On hyperboundedness and spectrum of Markov operators

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.