Let $P$ and $Q$ be two Markov transition kernel (or matrix) with the same stationary distribution $\pi$, furthermore assume they are all positive （an operator is positive if the spectrum are all non-negative. Define $P\succ Q$ in the sense that $P(x,y) \geq Q(x,y)$ for any $x\neq y$. I am wondering if $P^2 \succ Q^2$?
Tthe result is obviously not true without the positive assumption. Consider $P =\begin{pmatrix} 0&1\\ 1&0\\ \end{pmatrix}$ and $ Q = \begin{pmatrix} 0.5&0.5\\ 0.5&0.5\\ \end{pmatrix}$ gives us a counterexample.