The problem I'm struggling is the following:
Let $n$ be a positive integer and let $A=% \begin{pmatrix} B & C\\ C & B \end{pmatrix} \in\mathcal{M}_{2n}(\mathbb{R}_{+})$, where $B,C\in\mathcal{M}_{n} (\mathbb{R}_{+})$. I am interested in putting conditions on $B$ and $C$ such that the spectral radius of $A$ is less than $1$.
I think that the answer is that $B+C$ and $B-C$ have spectral radius less than $1$, but I'm not very familiar working with block matrices and I don't know how to prove it (I came up with this guess by working with the scalar case, when $B$ and $C$ are just nonnegative numbers).
I am also interested in computing the powers of $A$ in terms of the powers of $B$ and $C$, in the case when the spectral radius of $A$ is less than $1$ (is this similar to the case when $n=1$, or is there something fundamentally different?)
It's a partial answer, just for the first part of the question.
We have \begin{align} \det(A-XI)&=\det\pmatrix{B-XI&C\\C&B-XI}\\ &=\det\pmatrix{B-XI&C\\C+B-XI&B-XI+C}\\ &=\det\pmatrix{I&0\\0&B+C-XI}\det\pmatrix{B-XI&C\\I&I}\\ &=\det(B+C-XI)\det\pmatrix{B-C-XI&C\\0&I}\\ &=\det(B+C-XI)\det(B-C-XI), \end{align} hence $\sigma(A)=\sigma(B+C)\cup \sigma(B-C)$. We deduce that the spectral radius of $A$ is $<1$ if and only if so are those of $B+C$ and $B-C$.