Criteria of Spectral Gap for Markov Operators

Abstract

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 $$\Vert P{\Vert }_{\tau }:=\underset{R\to \infty }{\lim }\underset{\mu (f^2)\le 1}{\sup }\mu (f(Pf-R{)}^{+})<1.$$ This strengthens a conjecture of Simon and H$\varphi$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 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.

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