Compactness properties of perturbed sub-stochastic semigroups on L¹(μ). A preliminary version.

Abstract

We deal with positive c₀-semigroups {U(t);t≥0} of contractions in L¹(Ω;A,μ) with generator T where (Ω;A,μ) is an abstract measure space and provide a systematic approach of compactness properties of perturbed semigroups {e^{t("T-V")};t≥0} (or their generators) induced by singular and bounded below potentials V:(Ω;μ)→ℝ. The results are precised further for metric measure spaces (Ω,d,μ). 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 Ω_{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 suits c₀-semigroups with integral kernels, for instance most sub-markovian semigroups arising in the theory of Markov processes in general state spaces, and combines intimately the kernel of {U(t);t≥0} and the sublevel sets Ω_{M}. Indefinite potentials are also dealt with. We illustrate the relevance of some aspects of this theory by giving new compactness and spectral results on convolution semigroups, magnetic Schrödinger semigroups, weighted Laplacians (in particular the Poincaré inequality for probability measures e^{-Φ(x)}dx on ℝ^{N}) and Witten Laplacians on 1-forms