Uniform ergodicity and the one-sided ergodic Hilbert transform

Let $T$ be a bounded linear operator on a Banach space $X$ satisfying $\lVert T^n\rVert/n\rightarrow 0$. We prove that $T$ is uniformly ergodic if and only if the one-sided ergodic Hilbert transform $H(T)x:=\lim_{n\rightarrow \infty}\sum_{k=1}^nk^{-1}T^kx$ converges for every $x\in \overline{(I-T)X}$. When $T$ is a power-bounded (or more generally $(C,\alpha)$ bounded for some $0<\alpha<1$), then $T$ us uniformly ergodic if and only if the domain of $H$ equals $(I-T)X$.