  In this paper the general spectral properties of linear operators in Banach
spaces are studied. We find sufficient conditions on structure of Banach spaces
and resolvent properties that guarantee completeness of roots elements of
Schatten class operators. This approach generalizes the well known result for
operators in Hilbert spaces. In application, the boundary value problems for
the abstract equation of second order with variable coefficients are studied.
The principal part of the appropriate differential operator is not
self-adjoint. The discreetness of spectrum and completeness of root elements of
this operator are obtained.
