$J^\ast_{X,\lambda_t} $ and $J^\ast_{Y,\lambda_t} $ as $$J^\ast_{X,\lambda_t} \sim \mathcal N_C(0, \widehat{\Sigma}_{\lambda_t})  \ \ \mbox{and} \ \  J^\ast_{Y,\lambda_t} \sim \mathcal N_C(0,  \widehat{\Sigma}_{\lambda_t}),$$ where $ \widehat{\Sigma}_\lambda $ is the matrix obtained by replacing in $ \Sigma_\lambda$ the unknown spectral density kernel $ f_{X,\lambda}$ by its pooled estimator $ \widehat{f}_\lambda$. For $\sigma, \tau \in \{s_1,s_2,\ldots, s_k\}$, let $$p^\ast_{X,\lambda_t}(\sigma,\tau) = J^\ast_{X,\lambda_t}(\sigma)\overline{J}^\ast_{X,\lambda_t}(\tau)  \ \ \mbox{and} \ \ p^\ast_{Y,\lambda_t}(\sigma,\tau) = J^\ast_{Y,\lambda_t}(\sigma)\overline{J}^\ast_{Y,\lambda_t}(\tau)$$ while, for $ t=-1,-2, \ldots, -N$, set $$p^\ast_{X,\lambda_t}(\sigma,\tau)=\overline{p}^\ast_{X,-\lambda_t}(\sigma, \tau) \;\; \mbox{and} \;\;  p^\ast_{Y,\lambda_t}(\sigma,\tau)=\overline{p}^\ast_{Y,-\lambda_t}(\sigma,\tau).$$

3.    For $\sigma, \tau \in \{s_1,s_2,\ldots, s_k\}$, let $$\hat f^\ast_{X,\lambda}(\sigma,\tau)=\frac{1}{b T}\sum_{t=-N}^{N} W\left(\frac{\lambda-\lambda_t}{b}\right)\, \widehat p^\ast_{X, \lambda_t} (\sigma, \tau)$$ and $$\hat f^\ast_{Y,\lambda}(\sigma,\tau)=\frac{1}{b T}\sum_{t=-N}^{N} W\left(\frac{\lambda-\lambda_t}{b}\right)\, \widehat p^\ast_{Y, \lambda_t} (\sigma, \tau).$$

4.    Approximate the distribution of the test statistic $ \mathcal U_T $ defined in (\[eq:test-stat-main\]) by that of the bootstrap test statistic $ \mathcal U^\ast_{T,k} $, where $$\mathcal U^\ast_{T,k}=\frac{2\pi}{Tk^2}\sum_{\substack{l=-N \\l\neq0}}^{N}\sum_{i,j=1}^k \Big| \widehat{f}^\ast_{X,\lambda_l}(s_i,s_j)
    -\widehat{f}^\ast_{Y,\lambda_l}(s_i,s_j) \Big|^2.$$

[The number $k$ of points in the interval $[0,1]$ for which the $k$-dimensional complex-valued random vectors $ J^\ast_{X,\lambda_t}$ and $ J^\ast_{Y,\lambda_t}$ are generated can be set equal to the sampling points $s_j$, $j=1,2, \ldots, k$, at which the observed functional random elements $ X_t(s_j)$ and $ Y_t(s_j)$ are observed in reality. However, and as it is commonly done in functional data 