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Criteria of Spectral Gap for Markov Operators

Let $(E,\mathcal F,μ)$ be a probability space, and let $P$ be a Markov operator on $L^2(μ)$ with $1$ a simple eigenvalue such that $μP=μ$ (i.e. $μ$ is an invariant probability measure of $P$). Then $\hat P:=\ff 1 2 (P+P^*)$ has a spectral gap, i.e. $1$ is isolated in the spectrum of $\hat P$, if and only if $$\|P\|_τ:=\lim_{R\to\infty} \sup_{μ(f^2)\le 1}μ\big(f(Pf-R)^+\big)<1.$$ This strengthens a conjecture of Simon and H$ϕ$egh-Krohn on the spectral gap for hyperbounded operators solved recently by L. Miclo in \cite{M}. Consequently, for a symmetric, conservative, irreducible Dirichlet form on $L^2(μ)$, 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.