  The Cauchy problem for the von Neumann hierarchy of nonlinear equations is
investigated. One describes the evolution of all possible states of quantum
many-particle systems by the correlation operators. A solution of such
nonlinear equations is constructed in the form of an expansion over particle
clusters whose evolution is described by the corresponding order cumulant
(semi-invariant) of evolution operators for the von Neumann equations. For the
initial data from the space of sequences of trace class operators the existence
of a strong and a weak solution of the Cauchy problem is proved. We discuss the
relationships of this solution both with the $s$-particle statistical
operators, which are solutions of the BBGKY hierarchy, and with the
$s$-particle correlation operators of quantum systems.
