  Motivated by the problems of analytic hypoellipticity, we show that a special
family of compact non self-adjoint operators has a non-zero eigenvalue. We
recover old results by Christ,Hanges, Himonas, Pham-The-Lai and Robert proved
by using ordinary differential equations. We show our method applies to higher
dimensional cases, giving in particular a new class of hypoelliptic but not
analytic hypoelliptic operators.
