Let $(E,\mathcal E,\mu)$ be a probability space and $\kappa$ be a Markov kernel on $(E,\mathcal E)$ such that $\mu$ is reversible with respect to $\kappa$ and hence $$\kappa f:=\int\kappa(\;\cdot\;,{\rm d}y)f(y)\;\;\;\text{for }f\in L^2(\mu)$$ is self-adjoint.
Let $f\in L^2(\mu)$ with $f\ge0$ and $$f_0:=f-\int f\:{\rm d}\mu.$$ Assume $$\int(\kappa f_0)^2\le\rho^2\int f_0^2\:{\rm d}\mu\tag1$$ for some $\rho\in[0,1)$. I would like to show that $$\int(\kappa^nf_0)^2\:{\rm d}\mu\le\rho^{2n}\int f_0^2\:{\rm d}\mu\;\;\;\text{for all }n\in\mathbb N.\tag2$$ Are we able to show this?
Using Jensen's inequality, I'm only able to show that $$\int(\kappa^nf_0)^2\le\rho^2\int f^2\:{\rm d}\mu\tag3.$$