The density of hypercyclic operators on a Hilbert space

On a separable infinite dimensional complex Hilbert space, we show that the set of hypercyclic operators is dense in the strong operator topology, and moreover the linear span of hypercyclic operators is dense in the operator norm topology. Both results continue to hold if we restrict to only those hypercyclic operators with an infinite dimensional closed hypercyclic subspace. Our works make connections with the classical result on the nondenseness of cyclic operators in the operator norm topology, as well as the recent developments on hypercyclic subspaces.