Abstract: The authors establish the super-Poincare inequality on L^p-space with respect to a measure space, and obtain some necessary and sufficient conditions about semicompact and compact property of semigroup and the perturbation result. Meanwhile, a sufficient condition for super-Poincare inequality is shown, which generalizes some known results obtained on the L²-space. As applications, the essential spectrum of a class of non-symmetric diffusion operators on Riemannian manifold is studied.

Key words: Super-Poincare inequality, Compact semigroup, Asympotic kernel, Perturbation, Essential spectrum