Consider an ergodic Markov operator $M$ reversible with respect to a probability measure $mu$ on a general measurable space. We will show that if $M$ is bounded from ${cal L}^2(mu)$ to ${cal L}^p(mu)$, where $p>2$, then it admits a spectral gap. This result answers positively a conjecture raised by Høegh-Krohn and Simon 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. In general there is no quantitative link between hyperboundedness and spectral gap (except in the situation investigated by Wang), but there is one with another eigenvalue. In addition, the usual Cheeger and Buser inequalities will be extended to higher eigenvalues in the compact Riemannian setting.