  We study distributional properties of a quadratic form of a stationary
functional time series under mild moment conditions. As an important
application, we obtain consistency rates of estimators of spectral density
operators and prove joint weak convergence to a vector of complex Gaussian
random operators. Weak convergence is established based on an approximation of
the form via transforms of Hilbert-valued martingale difference sequences. As a
side-result, the distributional properties of the long-run covariance operator
are established.
