A remark on invariant subspaces of positive operators (Q2855913)

Let \(S, T, R, K\) be non-zero positive operators on a Banach lattice \(X\) such that (a) \(T\) is non-scalar and commutes with \(S\) and \(R\); (b) \(K\) is compact and \(R\leq K\). Under these hypotheses, it is proved that either \(S\) has an invariant closed ideal or \(T\) commutes with a non-zero compact operator, and therefore has a hyperinvariant subspace. An important consequence is as follows. Suppose that \(T, R, K\) are as before and \((S_{n})_{n}\) is a sequence of positive operators commuting with \(T\). Then there is a closed subspace invariant under \(T, R\) and all \(S_{n}\)'s. The paper also includes several examples discussing the optimality of the hypotheses.