Pseudospectra and Simultaneous Power Control

We prove that, for $M>0$ and $n,m\ge 2$, we can find two $10\times 10$ matrices $A$ and $B$ with identical pseudospectra such that we have simultaneously $\mathrm{\parallel }{A}^n\mathrm{\parallel }\mathrm{/}\mathrm{\parallel }{B}^n\mathrm{\parallel }>M$ and $\mathrm{\parallel }{A}^m\mathrm{\parallel }\mathrm{/}\mathrm{\parallel }{B}^m\mathrm{\parallel }>M$. We also prove that, under certain conditions, this result holds for two more general functions of $A$ and $B$.