Surjectivity of certain adjoint operators and applications

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 X0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_0$$\end{document} of the form X0=∑i=1n(αi·xi+βi·xi1+mi)∂∂xi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_0=\sum _{i=1}^{n}( \alpha _i \cdot x_i+\beta _i\cdot x_i^{1+m_i}) \frac{\partial }{\partial x_i}$$\end{document}, where αi,βi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _i, \beta _i $$\end{document} are positive and mi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_i$$\end{document} 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=X0++X0-+Z0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_0 = X_0^+ + X_0^- + Z_0$$\end{document}, such as X0x,y=Ax,y=A-x,A+y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ X_0\left( x,y\right) =A\left( x,y\right) =\left( A^{-}\left( x \right) ,A^{+}\left( y\right) \right) $$\end{document}, with A-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^-$$\end{document} (respectively, A+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ A^+ $$\end{document}) a symmetric matrix having eigenvalues λ<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \lambda < 0$$\end{document} (respectively, λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda >0 $$\end{document}) and Z0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_0$$\end{document} are germs infinitely flat at the origin. This sub-algebra admits a hyperbolic structure for the diffeomorphism ψt∗=(exp·tY0)∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi _{t*}=(exp\cdot tY_0)_*$$\end{document}. In a second step, we will show that the infinitesimal generator ad-X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ad_{-X}$$\end{document} is an epimorphism of this admissible Lie sub-algebra U. We then deduce, by our fundamental lemma, that U=E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U=E$$\end{document}.

• M. BENALILI in [1] has studied suitable spectral properties of the adjoint operators induced by appropriate perturbations of some hyperbolic linear vector fields of the form Y 0 = X + 0 + X − 0 + Z 0 , where Z 0 is k-flat in the unit ball. This paper is an extension on the basis of these works where 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 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 ψ t * = (ex p · 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.
Part I: Admissible hyperbolic-type algebra 2 Definitions
we have D α X (x) 1 + x 2 k/2 M r . We define on E a graduation of seminorms: so (E, · · · r ) is called a Fréchet space [6]. (b) Let G the Schwartz space, which is the vector space of class C ∞ functions on R n satisfying: G = { f ∈ C ∞ (R n )/∀ p ≥ 0, ∀x ∈ R n , ∀α ∈ N n ∃C p > 0 satisfying D α f (x) (1 + x 2 ) p ≤ C p }. G is a space where the Fourier transform exists as well as its inverse.
We define on G a graduation of seminorms as follows: so (G, · · · r ) is called a Fréchet space [6].

Smooth flow
(a) Adjoint diffeomorphisms we define the adjoint diffeomorphisms φ * t and (φ t ) * by Proprieties. For all X ∈ E, we associate the X −flow φ t = ex pt X, so (i) ad X (respectively, ad −X ) is an infinitesimal generator of the one-parameter group φ * t (respectively, (φ t ) * ) on E; that is, φ * t is a solution of the following dynamic system: That is to say: i.e. that ad −X is an infinitesimal generator of the one-parameter group (φ t ) * on E, and by the same reasoning, we will have Now, according to Property (i): -The same reasoning can be applied in the case: φ * t = (φ −t ) * .

(b) Smooth flow
Let X ∈ E and the X −flow φ t = ex pt X. We say that an adjoint flow φ * t decays tamely on E of degree r and base b if (i) φ * t preserves E ∀t ≥ 0, (ii) for any integer k ≥ b there is an integer l k = k + r and a strictly positive, continuous and decreasing function C k (t) defined on [0, ∞) satisfying: and the improper integral: Alternately, φ * t can be replaced by (φ t ) * = φ * −t according to the asymptotic behaviour of φ t [12].
2.3 Fréchet space with hyperbolic structure (a) Admissible algebra Definition 2.2 (Admissible algebra) Let U be a Lie-Fréchet sub-algebra of E. U is said to be admissible with respect to the vector field Y 0 = X 0 + Z 0 if and only if it satisfies the following conditions: (i) Let X 0 be a vector field on E such that is a real symmetric matrix of type (k × k) (respectively, l × l) with k + l = n, having strictly negative eigenvalues (respectively, strictly positive). The matrix A − satisfies: ∀m ≥ 2, ∀i = 1, k : ∃a R , a L > 0, And, respectively, for A + : ∀m ≥ 2, ∀i = 1, l : ∃b R , b L > 0, ρ 2 > 1 such that We adopt the following notation for the rest of this paper: (ii) There exists U 1 (respectively, U 2 ) a Lie-Fréchet sub-algebra of E containing the vector field is an infinitely flat germ at the origin, i.e. satisfying the following estimate: There exists M α > 0 such that (b) Hyperbolic structure Definition 2.3 Let E be a Fréchet space of vector fields, X ∈ E and the X −flow ψ t = ex pt X. E has a tame hyperbolic structure for (ψ t ) * if and only if ∃δ > 0 / [12].

Estimations
To show that U has a hyperbolic structure, we need some estimates.
3.1 Estimation of ex ptY + 0 and ex ptY − 0 (a) Estimation of ex pt X + 0 and ex pt X − 0 Lemma 3.1 Let X + 0 ∈ U 2 (respectively, X − 0 ∈ U 1 ) Then X + 0 -flow has for estimate, for all t 0 : respectively, for the X − 0 -flow: x e a L t (∀x ∈ R k ).
According to Wintner theorem [10], y, A + y ≤ d | y | 2 , y ∈ R l , where the constant d is the largest eigenvalue of the symmetric matrix A + . We get We finally have The same should be applicable to φ + −t by replacing t by (−t): On the other hand, applying the same reasoning to X − 0 shows that respectively, Solution of the following dynamic system: is complete, and the flow ψ − t (respectively, ψ + t ) would satisfy the following estimates: Proof Consider the equation below: If we put z = ζ 1−k 1 , we will have dz = (1 − k 1 ) ζ −k 1 dζ and the system becomes Similarly, we will have And by a similar reasoning, we come to

Lemma 3.3
The derivative of Y − 0 -flow (respectively, of Y + 0 -flow) has the following estimates for all t 0: (respectively, , and consider the following equation: We will have We will then have We can easily deduce And by a similar reasoning, we come to has the following estimates for all t 0, ∀l ∈ N : Proof We can generalize the previous result using a recurrence reasoning as follows: (1) For j = 0, 1, the result is verified.
(3) Let us show that this last property is true at the order j. Let the j th derivative of solution of the following dynamic system: Using the so-called resolvent transform method discussed, for example, in [4], Y. Domar studied the closed ideals in some Banach algebras [5]), we deduce that The integral in the preceding expression is well defined at the point and there are constants A l > 0 such that Since v is arbitrary, we can choose v = 1, and put Let K ⊂ R k a compact set, so I j converges uniformly when t tends to +∞ for all x ∈ K . As and using the recurrence hypotheses, we arrive at The integral I j is then uniformly convergent with respect to x when t tends to +∞. Consequently, there are constants M j = sup (1, M j (4) Conclusion: By similar reasoning, we shall have Example: Let K i be a compact of i , (i = 1, 2) Then ∃δ > 0, such that δ = min x∈K i x , we can then have the following absorbents: (ii) For every arbitrary positive constant ρ 2 , there is a constant m 2 = r · b R b L + ρ 2 > 0 and C 2 > 0 such that Proof (i) Consider the following diffeomorphism: For any integer r ≥ 0 and any multi-index β = (β 1 . . . ., β k ) ∈ N k of module |β| = β 1 + . . . + β k , we obtain for x ∈ R k : We will have Since ||φ − −t (x)|| ≤ ||x||e a L t , and posing z = e −t X − 0 x, then from which we obtain Let ρ 1 be an arbitrary positive constant Then there is m 1 = r. a L a R + ρ 1 > 0, we will have

And as the integral
For any integer r ≥ 0 and any multi-index β = (β 1 . . . ., β l ) ∈ N l of module |β | = β 1 + . . . + β l , we obtain We will have Since ||φ + t (y)|| ≤ ||y||e b R t , and posing z = e t X + 0 · y, from which we obtain Let ρ 2 be an arbitrary positive constant Then there is m 2 = r. b R b L + ρ 2 > 0, we will have From Lemma 3.1, y e b L t φ + t (y) , there is a constant C 2 > 0 such that And as the integral ∞ 0 e −t (ρ 2 .b L ) dt is convergent, (φ + −t ) * operates smoothly on U 2 .

(b) Wave operators Let
The wave operator is defined by then for all r ∈ N; ∀v ∈ R n such as v = 1, we have where ξ = ψ − s (x). According to Lemma 3.2, there exist constants M i > 0 such that ψ − s (x) K 1 ≤ M 1 e −sa R with K 1 a compact included in 1 and ∀k 1 > 1; we will then have By hypothesis (I) in Sect. 2.3, ∃ρ 1 > 1/k 1 a R − a L ≥ ρ 1 > 1, then from which, It follows that f − t and D f − t are infinite-uniformly bounded ∀t > 0; ∀r ∈ N. And by a similar reasoning, with K 2 a compact included in 2 and with the hypothesis (II) in Sect. 2.3 then, for all r ∈ N; ∀v ∈ R n such as v = 1, we have According to Lemma 3.2, ψ − −s (x) K 1 ≤ Me sa L , we deduce that there is a constant M > 0 such that We choose with ρ any positive constant.
D r As K 1 is arbitrary, we can find ε 1 > 0 and ρ > 0 such that It follows that f − −t and D r f − −t are infinite-uniformly bounded ∀t > 0; ∀r ∈ N. And by a similar reasoning, there is a constant M > 0 such that By choosing with ρ an arbitrary positive constant, we deduce that As K 2 is arbitrary, we can find ε 2 > 0 and ρ > 0 such that hence, We will then have, from equations (*) and (**).
As K 1 is arbitrary on 1 , so We proceed in the same way to demonstrate Admissible algebra with hyperbolic structure 4.1 (ex pt X 0 ) * decays tamely (ii) (φ t ) * decays tamely on U 1 and (φ −t ) * decays tamely on U 2 for all t ≥ 0. That is to say, ∀t ≥ 0, we have and let t ∈ R, ∀(x, y) ∈ R n = 1 ∪ 2 , we have We deduce that (φ t ) * is invariant on U, ∀t ∈ R.

Theorem 4.3 The admissible algebra U admits a hyperbolic structure for the flow ψ t * .
Proof According to Lemma 4.2, we deduce that U has a hyperbolic structure for ψ t .

Part II: applications
We will show, as an application, that the ideal of finite codimension extends over all the hyperbolic-type admissible algebra, and this using the following lemma: Fundamental lemma: ([8]) Let V be a finite codimension subspace of a R− vector space E and an endomorphism ψ on E such that 3. for all numbers b, c ∈ R such as b 2 − 4c < 0, the operator ψ 2 + bψ + c.id is surjective on V .
Then V = E.

Surjectivity of the operator ad Y 0 + bI
In this section, we study the surjectivity of some linear operators. Note by = ϕ 2 two adjoint endomorphisms and id the identity application.
By putting -First stage: Let us prove that W i is a solution of the equation: As ρ 1 is arbitrary, then It results Similarly, By adding bW 2 in both members, we will have We have As ρ 2 is arbitrary, then Therefore, -Second stage: Let us prove that W 1 is of class C ∞ on any compact. Let K 1 be a compact of 1 , and as according to Lemma 3.7, we will have As ρ 1 is arbitrary, then It follows that W 1 converges uniformly, ∀x ∈ 1 , where ultimately W 1 is of class C ∞ on any compact of 1 . The same reasoning is valid to prove that W 2 is of class C ∞ on any compact of 2 .
6 Surjectivity of the operator (ad Y 0 ) 2 + b.ad Y 0 + cI With -First stage: Let us show that W i is a solution of the equation As then the first member vanishes and Z 1 becomes We will have Consequently, )t is solution of the Cauchy problem: The operator ϕ 2 1 + bϕ 1 + c.id R k is, therefore, surjective on U ε 1 . The proof of the surjectivity of the operator ϕ 2 2 + bϕ 2 + c.id R l on U ε 2 is made in the same way. -Second stage: According to the previous lemma, W i , (i = 1, 2) are of class C ∞ on all compact. Lemma 6.2 For all b, c ∈ R, b 2 − 4.c < 0 the operator ϕ 2 + b.ϕ + c.id is surjective in U .
Proof The hypotheses of the fundamental lemma is satisfied thanks to Lemmas 4.2, 5.2 and 6.2, and, if in addition we have dim(E − U ) finite, then we deduce Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.