Characteristic invariant subspaces generated by a single vector

If f is an endomorphism of a finite dimensional vector space V over a field K then an invariant subspace X⊆V is called hyperinvariant (respectively, characteristic) if X is invariant under all endomorphisms (respectively, automorphisms) that commute with f. The characteristic hull of a subset W of V is defined to be the smallest characteristic subspace in V that contains W. It is known that characteristic subspaces that are not hyperinvariant can only exist when |K|=2. In this paper we study subspaces X which are the characteristic hull of a single element. In the case where |K|=2 we derive a necessary and sufficient condition such that X is hyperinvariant.