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Spectral gap for hyperbounded operators

Authors:	Feng-Yu Wang

Affiliation:	Department of Mathematics, Beijing Normal University, Beijing 100875, People's Republic of China

Abstract:	Let be a probability space, and a symmetric linear contraction operator on with and $Vert PVert _{L^2(mu)to L^4(mu)}<infty$ . We prove that $Vert PVert _{L^2(mu)to L^4(mu)}^4<2$ is the optimal sufficient condition for 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 -semigroup without a spectral gap.

Keywords:	Hyperboundedness ergodicity log-Sobolev inequality spectral gap

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