This question relates to Levin, Peres, and Wilmer’s book, and specifically to http://www.statslab.cam.ac.uk/~beresty/Articles/mixing2.pdf.
I am looking for a proof to the following claim:
I do not see how this is an obvious statement as
$$\lambda_1 \leq \lambda_* \Rightarrow \gamma = 1-\lambda_1 \geq 1 - \lambda_* = \gamma_* \Rightarrow \frac{1}{\gamma} \leq \frac{1}{\gamma_*} = t_{rel}$$
But for this statement to be obvious we need the inequality to be in the other direction.
We assume the chain in question is reversible, so $\lambda_1=1-\gamma$ is real. Consider $f$ which is a nonzero eigenfunction for $\lambda_1$. Note that $E_\pi(f)=0$. Then $$\mathcal E(f,f)=\langle (I-P)f,f \rangle=\gamma \langle f,f \rangle \,,$$ so $$\text{var}_\pi(f)= \frac1{\gamma} \mathcal E(f,f)\,$$