Description

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 (e^{t(« 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 L^1_- spaces, local weak compactness assumptions on unperturbed operators, « Dunford-Pettis » arguments and the assumption that the sublevel sets oméga_M : = {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 oméga_M. Indefinite potentials are also dealt with. Various applications to convolution semigroups, weighted Laplacians and Witten Laplacians on 1-forms are given.