A strictly weakly hypercyclic operator with a hypercyclic subspace

An interesting topic of study for a hypercyclic operator T:X→X on a topological vector space X has been whether X has an infinite\hyp{}dimensional, closed subspace consisting entirely, except for the zero vector, of hypercyclic vectors of T. These subspaces are called hypercyclic subspaces. It has been known that there is an operator T:H→H on a Hilbert space H that is hypercyclic with respect to the weak topology, but fails to be hypercyclic with respect to the norm topology. However, the natural question of whether such an operator T can have a hypercyclic subspace remains unsolved. In the present paper, we answer this question in the positive. Furthermore, we show that a continuous linear transformation L:B(H)→B(H) on the operator space B(H) can be hypercyclic with respect to the weak operator topology, but fail to be hypercyclic with respect to the strong operator topology.