Criteria of Spectral Gap for Markov Operators

2013/05/20 by Feng-Yu wang, wang, Feng-Yu · 2 citations
Mathematics · #Advanced Harmonic Analysis Research #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1305.4460

openalex publication_date 2013/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (E,\mathcal F,μ) be a probability space, and let P be a Markov operator on L²(μ) with 1 a simple eigenvalue such that μP=μ (i.e. μ is an invariant probability measure of P). Then P:=\ff 1 2 (P+P^*) has a spectral gap, i.e. 1 is isolated in the spectrum of P, if and only if ‖P‖_τ:=lim_R→∞ sup_{μ(f²)≤ 1}μ(f(Pf-R)⁺)lt;1. This strengthens a conjecture of Simon and Hϕegh-Krohn on the spectral gap for hyperbounded operators solved recently by L. Miclo in \citeM. Consequently, for a symmetric, conservative, irreducible Dirichlet form on 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.

Citations

• Logarithmic Sobolev Inequalities

Cited by

• Higher-order Cheeger inequalities for graphons
• Super and Weak Poincaré Inequalities for Sticky-Reflected Diffusion Processes

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