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We deal with positive C0-semigroups (U(t))t>0 of contractions in L( ;A; ) with generator T where ( ;A; ) 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 : ( ; ) ! R+. More precise results are given in metric measure spaces ( ; d; ). This new construction is based on several ingredients: new a priori estimates peculiar to Lspaces, local weak compactness assumptions on unperturbed operators, Dunford-Pettisarguments and the assumption that the sublevel sets M := fx;V (x) Mg are thin at innity with respect to (U(t))t>0. We show also how spectral gaps occur when the sublevel sets are not thin at innity. This formalism combines intimately the kernel of (U(t))t>0 and the sublevel sets M . Indenite potentials are also dealt with. Various applications to convolution semigroups, weighted Laplacians and Witten Laplacians on 1-forms are given.

Compactness properties of perturbed sub-stochastic C_0-semigroups on L^1() with applications to discreteness and spectral gaps