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Inventiones mathematicae

, Volume 200, Issue 1, pp 311–343 | Cite as

On hyperboundedness and spectrum of Markov operators

  • Laurent MicloEmail author
Article

Abstract

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 \(\mathbb {L}^2(\mu )\) to \(\mathbb {L}^p(\mu )\), where \(p>2\), then it admits a spectral gap. This result answers positively a conjecture raised by Høegh-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 for weighted 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.

Mathematical Subject Classification (2010)

Primary: 60J25 Secondary: 47A30 46E30 47A75 37A30 58J50 05C50 

Notes

Acknowledgments

I would like to express my gratefulness toward Feng-Yu Wang, who presented to me the conjecture of Høegh-Krohn and Simon and explained me his works on the subject, as well as to Bernard Helffer, who pointed out some shortages around (17) in a first version of this manuscript. Thanks also to the ANR STAB for its support.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institut de Mathématiques de Toulouse, UMR 5219Université de Toulouse and CNRSToulouse Cedex 9France
  2. 2.Institut de Mathématiques de ToulouseUniversité Paul SabatierToulouse Cedex 9France

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