Let $s \in \mathbb{R}$ such that $0<s<1$. Consider the fractional Laplacian $(-\Delta)^s$ in the real line defined via [Fourier series](https://en.wikipedia.org/wiki/Fourier_series) as follows: if $f:[-\pi,\pi] \subset \mathbb{R} \longrightarrow \mathbb{C}$ is written as
$$
f(x)=\sum_{n \in \mathbb{Z}} f_n e^{inx}
$$
then
$$
(-\Delta)^sf(x)=\sum_{n \in \mathbb{Z}} |n|^{2s} f_n e^{inx}.
$$

**Question.** Is true that
$$
\overline{f(x)}(-\Delta)^sf(x)=|(-\Delta)^{s/2}f(x)|^2? \tag{1}
$$

I thought in the following way: on the one-hand, we have
\begin{eqnarray}
\overline{f(x)}(-\Delta)^sf(x)= \sum_{n \in \mathbb{Z}}\overline{f_n e^{inx}}\cdot \sum_{n \in \mathbb{Z}}|n|^{2s} f_n   e^{inx}=   \sum_{n \in \mathbb{Z}} |n|^{2s} |f_n e^{inx}|^2. \tag{2}
\end{eqnarray}

In the other hand, 
\begin{eqnarray}
|(-\Delta)^{s/2}f(x)|^2=|(-\Delta)^{s/2}f(x)|\cdot |(-\Delta)^{s/2}f(x)|&=& \sum_{n \in \mathbb{Z}} |n|^{s} |f_n e^{inx}| \cdot \sum_{n \in \mathbb{Z}} |n|^{s} |f_n e^{inx}| \\
&=& \sum_{n \in \mathbb{Z}} |n|^{2s} |f_n e^{inx}|^2. \tag{3}
\end{eqnarray}

From $(2)$ and $(3)$ follows $(1)$. That is right?