Let B B be a real vector lattice and a Banach space under a semimonotonic norm. Suppose T T is a linear operator on B B which is positive and eventually compact, y y is a positive vector, and λ \lambda is a positive real. It is shown that ( λ I − T ) − 1 y {(\lambda I - T)^{ - 1}}y is positive if, and only if, y y is annihilated by the absolute value of any generalized eigenvector of T ∗ {T^\ast } associated with a strictly positive eigenvalue not less than λ \lambda . A strictly positive eigenvalue is a positive eigenvalue having an associated positive eigenvector. For the case of B = L p B = {L^p} this yields the result that ( λ I − T ) − 1 y ≧ 0 {(\lambda I - T)^{ - 1}}y \geqq 0 if, and only if, y y is almost everywhere zero on a certain set which depends on λ \lambda but is otherwise fixed.