Norm Preservers of Jordan Products
Electronic Journal of Linear Algebra
Norm preserver maps of Jordan product on the algebra M(n) of n x n complex matrices are studied, with respect to various norms. A description of such surjective maps with respect to the Frobenius norm is obtained: Up to a suitable scaling and unitary similarity, they are given by one of the four standard maps (identity, transposition, complex conjugation, and conjugate transposition) on M(n), except for a set of normal matrices; on the exceptional set they are given by another standard map. For many other norms, it is proved that, after a suitable reduction, norm preserver maps of Jordan product transform every normal matrix to its scalar multiple, or to a scalar multiple of its conjugate transpose.
Kuzma, Bojan; Lesnjak, Gorazd; Petek, Tatjana; Li, Chi-Kwong; and Rodman, Leiba, Norm Preservers of Jordan Products (2011). Electronic Journal of Linear Algebra, 22, 959-978.