Criteria of spectral gap for Markov operators

Let (E,F,μ)(E,F,μ) be a probability space, and let P   be a Markov operator on L2(μ)L2(μ) with 1 a simple eigenvalue such that μP=μμP=μ (i.e. μ is an invariant probability measure of P  ). Then Pˆ:=12(P+P⁎) has a spectral gap, i.e. 1 is isolated in the spectrum of Pˆ, if and only if‖P‖τ:=limR→∞supμ(f2)⩽1μ(f(Pf−R)+)<1. This strengthens a conjecture of Simon and Høegh-Krohn on the spectral gap for hyperbounded operators solved recently by L. Miclo in [10]. Consequently, for a symmetric, conservative, irreducible Dirichlet form on L2(μ)L2(μ), a Poincaré/log-Sobolev type inequality holds if and only if so does the corresponding defective inequality. Extensions to sub-Markov operators and non-conservative Dirichlet forms are also presented.