On classical adjoint-commuting mappings between matrix algebras

Let F be a field and let m and n be integers with m,n⩾3. Let Mn denote the algebra of n×n matrices over F. In this note, we characterize mappings ψ:Mn→Mm that satisfy one of the following conditions:1.|F|=2 or |F|>n+1, and ψ(adj(A+αB))=adj(ψ(A)+αψ(B)) for all A,B∈Mn and α∈F with ψ(In)≠0.2.ψ is surjective and ψ(adj(A-B))=adj(ψ(A)-ψ(B)) for every A,B∈Mn.Here, adjA denotes the classical adjoint of the matrix A, and In is the identity matrix of order n. We give examples showing the indispensability of the assumption ψ(In)≠0 in our results.