  In 1980, Gasymov showed that non-self-adjoint Hill operators with
complex-valued periodic potentials of the type $V(x)=\sum_{k=1}^{\infty} a_k
e^{ikx}$, with $\sum_{k=1}^{\infty}|a_k|<\infty$, have spectra $[0, \infty)$.
In this note, we provide an alternative and elementary proof of this result.
