Pseudospectra and Simultaneous Control

Maëva Ostermann

Published 2020-11-23Version 1

We prove that, under some conditions, for $N\ge 10$, $M>0$ and two functions $f$ and $g$ holomorphic in a domain $\Omega$, we can find two $N\times N$ matrices $A$ and $B$ with identical pseudospectra such that we have simultaneously $||f(A)||/||f(B)||>M$ and $||g(A)||/||g(B)||>M$. In particular, this is the case if $f(z)=z^n$ and $g(z)=z^m$, where $n,m\ge2$.