Let $P_1,P_2$ denote stochastic transition matrices on some countable set $I$. Consider $P_1,P_2$ as operators on $\ell^2(I)$.

Under which conditions can we show that for $t\in (0,1)$, $$\rho(tP_1+(1-t)P_2)\le t\rho(P_1)+(1-t)\rho(P_2),$$ where $\rho(\cdot )$ denotes the spectral radius???

If $P_1,P_2$ are selfadjoint, then the spectral radius can be replaced by the norm, and the inequality holds. If $P_1,P_2$ commute, then the inequality holds. But this seems very restrictive. I'm fine with the assumption that $P_1$ and $P_2$ have zeros at the same places.

Of course, it would be nice to consider this problem also for more general classes of Markov operators acting on some function space.

My favorite example is the case where $I$ is the free group $G$ of rank $d\ge2$ and the $P_1,P_2$ are given according to some measure on a set of generators, i.e. $P_i(g,g'):=\mu_i( g^{-1}g' ).$ In this case, the spectral radius of $P_i$ is strictly less than one, since the group is non-amenable (H. Kesten 1959). Note that non-zero constant functions are not in $\ell^2(G)$.

I'm interested in the case, where $G$ is non-amenable.

Update - about amenability If the group is $G$ is amenable, then for a large class of transition operators, the spectral radius is equal to one (e.g. uniform irreducibility and invariant measure bounded away from zero and infinity). Hence, under these conditions, all the spectral radii in the above inequality are equal to one. So, equality holds.

Example As an example, let $G$=$\mathbb{Z}$ and consider $P_1,P_2$ corresponding to the random walks, which go one step to the left resp. right with probabilities $p_i$ resp. $q_i$, $p_i+q_i=1$, $i=1,2$.

References I found some papers M.Zima about related properties for positive operators on partially ordered Banach spaces (see below). In there, some commutativity is required to prove the inequality in question. I don't want to assume commutativity. I would rather like to make use of the fact that our matrices are stochastic.

M.Zima, A theorem on the spectral radius of the sum of two operators and its applications, Bull. Austral. Math. Soc. 48 (1993), 427{434. MR 94j:47006

M.Zima, On the local spectral radius in partially ordered Banach spaces, Czechoslovak Math. J. 49 (1999), 835{841. MR 2001m:47011

and some more of the same author.

References Update In the following paper, Lax proved that the spectral radius on the set of $n\times n$ matrices with real eigenvalues is convex.

LAX, P. D. Differential equations, difference equations and matrix theory. Comm. Pure Appl. Math. 11 (1958), 175-194.