Let (M,g) be a closed Riemannian manifold of dimension n ≥ 3 and let f ∈ C^∞(M), such that the operator P_f := Δ_g + f is positive. If g is flat near some point p and f vanishes around p, we can define the mass of P_f as the constant term in the expansion of the Green function of P_f at p. In this paper, we establish many results on the mass of such operators. In particular, if f = \frac{n-2}{4(n-1)}\scal_g, i.e. if P_f is the Yamabe operator, we show the following result: assume that there exists a closed simply connected non-spin manifold M such that the mass is non-negative for every metric g as above on M, then the mass is non-negative for every such metric on every closed manifold of the same dimension as M.