  Motivated by the need to statistically quantify differences between modern
(complex) data-sets which commonly result as high-resolution measurements of
stochastic processes varying over a continuum, we propose novel testing
procedures to detect relevant differences between the second order dynamics of
two functional time series. In order to take the between-function dynamics into
account that characterize this type of functional data, a frequency domain
approach is taken. Test statistics are developed to compare differences in the
spectral density operators and in the primary modes of variation as encoded in
the associated eigenelements. Under mild moment conditions, we show convergence
of the underlying statistics to Brownian motions and construct pivotal test
statistics. The latter is essential because the nuisance parameters can be
unwieldy and their robust estimation infeasible, especially if the two
functional time series are dependent. In addition to these novel features, the
properties of the tests are robust to any choice of frequency band enabling
also to compare energy contents at a single frequency. The finite sample
performance of the tests are verified through a simulation study and are
illustrated with an application to fMRI data.
