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• W2085183103 abstract "Let E be a Polish space equipped with a probability measure μ on its Borel σ-field B, and π a non-quasi-nilpotent positive operator on Lp(E,B,μ) with 1<p<∞. Using two notions, tail norm condition (TNC for short) and uniformly positive improving property (UPI/μ for short) for the resolvent of π, we prove a characterization for the existence of spectral gap of π, i.e., the spectral radius rsp(π) of π being an isolated point in the spectrum σ(π) of π. This characterization is a generalization of M. Hino's result for exponential convergence of πn, where the assumption of existence of the ground state, i.e., of a nonnegative eigenfunction of π for eigenvalue rsp(π), in M. Hino's result, is removed. Indeed, under TNC only, we prove the existence of ground state of π. Furthermore, under the TNC, we also establish the finiteness of dimension of eigenspace of π for eigenvalue rsp(π) and a interesting finite triangularization of π, which generalizes L. Gross' famous result by removing his assumption of symmetry and weakening his assumption of hyperboundedness. Finally, we give several applications of the characterization for spectral gap to Schrödinger operators, some invariance principles of Markov processes, and Girsanov semigroups respectively. In particular, we present a sharp condition to guarantee the existence of spectral gap for Girsanov semigroups. Soient E un espace polonais, μ une mesure de probabilité sur sa tribu borelienne, et π un opérateur positif non-nilpotent sur Lp(E,μ) où 1<p<∞. En utilisant deux notions : condition de norme de queue (CNQ) et propriété de positivité améliorante uniforme pour la résolvante de π, nous établissons une caractérisation de l'existence du trou spectral de π, i.e., le rayon spectral rsp(π) est isolé dans le spectre de π. Cette caractérisation généralise un résultat antérieur de M. Hino sur la convergence exponentielle de πn, dont l'hypothèse d'existence de l'état fondamental est supprimée. De plus, sous la CNQ, nous démontrons que l'espace propre de π associé à rsp(π) est non-trivial et de dimension finie, ce qui améliore un résultat bien connu de L. Gross (1972) dans L2(E,μ), pour lequel la symétrie et l'hyper-bornitude de π etaient supposées. Finalement nous donnons quelques applications de cette caractérisation du trou spectral aux opérateurs de Schrödinger, aux principes d'invariance (faible ou forte) pour des processus de Markov et aux semigroupes de Girsanov, etc. En particulier nous obtenons une condition fine suffisante pour l'existence de trou spectral de semigroupes de Girsanov." @default.
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• W2085183103 date "2006-02-01" @default.
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• W2085183103 title "Spectral gap of positive operators and applications" @default.
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• W2085183103 doi "https://doi.org/10.1016/j.matpur.2004.11.004" @default.
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