Preservers of matrix pairs with bounded distance

Let m, n and k be positive integers such that 2 ⩽ k < n ⩽ m. Let V denote either the vector space of all m × n matrices over a field with at least three elements or the vector space of all n × n Hermitian matrices over a field F of characteristic ≠ 2 associated with an involution. We characterize surjective mappings T from V onto itself such that for every pair A, B ∈ V, rank(A − B) ⩽ k if and only if rank(T(A) − T(B)) ⩽k.