On hyperboundedness and spectrum of Markov operators

Laurent Miclo (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

Consider an ergodic Markov operator $M$ reversible with respect to a probability measure $\mu$ on a general measurable space. It is shown that if $M$ is bounded from $\LL^2(\mu)$ to $\LL^p(\mu)$, where $p>2$, then it admits a spectral gap. This result answers positively a conjecture raised by H{\o}egh-Krohn and Simon \cite{MR0293451} in a semi-group context. The proof is based on isoperimetric considerations and especially on Cheeger inequalities of higher order for weighted finite graphs recently obtained by Lee, Gharan and Trevisan \cite{2011arXiv1111.1055L}. It provides a quantitative link between hyperboundedness and an eigenvalue different from the spectral gap in general. In addition, the usual Cheeger inequality is extended to the higher eigenvalues in the compact Riemannian setting.

Mots clés: (Dirichlet) connectivity spectrum; Orlicz's norms; hypercontractivity.; hypercontractivity; Markov operators and semi-groups; hyperboundedness; spectral gap; Cheeger's inequalities

Domaines: Probabilités [math.PR]; Analyse fonctionnelle [math.FA]; Théorie spectrale [math.SP]

Laurent Miclo. On hyperboundedness and spectrum of Markov operators. Inventiones Mathematicae, 2015, vol. 200 (1), p. 311-343. ⟨10.1007/s00222-014-0538-8⟩. ⟨hal-00777146v3⟩