Criteria of Spectral Gap for Markov Operators

Let $(E,\mathcal F,\mu)$ be a probability space, and let $P$ be a Markov operator on $L^2(\mu)$ with $1$ a simple eigenvalue such that $\mu P=\mu$ (i.e. $\mu$ 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 $$\|P\|_\tau:=\lim_{R\to\infty} \sup_{\mu(f^2)\le 1}\mu\big(f(Pf-R)^+\big)<1.$$ This strengthens a conjecture of Simon and H$\phi$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 $L^2(\mu)$, a Poincar\'e/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.