  The linear operator $c + (-\Delta)^{\alpha/2}$, where $c > 0$ and
$(-\Delta)^{\alpha/2}$ is the fractional Laplacian on the periodic domain,
arises in the existence of periodic travelling waves in the fractional
Korteweg--de Vries equation. We establish a relation of the Green's function of
this linear operator with the Mittag--Leffler function, which was previously
used in the context of Riemann--Liouville's and Caputo's fractional
derivatives. By using this relation, we prove that Green's function is strictly
positive and single-lobe (monotonically decreasing away from the maximum point)
for every $c > 0$ and every $\alpha \in (0,2]$. On the other hand, we argue
from numerical approximations that in the case of $\alpha \in (2,4]$, the
Green's function is positive and single-lobe for small $c$ and non-positive and
non-single lobe for large $c$.
