Let $A, B \in \mathbb{R}^{n\times n}$ be two invertible matrices. I want to know if one can always find real scalars $\lambda_1, \ldots,\lambda_p$ such that
$$\rho\left(\prod_{i=1}^{p} (A- \lambda_i B) \right) < 1$$
where $\rho$ denotes the spectral radius.
Take $$ A = \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix},\quad B = \begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix}. $$ They are simoultaneously diagonalizable and the eigenvalues of $A+\lambda B$ are $1\pm\lambda \text i$, that have both norm at least 1. Notice that the eigenvalues of $\prod_{i=1}^{p} (A- \lambda_i B) $ are $\prod_{i=1}^{p} (1- \lambda_i \text i) $ and $\prod_{i=1}^{p} (1+ \lambda_i \text i) $, that have both norm at least 1.