  We decompose the Hilbert space of wave functions into two subspaces, and
assign to a given observable two effective representatives that act in the
model space. The first serves to determine some of the eigenvalues of the full
observable, while the second serves to determine its matrix elements, in any
basis in one of the subspaces, in terms of quantities pertaining to the model
space. We also show that if the Hamiltonian of a physical system possesses
symmetries then these symmetries continue to hold for its effective
representatives of the first type. Maximum information about the system can be
obtained in terms of two sets of effective representatives. The first set of
representatives is complete. Other observables that do not commute with all
members of the complete set have only one type of representative.
