  Let $T$ be a bounded linear operator on a Hilbert space $\mathcal{H}$, and
let $T \equiv V|T|$ be the polar decomposition of $T$. The mean transform of
$T$ is defined by $\widehat{T}:=\frac{1}{2}(V|T|+|T|V)$. In this paper we study
the iterates of the mean transform and we define the mean limit of an operator
as the limit (in the operator norm) of those iterates. We obtain new estimates
for the numerical range and numerical radius of the mean transform in terms of
the original operator. For the special class of unilateral weighted shifts we
describe the precise relationship between the spectral radius and the mean
limit, and obtain some sharp estimates.
