contrast to comparing mean functions, the problem of comparing the entire second order structure of two independent functional time series has been much less investigated. Notice that for i.i.d. functional data this problem simplifies to the problem of testing the equality of (the lag zero) covariance operators, see, e.g., Panaretos [*et al.*]{} (2010), Fremdt [*et al.*]{} (2012) (finite-dimensional projections), Pigoli [*et al.*]{} (2014) (distance measures) and Paparoditis and Sapatinas (2016) (fully functional). The same problem of testing the equality of the (lag-zero) covariance operators of two sets of independent functional time series has also been investigated by Zhang and Shao (2015) using a finite-dimensional projections approach.

However, the comparison of the entire second order structure of independent functional time series, is a much more involved problem due to the temporal dependence of the set of random elements considered. In describing the second order structure of functional time series, the spectral density operator, introduced in the functional set-up by Panaretos and Tavakoli (2013), is a very useful tool since it summarizes in a nice way the entire autocovariance structure of the underlying functional time series; see also Hörmann [*et al.*]{} (2015) and van Delft and Eichler (2018). It seems, therefore, very 