ON THE SPECTRUM OF A CLASS OF NON-SECTORIAL DIFFUSION OPERATORS

Abstract

In terms of the upper bounds of a second-order elliptic operator acting on specific Lyapunov-type functions with compact level sets, sufficient conditions are presented for the corresponding Dirichlet form to satisfy the Poincaré and the super-Poincaré inequalities. Here, the elliptic operator is assumed to be symmetric on L2(μ) with some probability measure μ. As applications, proofs are given for a class of (non-symmetric) diffusion operators generating C0-semigroups on L1(μ): that their Lp(μ)-essential spectrum is empty for p > 1. This follows since it is proved that their C0-semigroups are compact. 2000 Mathematics Subject Classification 47D07, 60H10.