  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 $f(\mu UAU^*)= f(A)$ 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
$[0,\infty)$ for any rank 1 nilpotent $X \in M_n$. Characterizations are given
for surjective maps $\phi$ 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.
