Generalised classical adjointcommuting mappings on matrix spaces / Ng Wei Shean - Ng, Wei Shean (2016)

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Let m, n be integers with m, n > 3, and let F and K be fields. We denote by Mn(F) the linear space of n × n matrices over F, Sn(F) the linear space of n × n symmetric matrices over F and Kn(F) the linear space of n × n alternate matrices over F. In addition, let F be a field with an involution −, we denote by Hn(F) the F−-linear space of n × n hermitian matrices over F and SHn(F) the F−-linear space of n×n skew-hermitian matrices over F where F− is a fixed field of F. We let adj A be the classical adjoint of a matrix A and In be the n × n identity matrix. In this dissertation, we characterise mappings ψ that satisfy one of the following conditions: (A1) ψ :Mn(F) →Mm(F) with either |F| = 2 or |F| > n + 1, and ψ(adj (A + αB)) = adj (ψ(A) + αψ(B)) for all A,B ∈Mn(F) and α ∈ F; (A2) ψ :Mn(F) →Mm(K) where ψ is surjective and ψ(adj (A − B)) = adj (ψ(A) − ψ(B)) for all A,B ∈Mn(F). Besides, we also study the structure of ψ on Hn(F), Sn(F), SHn(F) and Kn(F). We obtain a complete description of ψ satisfying condition (A1) or (A2) on Mn(F), Hn(F) and Sn(F) if ψ(In) 6= 0. If ψ(In) = 0, we prove that such mappings send all rank one matrices to zero. Clearly, ψ = 0 when ψ is linear. Some examples of nonlinear mappings ψ satisfying condition (A1) or (A2) with ψ(In) = 0 are given. In the study of ψ satisfying condition (A1) or (A2) on Kn(F), we obtain a nice structural result of ψ if ψ(A) = 0 for some invertible matrix A ∈ Kn(F). Some examples of nonlinear mappings ψ vanishing all invertible matrices are included. In the case of SHn(F), some examples of nonlinear mappings ψ satisfying condition (A1) or (A2) that send all rank onev matrices and invertible matrices to zero are given. Otherwise, a nice structural result of ψ is obtained.