Mémoires de la Société mathématique de France, n° 148. Compactness properties of perturbed sub-stochastic C0-semigroups on L1 (µ) with applications to discreteness and spectral gaps
We deal with positive C <sub>0</sub>-semigroups (U(t))<sub>t(...)0</sub> of contractions in L<sup>1</sup>(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 C <sub>0</sub>-semigroups ( e<sup>t(« T - V »)</sup> )<sub>t(...)0</sub> (or their generators) induced by singular potentials V : (oméga ; mu ) (...) (...)<sub>+</sub>. 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 <sup>1<sub>-</sub></sup> spaces, local weak compactness assumptions on unperturbed operators, « Dunford-Pettis » arguments and the assumption that the sublevel sets oméga<sub> M </sub> : = { x ; V(x) (...) M } are « thin at infinity with respect to ( U ( t ))<sub>t(...)0</sub> ». 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 ))<sub>t(...)0</sub> and the sublevel sets oméga<sub> M </sub>. Indefinite potentials are also dealt with. Various applications to convolution semigroups, weighted Laplacians and Witten Laplacians on 1-forms are given.
