Spectral properties of operators having dense orbits

In the following H will denote a separable, infinite-dimensional, complex Hilbert space. The term operator will always mean linear, bounded operator on H. By invariant subspace we mean closed, invariant linear manifold. For a given operator T, the set of all invariant subspaces of T will be denoted LatT, since obviously it is a lattice. The set of all operators commuting with T is denoted {T}\u27. A subspace will be called hyperinvariant for T if it is invariant under any operator in {T}\u27