  In this paper we consider evolution equations in the abstract Hilbert space
under the special conditions imposed on the operator at the right-hand side of
the equation. We establish the method that allows us to formulate the existence
and uniqueness theorem and find a solution in the form of a series on the root
vectors of the right-hand side. As an application we consider fractional
differential equations of various kinds. Such operators as the
Riemann-Liouville fractional differential operator, the Riesz potential, the
difference operator have been involved.
