Small transitive families of dense operator ranges (Q1811171)

In a complex, separable, infinite-dimensional Hilbert space, there exist 5 proper dense operator ranges with the property that every operator leaving each of them invariant is a scalar multiple of the identity. The algebra of operators leaving a pair of proper dense ranges invariant can have an infinite nest of invariant subspaces. A slight extension of Foias' theorem shows that it can also have a nontrivial reducing subspace.