  We propose a new measure for stationarity of a functional time series, which
is based on an explicit representation of the $L^2$-distance between the
spectral density operator of a non-stationary process and its best
($L^2$-)approximation by a spectral density operator corresponding to a
stationary process. This distance can easily be estimated by sums of
Hilbert-Schmidt inner products of periodogram operators (evaluated at different
frequencies), and asymptotic normality of an appropriately standardized version
of the estimator can be established for the corresponding estimate under the
null hypothesis and alternative. As a result we obtain a simple asymptotic
frequency domain level $\alpha$ test (using the quantiles of the normal
distribution) for the hypothesis of stationarity of functional time series.
Other applications such as asymptotic confidence intervals for a measure of
stationarity or the construction of tests for "relevant deviations from
stationarity", are also briefly mentioned. We demonstrate in a small simulation
study that the new method has very good finite sample properties. Moreover, we
apply our test to annual temperature curves.
