Gappedness Criterion in Markov Operators

• Gappedness criterion is a precise condition defined via a tail-norm that guarantees a spectral gap in the symmetrized Markov operator.
• It unifies various functional inequalities and extends classical hyperboundedness by controlling the concentration of L2-mass in tail regions.
• The criterion offers actionable insights for analyzing convergence rates, ergodicity, and the robustness of Dirichlet forms in both reversible and non-reversible settings.

The gappedness criterion, in the context of Markov operators and spectral theory, refers to a sharp, necessary and sufficient condition that guarantees a spectral gap at 1 for the symmetrized version of a Markov operator. This criterion, established by Wang, is formulated in terms of a weak “tail-norm” property and is significant for the theory of Markov semigroups, Dirichlet forms, and functional inequalities. It strictly extends classical criteria, such as hyperboundedness, and unifies the understanding of spectral gaps, defective and tight Poincaré inequalities, and related topics in ergodic theory and analysis.

1. Markov Operators, Spectral Gap, and Symmetrization

Let $(E,\mathcal F,\mu)$ be a probability space with Hilbert space $L^2(\mu)$. A linear operator $P:L^2(\mu)\to L^2(\mu)$ is a Markov operator if

• $P1 = 1$,
• $f \ge 0 \Rightarrow Pf \ge 0$,
• $\mu$ is $P$-invariant: $\mu(Pf) = \mu(f)$.

The symmetrization of $P$ is $\hat P := \frac{1}{2}(P+P^*)$, a self-adjoint Markov operator. The spectrum $\sigma(\hat P)$ is contained in $[-1,1]$. There is a spectral gap at 1 if $1$ is a simple eigenvalue and is isolated in $\sigma(\hat P)$, or equivalently, $\sup(\sigma(\hat P|_{H_0}))<1$ where $H_0 = \{ f \in L^2(\mu): \mu(f) = 0 \}$.

In the symmetric case, the spectral gap corresponds to the classical Poincaré inequality: $$\mu(f^2) - \mu(f)^2 \leq C \mu(f(1-P)f), \quad \forall f \in L^2(\mu).$$

2. Tail-Norm Definition and Interpretation

The critical notion underlying the gappedness criterion is Wang's tail-norm: $$\|P\|_\tau := \lim_{R \to \infty} \sup_{\mu(f^2) \leq 1} \mu\big( f(Pf - R)^+ \big)$$ where $(Pf-R)^+ := \max(Pf - R, 0)$ and the supremum runs over $f \in L^2(\mu)$ with $\mu(f^2)\leq 1$.

Intuitively, $\|P\|_\tau$ measures the maximal fraction of $L^2$-mass that $P$ can send into arbitrarily high “tails.” If $\|P\|_\tau < 1$, then uniformly in all normalized $f$, large values of $Pf$ can never concentrate more than a fraction of the $L^2$-mass, precluding heavy long-range transport by $P$.

3. Main Gappedness Theorem: Equivalence of Gap and Tail-Norm Condition

Wang’s main result provides a complete equivalence:

• The symmetrized Markov operator $\hat P$ has a spectral gap at 1 if and only if $\|P\|_\tau < 1$ (Wang, 2013).

This equivalence extends to several related norms:

• $\inf_{m \in \mathbb{N}} \|P^{2m-1}\|_\tau < 1$,
• $\inf_{m \in \mathbb{N}} \|P^m\|_{tail} < 1$, with $$\|Q\|_{tail} := \limsup_{R \to \infty} \sup_{\mu(f^2) \leq 1} \left( \mu( (|Q f| - R )^2 ) \right)^{1/2}.$$

Table: Equivalent Gappedness Criteria

Operator	Criterion for Spectral Gap
$\hat{P}$	$\sup \sigma(\hat{P}|_{H_0})<1$
$P$	$\|P\|_\tau <1$
$P^m$	$\inf_m \|P^{2m-1}\|_\tau <1$

All three conditions are mathematically equivalent.

4. Proof Outline and Key Steps

The proof proceeds by showing both directions:

• (Spectral gap $\Rightarrow$ tail-norm < 1): A Poincaré inequality for $\hat{P}$ produces a contraction that directly bounds $\|P\|_\tau$ away from 1.
• (Tail-norm < 1 $\Rightarrow$ spectral gap): The argument reduces to the symmetric case and leverages high-order isoperimetric (Cheeger) constants. A positive lower bound on these constants follows from the tail-norm estimate, which, via quantitative Cheeger-type inequalities and approximations by finite-state chains (Miclo's method), implies the existence of a spectral gap.

A key technical device is the use of higher-order isoperimetric constants

$$k_n := \inf_{ (A_1,...,A_n):\,\text{disjoint},\, \mu(A_k)>0} \max_{1\le k \le n} \frac{\mu(1_{A_k} S 1_{A_k^c})}{\mu(A_k)},$$

which govern the spectrum near 1. The tail-norm constraint controls these constants uniformly.

5. Extensions: Non-conservative Operators and Dirichlet Forms

Wang’s criterion admits powerful generalizations:

• Sub-Markov operators ($P 1 \leq 1$) with non-trivial kernel: The equivalence extends, replacing spectrum of $\hat P$ by spectrum of $P^*P$.
• Symmetric (possibly non-conservative) Dirichlet forms: For any irreducible symmetric Dirichlet form $(\mathcal{E}, D(\mathcal{E}))$, a defective Poincaré inequality

$$\mu(f^2) \leq C_1 \mathcal{E}(f, f) + C_2 \mu(|f|)^2$$

is equivalent to the true (tight) Poincaré inequality

$\mu(f^2) - \mu(f)^2 \leq C \mathcal{E}(f, f).$

Similarly, for log-Sobolev and general interpolation functionals $\operatorname{Var}_{\phi, \psi}(f)$, defective and tight forms are equivalent under irreducibility.

6. Impact on Functional Inequalities and the Simon–Høegh-Krohn Conjecture

The tail-norm criterion strictly generalizes classical results:

• Hyperboundedness ($P: L^2 \to L^{2+\epsilon}$) automatically implies $\|P\|_\tau = 0 <1$, but the converse is false. Many ergodic Markov operators with $\|P\|_\tau < 1$ are not hyperbounded.
• The Simon–Høegh-Krohn conjecture asserted that hyperboundedness forces a gap; this was confirmed by Miclo and subsumed by the tail-norm condition (Wang, 2013).
• In the theory of functional inequalities, there is no distinction between defective and tight Poincaré or log-Sobolev inequalities under irreducibility: the defective version holds if and only if the tight version holds. This closes a formerly ambiguous gap in the Dirichlet-form framework.

7. Summary and Significance

The gappedness criterion for Markov operators, as formulated through the tail-norm condition $\|P\|_\tau < 1$, provides a sharp, robust, and minimal test for the existence of a spectral gap. This test is both necessary and sufficient and unifies the landscape of spectral theory for Markov semigroups and functional inequalities in both symmetric and nonsymmetric settings (Wang, 2013).

This criterion is optimal and applies broadly, including non-reversible chains, sub-Markov cases, and symmetrizable and non-conservative Dirichlet forms. It delivers practical, checkable benchmarks for the analysis of convergence rates, ergodicity, and the robustness of inequalities in applied, probabilistic, and analytical settings.