Surjectivity of certain adjoint operators and applications
- PDF / 429,472 Bytes
- 22 Pages / 595.276 x 790.866 pts Page_size
- 55 Downloads / 223 Views
Amina Cherifi Hadjiat
Arabian Journal of Mathematics
· Azzeddine Lansari
Surjectivity of certain adjoint operators and applications
Received: 9 April 2020 / Accepted: 2 September 2020 / Published online: 15 September 2020 © The Author(s) 2020
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 n (αi · xi + βi · xi1+m i ) ∂∂xi , where αi , βi vector fields, restricted their study to fields X 0 of the form X 0 = i=1 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 Y0 = X 0+ + X 0− + Z 0 , such as X 0 (x, y) = A (x, y) = A− (x) , A+ (y) , with A− (respectively, A+ ) a symmetric matrix having eigenvalues λ < 0 (respectively, λ > 0) and Z 0 are germs infinitely flat at the origin. This sub-algebra admits a hyperbolic structure for the diffeomorphism ψt∗ = (ex p · tY0 )∗ . 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. Mathematics Subject Classification
53C05
1 Introduction The ideals of finite codimension in Lie algebras of vector fields have recently received a lot of attention. Some authors such as Pursel and Shanks [9], by studying the invertibility of the Lie bracket [X, Y ] = ad X (Y ) which is an infinitesimal generator of an one-parameter group t, γt = (exp t X )∗ , in Lie algebras containing a germ of vector fields X do not vanish at the origin O, have treated the finite-codimensional ideals of these algebras. This result has been prolonged in the Banach-Lie n algebras of vector fields infinitely flat at 0 containing germs which vanish at the origin of the form X 0 = i=1 (αi · xi + Z 0 (x)), where αi are of constant signs [2,3,8]. Among the motivations and possible applications of these results: • The properties of the injectivity of the exponential function of a vector field have given rise to the existence of the Fourier series [11]. • The properties of the surjectivity of the directional derivative of the exponential function have given rise to the existence of the inversibility of the exponential function through the Nash–Moser theorem where positive results were obtained first in a Fréchet space and then in a hyperbolic type Fréchet space by integrating the diffeomorphisms in the smooth flows [12]. • Boris Kolev in [7] studied the particular case of a Lie–Poisson canonical structure. A. Cherifi Hadjiat (B) Dynamic Systems and Applications Lab., Department of Mathematics, University of Tlemcen, Tlemcen, Algeria E-mail: [email protected] A. Lansari Department of Mathematics, University of Tlemcen, Tlemcen, Algeria E-mail: [email protected]
123
568
Arab. J. Math. (2020) 9:567–588
• M. BENALILI in [1] has studied suitable spectral properties of the adjoint operators induced by appropriate perturba
Data Loading...
