Maps preserving spectral radius, numerical radius, spectral norm

Abstract

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 ◦ B∥ = ∥f(A) ◦ f(B)∥ for all matrices A and B.