Ergodic characterization of van der Corput sets

We prove an analogue to the well-known equivalence of intersective sets and Poincaré recurrent sets, in a stronger setting. We show that a set D is van der Corput, if and only if for each Hilbert space H, unitary operator U, and ${x in H}$ such that the projection of x to the kernel of (U ? I) is nonvanishing, there exists ${d in D,}$ such that (U ^d x, x)≠ 0. We also characterize the smallest such d.