Irreducible semigroups of functionally positive nilpotent operators

Abstract:

For each irrational number $\theta \in (0,1)$, we construct a semigroup ${\mathcal {S}_\theta }$ of nilpotent operators on ${\mathcal {S}^2}([0,1])$ that are also partial isometries and positive in the sense that the operator maps nonnegative functions to nonnegative functions. We prove that each semigroup ${\mathcal {S}_\theta }$ is discrete in the norm topology and hence norm-closed and that the weak closure of ${\mathcal {S}_\theta }$ is independent of ${\mathcal {S}_\theta }$. We show that each semigroup ${\mathcal {S}_\theta }$ has no nontrivial invariant subspaces.