On the one-sided ergodic Hilbert transform

Let T be a unitary contraction on a Hilbert space X such that X = (I- T)X. We answer two questions related to the strongly continuous semi group {(I-Tt : r 2: 0}, studied in [DL]. We show that the domain of the infinitesimal generator G is precisely the set of functions f for which the one sided ergodic Hilbert transform I;~=l ~ converges. We also show that the domain of G is not Uo<<><l (I-T)"' X. The tools used are essentially of a spectral nature.