Minimal Vectors of Positive Operators

We use the method of minimal vectors to prove that certain classes of positive quasinilpotent operators on Banach lattices have invariant subspaces. We say that a collection of operators F on a Banach lattice X satisfies condition (*) if there exists a closed ball B(x0,r) in X such that x0 ≥ 0 and ∥x0∥ > r, and for every sequence (xn) in B(x0,r) ∩ [0,x0] there exists a subsequence ${\mathrm{x}}_{{\mathrm{n}}_{\mathrm{i}}}$ and a sequence Ki ∈ F such that ${\mathrm{K}}_{\mathrm{i}}{\mathrm{x}}_{{\mathrm{n}}_{\mathrm{i}}}$ converges to a non-zero vector. Let Q be a positive quasinilpotent operator on X, one-to-one, with dense range. Denote 〈Q] = {T ≥ 0 | TQ ≤ QT}. If either the set of all operators dominated by Q or the set of all contractions in 〈Q] satisfies (*), then 〈Q] has a common invariant subspace. We also show that if Q is a one-to-one quasinilpotent interval preserving operator on C0(Ω), then 〈Q] has a common invariant subspace.