value, such as all $f \in \ell^2_0(V, \pi)$. Take $f \in \ell^2_0(V, \pi)$. Notice that $\sum_{x, y \in V} c_2(x, y) = \sum_{x \in V} \pi(x) = 1$. Thus, we have from \[e\].1/ that $$(2 a c)^2 \|f\|_\infty^2
\le
{1\over2} \sum_{x, y \in V} c_2(x, y) [f(x) - f(y)]^2
= \bigip{ (I-Q^2) f, f } 
\,.
\label e.sqbd$$ Alternatively, we may apply \[e\].1/ to the function $\sgn(f)
f^2$. Using the trivial inequality $$|\sgn(s) s^2 - \sgn(t) t^2| \le |s - t| \cdot \big(|s| + |t|\big)
\,,$$ valid for any real numbers $s$ and $t$, we obtain that (2 a c)\^2 f\_\^4 &([12]{} \_[x, y V]{} c\_2(x, y) |f(x) - f(y)| (|f(x)| + |f(y)|) )\^2 & .

Putting both these estimates together, we get $$2 (a c)^2 \max \{ 2 \|f\|_\infty^2,\, \|f\|_\infty^4 \}
\le \bigip{ (I-Q^2) f, f }$$ for $\ip{f, f} = 1$. As before, this implies that $$(k+1) 2 (a c)^2 \max \big\{ 2 \|Q^k f\|_\infty ^2,\, \|Q^k f\|_\infty ^4 \big\}
\le
1
\,,$$ which shows that the norm of $Q^k : \ell^2_0(V, \pi) \to \ell^\infty(V)$ is bounded by $$\alpha_k 
:=
\min \big\{ [(2 a c)^2 (k+1)]^{-1/2},\, [(2k+2) (a c)^2]^{-1/4}\big\}
\,.$$ Let $T :
\ell^2(V, \pi) \to \ell^2_0(V, \pi)$ 