Surjectivity of certain adjoint operators and applications

Abstract

AbstractThis 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 $$X_0$$ X 0 of the form $$X_0=\sum _{i=1}^{n}( \alpha _i \cdot x_i+\beta _i\cdot x_i^{1+m_i}) \frac{\partial }{\partial x_i}$$ X 0 = ∑ i = 1 n ( α i · x i + β i · x i 1 + m i ) ∂ ∂ x i , where $$\alpha _i, \beta _i $$ α i , β i are positive and $$m_i$$ 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 $$Y_0 = X_0^+ + X_0^- + Z_0$$ Y 0 = X 0 + + X 0 - + Z 0 , such as $$ X_0\left( x,y\right) =A\left( x,y\right) =\left( A^{-}\left( x \right) ,A^{+}\left( y\right) \right) $$ X 0 x , y = A x , y = A - x , A + y , with $$A^-$$ A - (respectively, $$ A^+ $$ A + ) a symmetric matrix having eigenvalues $$ \lambda < 0$$ λ < 0 (respectively, $$\lambda >0 $$ λ > 0 ) and $$Z_0$$ Z 0 are germs infinitely flat at the origin. This sub-algebra admits a hyperbolic structure for the diffeomorphism $$\psi _{t*}=(exp\cdot tY_0)_*$$ ψ t ∗ = ( e x p · t Y 0 ) ∗ . In a second step, we will show that the infinitesimal generator $$ad_{-X}$$ a d - X is an epimorphism of this admissible Lie sub-algebra U. We then deduce, by our fundamental lemma, that $$U=E$$ U = E .