Range-compatible homomorphisms on matrix spaces

Abstract

Let K be a (commutative) field, and U and V be finite-dimensional vector spaces over K. Let S be a linear subspace of the space L(U,V) of all linear operators from U to V. A map F:S→V is called range-compatible when F(s)∈Ims for all s∈S. Obvious examples of such maps are the evaluation maps s↦s(x), with x∈U.In this article, we classify all the range-compatible group homomorphisms on S provided that codimL(U,V)S≤2dim⁡V−3, unless K has cardinality 2 and codimL(U,V)S=2dim⁡V−3. Under those assumptions, it is shown that the linear range-compatible maps are the evaluation maps, and the above upper-bound on the codimension of S is optimal for this result to hold.As an application, we obtain new sufficient conditions for the algebraic reflexivity of an operator space and, with the above conditions on the codimension of S, we give an explicit description of the range-restricting and range-preserving homomorphisms on S.