 
Summary: Contemporary Mathematics
Volume 00, 2007
On the onesided ergodic Hilbert transform
Idris Assani and Michael Lin
Abstract. 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  T)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
n=1
T n
f
n
converges. We also show that
the domain of G is not 0<<1(I  T)X. The tools used are essentially of a
spectral nature.
1. Introduction
In 1937, Marcinkiewicz and Zygmund [MZ] proved the following.
Theorem A. Let {n} be an i.i.d. sequence with E(1) < and E(1) = 0.
If 1 is symmetric, or E(1 log+
