Invariant subspaces of super left-commutants

For a positive operator $Q$ on a Banach lattice, one defines $\langle Q]=\{T\geq 0\,:\,~TQ\leq QT\}$ and $[Q\rangle=\{T\geq 0\,:\, TQ\geq QT\}$. There have been several recent results asserting that, under certain assumptions on $Q$, $[Q\rangle$ has a common invariant subspace. In this paper, we use the technique of minimal vectors to establish similar results for $\langle Q]$.