and, similarly, $\mathcal F_{Y, \lambda}$ by an integral operator $\widehat{\mathcal F}_{Y,\lambda}$ induced by right-integration with the kernel $$\hat f_{Y,\lambda}(\sigma,\tau)=\frac{1}{b T}\sum_{t=-N}^{N} W\left(\frac{\lambda-\lambda_t}{b}\right)\, \widehat p_{Y, \lambda_t} (\sigma, \tau),  \;\; \mbox{for all} \;\; \sigma,\tau \in [0,1].$$ Here, $N=[(T-1)/2]$ and $\lambda_t=2\pi t/T, \;t=-N,\dots, N,$ denote the Fourier frequencies. Further $b=b_T>0$ is an asymptotically vanishing bandwidth and $W$ denotes a weight function. Moreover, as in Panaretos and Tavakoli (2013), $$\widehat p_{X, \lambda}(\sigma,\tau)=\frac{1}{2\pi T}\sum_{s_1,s_2=1}^T X_{s_1}(\sigma)X_{s_2}(\tau)\, \exp(i\lambda( s_1- s_2)),  \;\; \mbox{for all} \;\; \sigma,\tau \in [0,1],$$ and $$\widehat p_{Y, \lambda}(\sigma,\tau)=\frac{1}{2\pi T}\sum_{s_1,s_2=1}^T Y_{s_1}(\sigma)Y_{s_2}(\tau)\, \exp(i\lambda( s_1- s_2)),  \;\; \mbox{for all} \;\; \sigma,\tau \in [0,1],$$ denote the periodogram kernels based on $X_1,\dots, X_T$ and $Y_1,\dots, Y_T$, respectively.

For the hypothesis testing problem (\[eq:testM\]), we propose the following test statistic which evaluates the distance between the estimated spectral density operators, i.e., $${\mathcal U_T}=\int_{-\pi}^\pi\|\widehat{\mathcal F}_{X,\lambda}-\widehat{\mathcal F}_{Y,\lambda}\|_{HS}^2\, d\lambda,
\label{eq:test-stat-main}$$ where $\|\cdot\|_{HS}$ denotes the Hilbert Schmidt norm on $L^2$. The following theorem shows asymptotic normality of the suitably normalized test statistic $\mathcal U_T$ under ${\mathcal H_0}$.

\[t.asy-test\] Suppose that the stretches of observations $X_1,\dots, X_T$ and $Y_1,\dots, Y_T$ stem from the two independent functional linear processes $(X_t)_{t\in\Z}$ and $(Y_t)_{t\in \Z}$, respectively, satisfying Assumption 1. Moreover, assume that

-   $b\sim T^{-\nu }$ for some $\nu\in 