paper is organized as follows. Section 2 contains the main assumptions on the underlying functional linear processes and states the hypothesis testing problem under study. Section 3 is devoted to the suggested test statistic and its asymptotic behavior while Section 4 discusses a frequency domain bootstrap procedure to estimate the distribution of the test statistic under the null. Asymptotic validity of the bootstrap procedure is established and consistency of the bootstrap-based test under the alternative is also proved. Section 5 contains numerical simulations and an application to meteorological functional time series. Auxiliary results and proofs of the main results are deferred to the Appendix.

Assumptions and the Testing Problem {#S2}
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Suppose that observations $X_1,\dots, X_T$ and $Y_1,\dots, Y_T$ stem from two independent functional linear processes $(X_t)_{t\in\Z}$ and $(Y_t)_{t\in\Z}$ satisfying the following assumption.

[*Assumption 1*]{} : $(X_t)_{t\in\Z}$ and $(Y_t)_{t\in\Z}$ are independent functional linear processes, given by $$\label{eq.lin-proc}
X_t=\sum_{j\in \Z} a_j \varepsilon_{t-j}\quad\text{and}\quad Y_t=\sum_{j\in \Z} b_j e_{t-j},\; t\in \Z,$$ with values in $L^2_\R([0,1], \mu)$, where $\mu$ denotes the Lebesgue measure. The innovations $(\varepsilon_t)_{t \in \Z}$ and $(e_t)_{t \in \Z}$ are two i.i.d. mean zero Gaussian processes with values in $L^2_\R([0,1], \mu)$. The (real-valued) coefficients $(a_j)_{j \in \Z}$ and $(b_j)_{j \in \Z}$ satisfy 