appealing to develop a spectral approach for testing equality of the second order properties of two functional time series. A testing procedure into this direction has been recently proposed by Tavakoli and Panaretos (2016) where projections on finite dimensional spaces of the differences of the estimated spectral density operators of the two functional time series have been used. Although projection-based tests have the advantage to lead to manageable limiting distributions, and can be very powerful when the deviations from the null are captured by the finite-dimensional space projected, such tests have no power for alternatives which are orthogonal to the projection space. It is, therefore, important to develop a fully functional test for the testing problem at hand. Notice that in the finite-dimensional case, i.e., for (univariate or multivariate) real-valued time series, such tests have been developed among others by Eichler (2008) and Dette and Paparoditis (2009).

The contribution of this paper is twofold. First, we develop a fully functional approach for testing equality of the entire second order structure between two independent functional linear processes. The testing procedure proposed, evaluates, for each frequency, the Hilbert-Schmidt norm between the (estimated) spectral density operators of the functional linear process at hand. 