In many cases the self-adjoint projection of a Lebesgue space L2(dx) onto a closed subspace is also bounded on a weighted space L2(wdx) . Our main result is that in this case certain self-adjoint projections on weighted spaces are bounded on L2(dx) . The analysis also produces an invertibility criterion for certain Toeplitz operators. The proof is based on analysis of a perturbation series and hence is valid in fairly general circumstances.