On weak orbits of operators

Let T be a completely nonunitary contraction on a Hilbert space H with r ( T ) = 1 . Let a n > 0 , a n → 0 . Then there exists x ∈ H with | 〈 T n x , x 〉 | ⩾ a n for all n. We construct a unitary operator without this property. This gives a negative answer to a problem of van Neerven.