Spectral gap for hyperbounded operators

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.