  In this paper we deal with a second order multidimensional fractional
differential operator. We consider a case where the leading term represented by
the uniformly elliptic operator and the final term is the Kipriyanov operator
of fractional differentiation. We conduct classification of such a type of
operators by belonging of their resolvent to the Schatten-von Neumann class and
formulate the sufficient condition for completeness of the root functions
system. Finally we obtain an asymptotic formula.
