Compactness properties of perturbed sub-stochastic C 0 -semigroups on L 1 () with applications to discreteness and spectral gaps - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2016

Compactness properties of perturbed sub-stochastic C 0 -semigroups on L 1 () with applications to discreteness and spectral gaps

Résumé

We deal with positive C 0-semigroups (U (t)) t>0 of contractions in L 1 ((; A;) with generator T where ((; A;) is an abstract measure space and provide a systematic approach of compactness properties of perturbed C 0-semigroups e t(" T V ") t0 (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 L 1-spaces, local weak compactness assumptions on unperturbed operators, " Dunford-Pettis " arguments and the assumption that the sublevel sets M := fx; V (x) M g 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 Lapla-cians and Witten Laplacians on 1-forms are given.
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Dates et versions

hal-01278274 , version 1 (24-02-2016)

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  • HAL Id : hal-01278274 , version 1

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Mustapha Mokhtar-Kharroubi. Compactness properties of perturbed sub-stochastic C 0 -semigroups on L 1 () with applications to discreteness and spectral gaps. 2016. ⟨hal-01278274⟩
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