Nonsurjective maps between rectangular matrix spaces preserving disjointness, triple products, or norms

Let Mm,n be the space of m×n real or complex rectangular matrices. Two matrices A,B∈Mm,n are disjoint if A∗B=0n and AB∗=0m. We show that a linear map Φ:Mm,n→Mr,s preserving disjointness exactly when Φ(A)=U⎛⎜⎝A⊗Q1000At⊗Q2000⎞⎟⎠V,∀A∈Mm,n, for some unitary matrices U∈Mr,r and V∈Ms,s, and positive diagonal matrices Q1,Q2, where Q1 or Q2 may be vacuous. The result is used to characterize nonsurjective linear maps between rectangular matrix spaces preserving (zero) JB∗-triple products, the Schatten p-norms or the Ky--Fan k-norms.