Pseudospectra and Simultaneous Power Control

Abstract

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 $\Vert A^n\Vert /\Vert B^n\Vert >M$ and $\Vert A^m\Vert /\Vert B^m\Vert >M$. We also prove that, under certain conditions, this result holds for two more general functions of $A$ and $B$.