  For many-particle systems defined on lattices we investigate the global
structure of effective Hamiltonians and observables obtained by means of a
suitable basis transformation. We study transformations which lead to effective
Hamiltonians conserving the number of excitations. The same transformation must
be used to obtain effective observables. The analysis of the structure shows
that effective operators give rise to a simple and intuitive perspective on the
initial problem. The systematic calculation of n-particle irreducible
quantities becomes possible constituting a significant progress. Details how to
implement the approach perturbatively for a large class of systems are
presented.
