On a normed space X ordered by a cone K we consider a continuous linear operator A from X to X of the following kind: If a positive continuous functional f attains 0 on some positive element x, then f(Ax) is greater or equal to 0. If X is a vector lattice, then such operators can be represented as sI + B, where B is a positive operator, I is the identity and s is a real number. We generalize this assertion for weaker assumptions on X, using the Riesz decomposition property.