On operators with orbits dense relative to nontrivial subspaces

In the present paper we consider bounded linear operators which have orbits dense relative to nontrivial subspaces. We give nontrivial examples of such operators and establish many of their basic properties. An example of an operator which has an orbit dense relative to a certain subspace but is not subspace-hypercyclic for this subspace is given. This, in turn, provides a new answer to a question posed in [18]. Other hypercyclic-like properties of such operators are also considered.