Spectral gap of positive operators and applications

Abstract

Let E be a Polish space equipped with a probability measure μ on its Borel σ-field B , and π a non-quasi-nilpotent positive operator on L p ( E , B , μ ) with 1 < p < ∞ . Using two notions, tail norm condition (TNC for short) and uniformly positive improving property ( UPI / μ for short) for the resolvent of π, we prove a characterization for the existence of spectral gap of π, i.e., the spectral radius r sp ( π ) of π being an isolated point in the spectrum σ ( π ) of π. This characterization is a generalization of M. Hino's result for exponential convergence of π n , where the assumption of existence of the ground state, i.e., of a nonnegative eigenfunction of π for eigenvalue r sp ( π ) , in M. Hino's result, is removed. Indeed, under TNC only, we prove the existence of ground state of π. Furthermore, under the TNC, we also establish the finiteness of dimension of eigenspace of π for eigenvalue r sp ( π ) and a interesting finite triangularization of π, which generalizes L. Gross' famous result by removing his assumption of symmetry and weakening his assumption of hyperboundedness. Finally, we give several applications of the characterization for spectral gap to Schrödinger operators, some invariance principles of Markov processes, and Girsanov semigroups respectively. In particular, we present a sharp condition to guarantee the existence of spectral gap for Girsanov semigroups.