  We develop a rigorous formalism for the description of the evolution of
observables in quantum systems of particles. We construct a solution of the
initial-value problem to the quantum dual BBGKY hierarchy of equations as an
expansion over particle clusters whose evolution are governed by the
corresponding-order dual cumulant (dual semi-invariant) of the evolution
operators of finitely many particles. For initial data from the space of
sequences of bounded operators the existence and uniqueness theorem is proved.
