Preservers of Unitary Similarity Functions on Lie Products of Matrices

Denote by $M_n$ the set of $n\times n$ complex matrices. Let $f: M_n \rightarrow [0,\infty)$beacontinuousmapsuchthat$f(\mu UAU^*)= f(A)$ 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 $[0,\mathrm{\infty })$ for any rank 1 nilpotent $X\in M_n$. Characterizations are given for surjective maps $\varphi$ on $M_n$ satisfying $f(AB-BA) = f(\phi(A)\phi(B)-\phi(B)\phi(A))$. 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.