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Contemporary Mathematics Volume 00, 2007

Summary: Contemporary Mathematics
Volume 00, 2007
On the one-sided 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
T 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+


Source: Assani, Idris - Department of Mathematics, University of North Carolina at Chapel Hill


Collections: Mathematics