Nonlinear eigenvalues and analytic hypoellipticity

Motivated by the problem of analytic hypoellipticity, we show that a special family of compact non-self-adjoint operators has a nonzero eigenvalue. We recover old results obtained by ordinary differential equations techniques and show how it can be applied to the higher dimensional case. This gives in particular a new class of hypoelliptic, but not analytic hypoelliptic operators.