| Contents |
| We deal with positive semigroups of contractions on L1 spaces over abstract measure spaces and provide a systematic approach of compactness properties of perturbed semigroups induced by perturbing the generator by singular and bounded below potentials. The results are precised further on L1 spaces over metric measure spaces. This new theory relies 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 of the potential are "thin at infinity". We show also how spectral gaps occur when the sublevel sets are not "thin at infinity". Various applications, in particular to convolution semigroups, are given. |
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