Preservers of Unitary Similarity Functions on Lie Products of Matrices

Abstract

Denote by $M_n$ the set of $n\times n$ complex matrices. Let $f:M_n\to [0,\infty )$ be a continuous map such that $f(\mu UA{U}^{\ast })=f(A)$ for any complex unit $\mu$, $A\in M_n$ and unitary $U\in M_n$, $f(X)=0$ if and only if $X=0$ and the induced map $t\mapsto f(tX)$ is monotonic increasing on $[0,\infty )$ for any rank 1 nilpotent $X\in M_n$. Characterizations are given for surjective maps $\varphi$ on $M_n$ satisfying $f(AB-BA)=f(\varphi (A)\varphi (B)-\varphi (B)\varphi (A))$. The general theorem are then used to deduce results on special cases when the function is the pseudo spectrum and the pseudo spectral radius, that answers a question of Molnar raised at the 2014 CMS summer meeting.

16 pages