March 31, 2008
Guy Cohen
Title: The one-sided ergodic Hilbert transform of normal contractions
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Let $T$ be a normal contraction on a Hilbert space $H$. For $f\in H$ we study the convergence of the one-sided ergodic Hilbert transform $\lim_n \sum_{k=1}^n \frac{T^k f}k$. We prove that weak and strong convergence are equivalent, and show that the convergence is equivalent to convergence of the series $\sum_{n=1}^\infty \frac {\log n\|\sum_{k=1}^nT^kf\|^2}{n^3}$. When $H=\overline{(I-T)H}$, the transform is shown to be precisely minus the infinitesimal generator of the strongly continuous semi-group $\{(I-T)^r\}_{r\ge 0}$. The equivalence of weak and strong convergence of the transform is proved also for $T$ an isometry or the dual of an isometry. For a general contraction $T$, we obtain that convergence of the series $\sum_{n=1}^\infty\frac{\langle T^n f,f\rangle \log n}n$ implies strong convergence of $\sum_{n=1}^\infty \frac{T^nf}n$.