Surjectivity of certain adjoint operators and applications

Abstract

This paper is an extension and generalization of some previous works, such as the study of M. Benalili and A. Lansari. Indeed, these authors, in their work about the finite co-dimension ideals of Lie algebras of vector fields, restricted their study to fields X_0 of the form X_0=sum _{i=1}^{n}( alpha _i cdot x_i+beta _icdot x_i^{1+m_i}) frac{partial }{partial x_i}, where alpha _i, beta _i are positive and m_i are even natural integers. We will first study the sub-algebra U of the Lie-Fréchet space E, containing vector fields of the form Y_0 = X_0^+ + X_0^- + Z_0, such as X_0left( x,yright) =Aleft( x,yright) =left( A^{-}left( x right) ,A^{+}left( yright) right) , with A^- (respectively, A^+ ) a symmetric matrix having eigenvalues lambda < 0 (respectively, lambda >0 ) and Z_0 are germs infinitely flat at the origin. This sub-algebra admits a hyperbolic structure for the diffeomorphism psi _{t*}=(expcdot tY_0)_*. In a second step, we will show that the infinitesimal generator ad_{-X} is an epimorphism of this admissible Lie sub-algebra U. We then deduce, by our fundamental lemma, that U=E.