I am reading a paper (link) which discusses discrete-time Markov processes on a state space $\Omega \subseteq \mathbb R^n$.
If $\pi(x, y)$ and $\pi_T(x, y)$ are both transition kernels, then consider $$ \hat \pi(x, y) = \int_{\Omega} \pi(x, z) \pi_T(z, y) dz. $$
The paper asserts (p. 5) that
...if we assume existence of a spectral gap for both $\pi$ and $π_T$ and denote the leading eigenvalues of these kernels by $\hat \lambda < 1, \lambda < 1$, and $λ_T < 1$, respectively, we have $\hat \lambda ≤ \lambda \lambda_T$.
Question: Why is this? I have found posts like this one that look like they might be relevant, but tey are for matrices, whereas I think I am dealing with general kernels.
Thanks.
