Spectral gap for hyperbounded operators

Abstract

Let ( E , F , μ ) (E,\mathcal F,\mu ) be a probability space, and P P a symmetric linear contraction operator on L 2 ( μ ) L^2(\mu ) with P 1 = 1 P1=1 and ‖ P ‖ L 2 ( μ ) → L 4 ( μ ) > ∞ \|P\|_{L^2(\mu )\to L^4(\mu )}>\infty . We prove that ‖ P ‖ L 2 ( μ ) → L 4 ( μ ) 4 > 2 \|P\|_{L^2(\mu )\to L^4(\mu )}^4>2 is the optimal sufficient condition for P 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 C_0 -semigroup without a spectral gap.