  Given $N\ge2$ closed subspaces $M_1,\dotsc, M_N$ of a Hilbert space $X$, let
$P_k$ denote the orthogonal projection onto $M_k$, $1\le k\le N$. It is known
that the sequence $(x_n)$, defined recursively by $x_0=x$ and
$x_{n+1}=P_N\cdots P_1x_n$ for $n\ge0$, converges in norm to $P_Mx$ as
$n\to\infty$ for all $x\in X$, where $P_M$ denotes the orthogonal projection
onto $M=M_1\cap\dotsc\cap M_N$. Moreover, the rate of convergence is either
exponentially fast for all $x\in X$ or as slow as one likes for appropriately
chosen initial vectors $x\in X$. We give a new estimate in terms of natural
geometric quantities on the rate of convergence in the case when it is known to
be exponentially fast. More importantly, we then show that even when the rate
of convergence is arbitrarily slow there exists, for each real number
$\alpha>0$, a dense subset $X_\alpha$ of $X$ such that
$\|x_n-P_Mx\|=o(n^{-\alpha})$ as $n\to\infty$ for all $x\in X_\alpha$.
Furthermore, there exists another dense subset $X_\infty$ of $X$ such that, if
$x\in X_\infty$, then $\|x_n-P_Mx\|=o(n^{-\alpha})$ as $n\to\infty$ for all
$\alpha>0$. These latter results are obtained as consequences of general
properties of Ritt operators. As a by-product, we also strengthen the
unquantified convergence result by showing that $P_M x$ is in fact the limit of
a series which converges unconditionally.
