Is the average of two Schur stable matrices also stable?

Question

Let $A$ and $B$ be arbitrary real matrices of the same dimension. If they are both Schur stable for discrete-time systems, i.e., all eigenvalues of $A$ and $B$ have norm strictly less than one, can we know whether the matrix $\frac{A}{2}+\frac{B}{2}$ is Schur stable? Some counterexamples or references are appreciated.

I know that the sum of two Hurwitz stable matrices is not necessary to be Hurwitz stable. I also notice that if the two Hurwitz stable matrices are commuting, their sum must be stable. If the answer to my question is no, is there a similar condition for Schur stable matrices.

Thanks.

Answer 1

In general no. However, it is the case when

1. $A$ and $B$ are normal.
2. There exists $T$ such that $T^{-1}AT$ and $T^{-1}BT$ are both upper triangular.
3. $AB=BA$.

Also, see my answer here.

For a counter example in the general case let $A=\begin{bmatrix}1/2&1\\0&1/2\end{bmatrix}$ and $B=A^T$.