Additive preservers of rank-additivity on matrix spaces

AbstractLet F be a field. Let V denote the vector space of all m×n matrices over F or the vector space of all n×n symmetric matrices over F of characteristic not 2 or 3. For each fixed positive integer s⩾2, let Qs denote the set of all matrix pairs (A, B) in V such that rank(A+B)=rank(A)+rank(B)⩽s. We characterize additive mappings ψ on V such that (ψ(A),ψ(B))∈Qs whenever (A,B)∈Qs for a fixed s. We also describe the structure of linear mappings from the space of n×n matrices over F to the space of p×q matrices over F that preserve rank-additivity, where charF≠2