On the Existence of the Ergodic Hilbert Transform

Abstract

Let $u$ be a unitary operator acting in $\mathbb{L}_2(\Omega, F, p)$, where $p$ is a probability measure. We prove that the limit $\lim_{n\rightarrow\infty}\sum_{0 < |k| \leq n} u^k f/k$ exists almost surely, for every $f \in \mathbb{L}_2(\Omega, F, p)$ if and only if the limit $\lim_{n\rightarrow\infty} n^{-1}\sum^{n-1}_{k=0}u^kf$ exists almost surely, for every $f \in \mathbb{L}_2(\Omega, F, p)$.