Common hypercyclic vectors for the unitary orbit of a hypercyclic operator

For a separable, infinite dimensional Hilbert space, it was recently shown by the authors that the similarity orbit of a hypercyclic operator contains a path of operators which is dense in the operator algebra with the strong operator topology, and yet the set of common hypercyclic vectors for the entire path is a dense Gδ set. Motivated by that result, we show in the present paper that the unitary orbit of any hypercyclic operator contains a path of operators whose closure contains the entire unitary orbit with the strong operator topology, and yet every nonzero vector in the linear span of the orbit of a given hypercyclic vector is a common hypercyclic vector for the entire path.