Classical adjoint-commuting mappings on Hermitian and symmetric matrices

Let m and n be integers such that m,n⩾3, and let F and K be fields which possess involutions − of F and ∧ of K, respectively. Let HnF be the K--linear space n×n Hermitian matrices over F. In this note, we address the general description of mappings ψ satisfying one of the following conditions:1.ψ:Hn(F)→Hm(K), with either |K∧|=2, or |F-||K∧|>3 and F and K of characteristic ≠2 if − and ∧ are the identity maps, and ψ is surjective satisfying ψ(adj(A-B))=adj(ψ(A)-ψ(B)) for every A,B∈Hn(F).2.ψ:Hn(F)→Hm(F), with either |F-|=2 or |F-|>n+1, and ψ(adj(A+αB))=adj(ψ(A)+αψ(B)) for every A,B∈Hn(K) and α∈F-.3.ψ:Hn(F)→Hm(K) is additive with adjψ(A)=ψ(adjA) for every A∈Hn(F). Here, and are the fixed fields with respect to the involutions − of F and ∧ of K, respectively, and adjA denotes the classical adjoint of the matrix A.