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On hyperboundedness and spectrum of Markov operators. (English) Zbl 1312.47016

Summary: 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 B. Simon and R. Høegh-Krohn [J. Funct. Anal. 9, 121–180 (1972; Zbl 0241.47029)] 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 J. R. Lee et al. [in: Proceedings of the 44th annual ACM symposium on theory of computing, STOC 2012. New York, NY, USA, May 19–22, 2012. New York, NY: Association for Computing Machinery (ACM). 1117–1130 (2012; Zbl 1286.05091)]. 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.

MSC:

47A35 Ergodic theory of linear operators
37A30 Ergodic theorems, spectral theory, Markov operators
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
60J25 Continuous-time Markov processes on general state spaces
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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