AbstractLet 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 Lp(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 rsp(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 πn, where the assumption of existence of the ground state, i.e., of a nonnegative eigenfunction of π for eigenvalue rsp(π), in M. Hino's result, is removed. Indeed, under TNC only, we prove the ex...