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Let n be an even integer with n 4 . In this note we study classical adjoint commuting mappings ψ on the space of n×n alternate matrices, and on the space of n×n skew-Hermitian matrices with respect to a proper involution, satisfying one of the following conditions: • ψ(adj (A+αB)) = adj (ψ(A)+αψ(B)) • ψ(adj (A−B)) = adj (ψ(A)−ψ(B)) and ψ is surjective for scalar α and matrices A,B in each respective matrix spaces. Here, adj A denotes the classical adjoint of a matrix A . Mathematics subject classification (2010): 15A03, 15A04, 15A86.

Classical adjoint commuting mappings on alternate matrices and skew-Hermitian matrices