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Spectral gap for hyperbounded operators

Author: Feng-Yu Wang
Journal: Proc. Amer. Math. Soc. 132 (2004), 2629-2638
MSC (2000): Primary 47D07, 60H10
Published electronically: April 8, 2004
MathSciNet review: 2054788
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Abstract: Let $(E,\mathcal F,\mu)$ be a probability space, and $P$ a symmetric linear contraction operator on $L^2(\mu)$ with $P1=1$ and $\Vert P\Vert _{L^2(\mu)\to L^4(\mu)}<\infty$. We prove that $\Vert P\Vert _{L^2(\mu)\to L^4(\mu)}^4<2$ is the optimal sufficient condition for $P$ to have a spectral gap. Moreover, the optimal sufficient conditions are obtained, respectively, for the defective log-Sobolev and for the defective Poincaré inequality to imply the existence of a spectral gap. Finally, we construct a symmetric, hyperbounded, ergodic contraction $C_0$-semigroup without a spectral gap.

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Additional Information

Feng-Yu Wang
Affiliation: Department of Mathematics, Beijing Normal University, Beijing 100875, People’s Republic of China

Keywords: Hyperboundedness, ergodicity, log-Sobolev inequality, spectral gap
Received by editor(s): October 15, 2002
Received by editor(s) in revised form: June 3, 2003
Published electronically: April 8, 2004
Additional Notes: Supported in part by NNSFC(10025105, 10121101), TRAPOYT and the 973-Project
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2004 American Mathematical Society

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