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Analyzing the behavior at infinity of the sequence of eigenvalues given by matrix symbol of a invariant operator with respect to a fixed elliptic operator, we obtain necessary and sufficient conditions to ensure that perturbations of globally hypoelliptic operators continue to have this property. As an application, we recover classical results about perturbations of constant vector fields on the torus and extend them for more general classes of perturbations. Additionally, we construct examples of low order perturbations that destroy the global hypoellipticity, in the presence of diophantine phenomena.Contents 15 6. Analytic perturbations of vector fields on the torus 19 References 24

The main goal of this paper is to address global hypoellipticity issues for the class of first order pseudo-differential operators L = D t + C(t, x, D x ), where (t, x) ∈ T × M , T is the one-dimensional torus, M is a closed manifold and C(t, x, D x ) is a first order pseudo-differential operator on M , smoothly depending on the periodic variable t. In the case of separation of variables, when C(t, x, D x ) = a(t)p(x, D x ) + ib(t)q(x, D x ), we give necessary and sufficient conditions for the global hypoellipticity of L. In particular, we show that the famous (P) condition of Nirenberg-Treves is neither necessary nor sufficient to guarantee the global hypoellipticity of L.

This note presents an investigation on the global hypoellipticity problem forCauchyis a pseudo-differential operator on T n and c j = c j (t), a smooth, complex valued function on T. The main goal of this investigation consists in establishing connections between the global hypoellipticity of the operators L and its normal form L 0 = m j=1 (D t + c 0,j P j (D x )). In order to do so, the problem is approached by combining Hörmander's and Siegel's conditions on the symbols of the operators L j = D t + c j (t)P j (D x ).

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