On hyperboundedness and spectrum of Markov operators

Consider an ergodic Markov operator M reversible with respect to a probability measure µ on a general measurable space. It is shown that if M is bounded from L2pµq to Lppµq, where p ą 2, then it admits a spectral gap. This result answers positively a conjecture raised by Høegh-Krohn and Simon [33] 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 [25]. It provides a quantitative link between hyperboundedness and an eigenvalue different from the spectral gap in general. In addition, the usual Cheeger and Buser inequalities are extended to higher eigenvalues in the compact Riemannian setting