Starting from a decomposition of the Hamiltonian H( x) of the nuclear many-body problem in the form H( x) = H 0 + xV, where H 0 is a shell-model Hamiltonian, V the residual interaction, and x a strength parameter, we introduce a general effective interaction W( x) describing the interaction of nucleons within a shell, and the associated effective operators A ̂ (x) . We display some properties of these operators. From a particular choice of W( x) we obtain the expressions introduced earlier by several authors. The convergence of the expansions for W( x) and A ̂ (x) in powers of x is investigated. It is shown that W( x) and A ̂ (x) are holomorphic in a domain of the complex x-plane including the point x = 0. With the help of a generalization of the von Neumann-Wigner noncrossing rule, we exhibit the nature of the common singularity of W( x) and A ̂ (x) which is closest to the origin and thus defines the radius r 0 of convergence of the expansions of W and A ̂ . It is shown that r 0 is unaffected by the cancellation of unlinked diagrams. A criterion of consistency is established, which shows that most of the practical calculations of W lead to results which are inconsistent with the definition of W.