$\sum_{j \in \Z}|j|^{1/2}(|a_j|+|b_j|)<\infty$.

We are interested in testing for equality of the second order structure between the two functional linear processes given in (\[eq.lin-proc\]). It turns out that a spectral approach to this problem is very appealing. First, note that we can define a spectral density operator in the sense of Panaretos and Tavakoli (2013) in the present set up generalizing the concept of spectral densities for univariate time series and spectral density matrices for multivariate time series. Here and in the sequel, we will abbreviate $L^2_\R([0,1]^d,\mu)$ by $L^2$ if the dimension $d$ becomes clear from the context.

\[l.ex-SDO\] Suppose that $(X_t)_{t\in\Z}$ and $(Y_t)_{t\in \Z}$ are functional linear processes satisfying Assumption 1. Then, for arbitrary $\lambda\in (-\pi,\pi]$, $$f_{X,\lambda}(\cdot,\cdot)=\frac{1}{2\pi}\sum_{t\in\Z} \exp(-i\lambda t) \, r_{X,t}(\cdot,\cdot)\quad\text{and}\quad f_{Y,\lambda}(\cdot,\cdot)=\frac{1}{2\pi}\sum_{t\in\Z} \exp(-i\lambda t)\, r_{Y,t}(\cdot,\cdot)$$ with $r_{X,t}$ and $r_{Y,t}$ denoting the autocovariance kernels of $X$ and $Y$ at lag $t$, respectively, converge absolutely in $L^2$. Moreover, for all $\sigma,\tau \in [0,1]$, $$r_{X,t}(\sigma, \tau)=\int_{(-\pi,\pi]} f_{X,\lambda}(\sigma,\tau)\, e^{it\lambda}\, d\lambda\quad\text{and}\quad
r_{Y,t}(\sigma, \tau)=\int_{(-\pi,\pi]} f_{Y,\lambda} (\sigma, \tau)\, e^{it\lambda}\, d\lambda\quad
\forall t \in \mathbb{Z},$$ where equality holds in $L^2$. The operators ${\mathcal F_{X,\lambda}}$ and ${\mathcal F_{Y,\lambda}}$, induced by right integration of $f_{X,\lambda}$ and $f_{Y,\lambda}$, are self-adjoint, nonnegative definite and it holds $$\mathcal F_{X,\lambda}=\frac{1}{2\pi}\sum_{t\in\Z} e^{-i\lambda t} \mathcal 