  We consider infinite-dimensional Hilbert space-valued random variables that
are assumed to be temporal dependent in a broad sense. We prove a central limit
theorem for the moving block bootstrap and for the tapered block bootstrap, and
show that these block bootstrap procedures also provide consistent estimators
of the long run covariance operator. Furthermore, we consider block
bootstrap-based procedures for fully functional testing of the equality of mean
functions between several independent functional time series. We establish
validity of the block bootstrap methods in approximating the distribution of
the statistic of interest under the null and show consistency of the block
bootstrap-based tests under the alternative. The finite sample behaviour of the
procedures is investigated by means of simulations. An application to a
real-life dataset is also discussed.
