be the orthogonal projection $T f := f - \ip{f, \bfo} \bfo$. Given what we have shown, we see that the norm of $Q^k T : \ell^2(V, \pi) \to \ell^\infty(V)$ is bounded by $\alpha_k$. By duality, the same bound holds for $T Q^k : \ell^1(V, \pi) \to \ell^2(V, \pi)$. As before, we deduce that the norm of $Q^k T Q^k : \ell^1(V, \pi) \to \ell^\infty(V)$ is at most $\alpha_k^2$ and the norm of $Q^k T Q^{k+1} : \ell^1(V, \pi) \to \ell^\infty$ is at most $\alpha_k \alpha_{k+1}$. Applying these inequalities to $f := \II x/\pi(x)$ gives \[e\].finb/.

r.rate In the infinite case, we do not actually need to assume that $a > 0$. That is, suppose that $Q$ is a transition matrix of a Markov chain that is reversible with respect to a positive measure $\pi$. Assume that $c := \inf \big\{ \pi(x) Q(x, y) \st x \ne y \hbox{ and }
Q(x, y) > 0 \big\} > 0$. If $\pi$ is infinite, then $$Q^k(x, x)/\pi(x) \le {4 \over c \sqrt{k+1}}
\,.
%\label e.better$$ To see this, we have only to establish that $$|f(x_0)|
\le
(1/c)\sum_{x, y \in V} c_2(x, y) |f(x) - f(y)|
%\,.
%\label e.2$$ as a substitute for 