  Let $T$ be a bounded linear operator on a Banach space $X$ satisfying
$\|T^n\|/n \to 0$. We prove that $T$ is uniformly ergodic if and only if the
one-sided ergodic Hilbert transform $H_Tx:= \lim_{n\to\infty} \sum_{k=1}^n
k^{-1}T^k x$ converges for every $x \in \overline{(I-T)X}$. When $T$ is
power-bounded (or more generally $(C,\alpha)$ bounded for some $0< \alpha <1$),
then $T$ is uniformly ergodic if and only if the domain of $H_T$ equals
$(I-T)X$. We then study rotational uniform ergodicity -- uniform ergodicity of
every $\lambda T$ with $|\lambda|=1$, and connect it to convergence of the
rotated one-sided ergodic Hilbert transform, $H_{\lambda T}x$.
  In the Appendix we prove that positive isometries with finite-dimensional
fixed space on infinite-dimensional Banach lattices are never uniformly
ergodic. In particular, the Koopman operators of ergodic, even non-invertible,
probability preserving transformations on standard spaces are never uniformly
ergodic.
