Abstract
We characterize multiplicative maps φ on semigroups of square matrices satisfying φ( P)⊆ P for matrix sets P , such as rank k (idempotent) matrices, totally nonnegative matrices, P 0 matrices, M 0 matrices, positive semidefinite matrices, Hermitian matrices, normal matrices, and contractions. We also characterize multiplicative maps φ satisfying φ( g( X))= φ( X) for various functions g on square matrices, such as the spectrum, spectral radius, numerical range, numerical radius, and matrix norms.
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