Let $(E,\mathcal E,\pi)$ be a probability space, $$L^2_0(\mu):=\left\{f\in L^2(\mu):\mu f=0\right\}$$ and $\kappa$ be a Markov kernel on $(E,\mathcal E)$ such that $\mu$ is reversible with respect to $\kappa$. We may treat $P$ as a self-joint linear contraction (operator norm at most $1$) on $L^2_0(\mu)$ via $$Pf:=\int\kappa(\;\cdot\;,{\rm d}y)f(y).$$ Assume that the spectrum $\sigma(P)$ is contained in $[-1,1)$ and hence $L:=1-P$ is bijective with and $L^{-1}$ is bounded. I'm interested in the quantity $$\sup_{\substack{f\:\in\:L^2_0(\pi)\\\left\|f\right\|_{L^2(\pi)}\:=\:1}}\langle Lf,f\rangle_{L^2(\pi)}.\tag1$$
If $E$ is finite, say $E=\{1,\ldots,n\}$, and $\mathcal E=2^E$, we may treat $\pi$ as a vector in $\mathbb R^n$ and $(1)$ is equal to (see definitions below) $$\max_{\substack{z\:\in\:\mathbb R^{n-1}\\|z|\:=\:1}}\left\langle\left(I_{n-1}-V_2^\#PV_2\right)^{-1}z,z\right\rangle\tag2.$$ Let $R(P;\pi):=\left(I_{n-1}-V_2^\#PV_2\right)^{-1}$. I've read in a paper that the maximizer of $(2)$ is attained at the unit eigenvector associated with largest eigenvalue $\lambda_{\text{max}}$ of the symmetric matrix $R(P;\pi)+{R(P;\pi)}^T$ and the optimal objective value is the logarithmic norm $\frac{\lambda_{\text{max}}}2$ of $R(P;\pi)$.
Question: Are we able to generalize this result?
