= Absolute L2 spectral gap of a reversible Markov chain
{title2=$\|P\|_{L^2_0(\mu)}<1$}
For a <reversible Markov chain> with invariant law $\mu$, its <Markov kernel> acts as a <self-adjoint operator> $P$ on $L^2(\mu)$. An absolute <spectral gap> means that $\|P\|$ on the mean-zero functions is at most $r<1$. <Geometric ergodicity> implies this gap for a reversible chain. Indeed, for bounded mean-zero $g$ supported where the geometric prefactor is bounded, the total-variation estimate gives $\langle g,P^{2n}g\rangle\leq C_g r^{2n}$. The <spectral theorem for normal operators> puts its spectral measure in $[-r,r]$. Such test functions are dense in the mean-zero subspace.
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