Continuous commutativity and spectrum preservers on normal matrices are conjugations or transpose conjugations, while on semisimple and general matrices they admit more exotic forms determined by the local regularity of complex conjugation near coincident eigenvalues.
Characterization of Jordan homomorphism onM n using preserving properties.Linear Algebra Appl., 269:33–46, 1998
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Continuous spectrum- and commutativity-preserving maps on normal matrices with spectra in a given interval image are conjugations, transpose conjugations, or spectrum orderings with fixed eigenspaces.
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Eigenvalue collision and exotic preservers on semisimple operators
Continuous commutativity and spectrum preservers on normal matrices are conjugations or transpose conjugations, while on semisimple and general matrices they admit more exotic forms determined by the local regularity of complex conjugation near coincident eigenvalues.
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The combinatorics of permuting and preserving curve-bound spectra
Continuous spectrum- and commutativity-preserving maps on normal matrices with spectra in a given interval image are conjugations, transpose conjugations, or spectrum orderings with fixed eigenspaces.