This paper is devoted to the development of an operator formulation of the recent extension of the centroid molecular dynamics method [J. Chem. Phys. 110, 3647 (1999); 111, 5303 (1999)] to boson and fermion systems. An operator calculus is used to rederive the basic equations of centroid dynamics. The following generalization to the case of systems of many indistinguishable particles is based on the use of a projection operator. Two different definitions of the quasi-density operator for bosonic and fermionic systems are suggested. The first definition allows an exact evaluation of equilibrium properties for systems with exchange effects using classical-like molecular dynamics calculations. The second one provides a formal justification of Bose-Einstein/Fermi-Dirac centroid dynamics with the same set of approximations as for Boltzmann statistics, and can be used to extract quantum dynamical information. In this case, the corresponding centroid correlation function can be related to a double Kubo transformed quantum mechanical one.