditions in the space variables. The order of the differentiation operator $\frac{\partial ^{2p} }{\partial t^{2p}}$ is assumed even. Note that $A(x,D)$ in the space variables is an arbitrary elliptic operator with some rather general boundary operators $B_j$ obeying the conventional Agmon conditions. The Agmon conditions ensure the existence of a complete orthonormal system of eigenfunctions (in $L_2(\Omega)$) provided that $\Omega$ is a bounded domain with sufficiently smooth boundary.