On the one-sided ergodic Hilbert transform

Let T be a unitary contraction on a Hilbert space X such that X = <((I - T)X))over bar >. We answer two questions related to the strongly continuous semi group {(I - Tau)(r) : r >= 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 Sigma(infinity)(n = 1) (n)/(Tau nf) converges. We also show that the domain of G is not boolean OR(0 <alpha < 1) (I - Tau)(alpha) X. The tools used are essentially of a spectral nature.