  Consider a stationary, linear Hilbert space valued process. We establish
Berry-Essen type results with optimal convergence rates under sharp dependence
conditions on the underlying coefficient sequence of the linear operators. The
case of non-linear Bernoulli-shift sequences is also considered. If the
sequence is $m$-dependent, the optimal rate $(n/m)^{1/2}$ is reached. If the
sequence is weakly geometrically dependent, the rate $(n/\log n)^{1/2}$ is
obtained.
