Spectral gap of positive operators and applications

Let E be a Polish space equipped with a probability measure mu on its Borel sigma-field B, and pi a non-quasi-nilpotent positive operator on L-P (E, B, mu) with 1 < p < infinity. Using two notions, tail norm condition (TNC for short) and uniformly positive improving property (UPI/mu for short) for the resolvent of pi, we prove a characterization for the existence of spectral gap of pi, i.e., the spectral radius r(sp) (pi) of pi being an isolated point in the spectrum sigma (pi) of pi. This characterization is a generalization of M. Hino's result for exponential convergence of pi(n) where the assumption of existence of the ground state, i.e., of a nonnegative eigenfunction of pi for eigenvalue r(sp)(pi), in M. Hino's result, is removed. Indeed, under TNC only, we prove the existence of ground state of pi. Furthermore, under the TNC, we also establish the finiteness of dimension of eigenspace of pi for eigenvalue r(sp)(,pi) and a interesting finite triangularization of pi, 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 Schrodinger 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. (c) 2005 Elsevier SAS. All rights reserved.