Effective Hamiltonians and effective operators produce, respectively, exact energies and matrix elements of a time-independent operator A for a finite number of eigenstates of a time-independent Hamiltonian H. We obtain degenerate and quasidegenerate perturbative expressions for the particularly useful canonical effective operator A(C) through second order in perturbation theory. The corresponding A(C) diagrammatic expressions are derived for the case where A(C) acts in a complete finite space. Our first order results have been used previously for ab initio computations of dipole and transition dipole moments in diatomic hydrides and for testing the assumptions in semiempirical methods for dipole properties. A brief discussion is also provided on the computational labors required by first and second order A(C) many-body calculations, the derivation of A(C) diagrams when Ac acts in an incomplete finite space, and on the derivation of diagrammatic rules for A(C) in arbitrary perturbation order.