Effective Hamiltonians and effective operators act on a restricted model space to give the same energies and matrix elements as those of the full Hamiltonian and operators between the corresponding true eigenstates. For the effective Hamiltonian there are two "obvious" choices: the simplest non-Hermitian effective Hamiltonian and the canonical Hermitian effective Hamiltonian. In this paper, we derive a perturbative effective operator which works together with the non-Hermitian effective Hamiltonian, prove that it can be expanded with only connected diagrams, and show how to construct the connected diagrams easily from the diagrams of the effective Hamiltonian by substitution of vertices. This effective operator is much simpler than the Hermitian effective operator and therefore is expected to be more suitable for ab initio calculations.