Cohomologie de l'algèbre de Lie des opérateurs différentiels sur une variété, à coefficients dans les fonctions

The space of local q-cochains of the Lie algebra of differential operators on a manifold, with coefficients in the space of functions, is naturally graded. The homogeneous terms of a cochain are totally ordered and the derivatives may be symbolized by linear forms on ø^m. This leads to a method giving the first three cohomology spaces.     

Comparaison des semi-groupes et des résolvantes d'ordre α associés à des opérateurs différentiels de type divergence

We prove that the densities of the semi-groups of order , 0<<1 associated with differential operators of second order and of divergence type, and the density of Riesz semi-groups of order are comparables.We give a necessary and sufficient condition such that the semi-group of order and its resolvent family and their perturbated with a nonnegative and regular Radon measure are comparables.When =1, we prove that the semi-group of brownian motion and its perturbated with a radial and nonnegative measure are comparables if and only if the measure generates a bounded potential, but the result is not true if the measure is not radial.

Ce travail est soutenu par la Fondation Nationale pour la Recherche Scientifique.     

Spectre discret des opérateurs périodiques perturbés par un opérateur différentiel à coefficients décroissants

We study the discrete spectrum arising in the spectral gaps of an elliptic periodic differential operator, P, perturbed by a differential operator, Q, with decaying coefficient. In the semi-classical regime (Q = Q(h), h0) (resp. the large-coupling constant: Q = λQ, λ → + ∞,), we obtain an asymptotic expansion in powers of h (resp. λ) of trƒ(P + Q(h)) (resp. trƒ(P + λQ)) and we give explicitly the leading term. Here ƒ ε C₀^∞(I) where I is an interval disjoint from the spectrum of P. We apply these results to study the asymptotic distribution of eigenvalues.     

Semi-groupes sous-Markoviens engendrés par des opérateurs matriciels

We characterize generators of sub-Markovian semigroups onL ^p () by a version of Kato's inequality. This will be used to show (under precise assumptions) that the semigroup generated by a matrix operatorA=(A _ij )_1i,jn on (L ^p ()) ⁿ is sub-Markovian if and only if the semigroup generated by the sum of each rowA _i 1+...+A _in (1in), is sub-Markovian. The corresponding result on (C ₀(X)) ⁿ characterizes dissipative operator matrices.

     

Moyennabilité des groupes dénombrables et actions sur les espaces de Cantor

We answer a question of R. Grigorchuk by showing the following characterization: for a countable group Γ to be amenable, it is necessary and sufficient that any continuous action of Γ on the Cantor set has an invariant probability measure. The proof uses an easy variation of a classical result of Alexandroff and Urysohn: any metrisable Γ-space is an equivariant surjective image of a Γ-Cantor set.     

Sur les opérateurs précompacts positifs

We characterize complete Hausdorff locally convex solid lattices (E,τ) satisfying the following property: for all operators $\text{S,T}\text{:E→E}$ such that 0?S?T and T precompact, the operator S² is precompact. To cite this article: B. Aqzzouz, R. Nouira, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Opérateurs pseudo-Fredholm dans les espaces de banach

Résumé  Les opérateurs pseudo-Fredholm ont été introduits et étudiés par M. Mbekhta dans le cas des espaces de Hilbert. Dans ce travail on se propose d’étendre l’étude de cette classe au cadre plus général des espaces de Banach. Notamment, nous montrons que cette classe reste stable par des petites perturbations. Ce dernier résultat donne une généralisation à un Théorème de P. Aiena et M. Mbekhta

---