Preservers of Unitary Similarity Functions on Lie Products of Matrices
Jianlian Cui, Chi-Kwong Li, Yiu‐Tung Poon · arXiv (Cornell University) · 2014
Denote by $M_n$ the set of $n\times n$ complex matrices. Let $f: M_n \rightarrow [0,\infty)$ be a continuous map such that $f(μUAU^*)= f(A)$ for any complex unit $μ$, $A \in M_n$ 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 $ϕ$ on $M_n$ satisfying $f(AB-BA) = f(ϕ(A)ϕ(B)-ϕ(B)ϕ(A))$. 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.