Given that $(\psi_i)_0^{|\chi|-1}$ is an orthonormal basis in $\ell^2(\pi)$ of real eigenfunctions of $K$ where $K$ is a Markov operator, then why is $\sum_0^{|\chi|-1} \psi_i^2(x)\pi(x) = 1$?
If we were summing over $x$, then it would make sense since $\psi_i$ is of unit norm. This time though, we're summing over $i$ so I don't see why the above still holds. I think it's false, but I want to make sure.
And the source of this is here: http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.aoap/1034968224
(At the end of page 702.)
I'm also stuck trying to prove corollary 2.2 in the paper, which is $\|h_t^x-1\|_2^2 \leq \frac{1}{\pi(x)}e^{-t}+\|k_x^{[t/2]}-1\|_2^2.$ Help would be much appreciated!
UPDATE I GOT IT.
However, I would still appreciate someone explaining to me why $\left\|H_t\right\|_{2\to \infty}=\sup_x h_{2t}(x,x)>1$ for all finite $t>0$, yet the limit as $t\to \infty$ is $1.$ That's unclear to me. Also why is $\left\|H_t\right\|_{2\to q}$ decreasing with respect to $t.$ I'd especially appreciate an intuitive explanation.