Serie : Mémoires de la Société mathématique de France. Vol 148
Paru le 15/06/2016 | Broché IV-87 pages
Professionnels
We deal with positive C0-semigroups (U(t))t(...)0 of contractions in L1(oméga ;A, mu) with generator T where (oméga ; A, mu) is an abstract measure space and provide a systematic approach of compactness properties of perturbed C0-semigroups (et(« T - V »))t(...)0 (or their generators) induced by singular potentials V : (oméga ; mu) (...) (...)+. More precise results are given in metric measure spaces (oméga, d, mu). This new construction is based on several ingredients : new a priori estimates peculiar to L1- spaces, local weak compactness assumptions on unperturbed operators, « Dunford-Pettis » arguments and the assumption that the sublevel sets omégaM : = {x ; V(x) (...) M} are « thin at infinity with respect to (U(t))t(...)0 ». We show also how spectral gaps occur when the sublevel sets are not « thin at infinity ». This formalism combines intimately the kernel of (U(t))t(...)0 and the sublevel sets omégaM. Indefinite potentials are also dealt with. Various applications to convolution semigroups, weighted Laplacians and Witten Laplacians on 1-forms are given.
