I am stopped by the following algebra, which appears in the third line of page 3 of the file https://projecteuclid.org/euclid.aos/1366980558#supplemental
Let $X$ be a functional stochastic process and let $\phi^{(T)}(\tau) = \int_0^1p_{\omega}^{(T)}(\tau,\sigma)\times\phi(\sigma) d\sigma$, where $p_{\omega}^{(T)}(\tau,\sigma) = T^{-1} \sum_{t,s=0}^{T-1}e^{-i\omega(t-s)}\times X_t(\tau)\times X_s(\sigma)$, and $\phi \in L^2[[0,1],\mathbb{C}] $(that is, $\phi$ is a complex-valued square integrable function.), and $X_t$ is a random element of $L^2[[0,1],\mathbb{C}]$ with the parameterization $\alpha \rightarrow X_t(\alpha)\in\mathbb{C} $ for $ \alpha \in [0,1].$ (that is, $X_t$ is a Hilbert space-valued random variable.)
Here, the inner product on $L^2[[0,1],\mathbb{C}]$ is given by $<x,y> = \int_0^1x(t)\overline{y(t)}dt$ and the norm is induced by the inner product. Also, it is assumed that $E\|X_t\|^4 <\infty$. How can I show that $E\|\phi^{(T)}\|^2 <\infty$?