ELA, Volume 16, pp. 347-365, October 2007, abstract.

Maps Preserving Spectral Radius, Numerical radius, Spectral norm

Chi-Kwong Li, Edward Poon, and Ashwin Rastogi

Characterizations are obtained for Schur (Hadamard) multiplicative
maps on complex matrices preserving the spectral radius, numerical
radius, or spectral norm. Similar results are obtained for maps
under weaker assumptions. Furthermore, a characterization is given
for maps f satisfying ||A o B|| = ||f(A) o f(B)|| for
all matrices A and B.