  We consider the asymptotic normality in $L^2$ of kernel estimators of the
long run covariance kernel of stationary functional time series. Our results
are established assuming a weakly dependent Bernoulli shift structure for the
underlying observations, which contains most stationary functional time series
models, under mild conditions. As a corollary, we obtain joint asymptotics for
functional principal components computed from empirical long run covariance
operators, showing that they have the favorable property of being
asymptotically independent.
