Maps preserving spectral radius, numerical radius, spectral norm
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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.