  We propose a general white noise test for functional time series based on
estimating a distance between the spectral density operator of a weakly
stationary time series and the constant spectral density operator of an
uncorrelated time series. The estimator that we propose is based on a kernel
lag-window type estimator of the spectral density operator. When the observed
time series is a strong white noise in a real separable Hilbert space, we show
that the asymptotic distribution of the test statistic is standard normal, and
we further show that the test statistic diverges for general serially
correlated time series. These results recover as special cases those of Hong
(1996) and Horv\'ath et al. (2013). In order to implement the test, we propose
and study a number of kernel and bandwidth choices, including a new data
adaptive bandwidth, as well as data adaptive power transformations of the test
statistic that improve the normal approximation in finite samples. A simulation
study demonstrated that the proposed method has good size and improved power
when compared to other methods available in the literature, while also offering
a light computational burden.
