On hyperboundedness and spectrum of Markov operators

• 成果类型: Article
• 署名作者: Miclo, Laurent
• 署名单位: Universite Federale Toulouse Midi-Pyrenees (ComUE); Universite de Toulouse; Institut National des Sciences Appliquees de Toulouse; Universite Toulouse III - Paul Sabatier; Centre National de la Recherche Scientifique (CNRS); Universite de Toulouse; Universite Toulouse III - Paul Sabatier
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-014-0538-8
• 发表日期: 2015
• 页码: 311-343
• 关键词: nodal domains; inequalities; gap; asymptotics; bounds

摘要

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 L-2(mu) to L-p(mu), where p > 2, then it admits a spectral gap. This result answers positively a conjecture raised by Hoegh-Krohn and Simon (J. Funct. Anal. 9:121-80, 1972) in the more restricted semi-group context. The proof is based on isoperimetric considerations and especially on Cheeger inequalities of higher order forweighted finite graphs recently obtained by Lee et al. (Proceedings of the 2012 ACM Symposium on Theory of Computing, 1131-1140, ACM, New York, 2012). 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 and the exponential behaviors of the small eigenvalues of Witten Laplacians at small temperature are recovered.