Preservers of unitary similarity functions on Lie products of matrices

Denote by Mn the set of n×n complex matrices. Let f:Mn→[0,∞) be a continuous map such that f(μUAU⁎)=f(A) for any complex unit μ, A∈Mn and unitary U∈Mn, f(X)=0 if and only if X=0 and the induced map t↦f(tX) is monotonically increasing on [0,∞) for any rank one nilpotent X∈Mn. Characterization is given for surjective maps ϕ on Mn satisfying f(AB−BA)=f(ϕ(A)ϕ(B)−ϕ(B)ϕ(A)). The general theorem is then used to deduce results on special cases when the function is the pseudo spectrum and the pseudo spectral radius.