Quasi-range-compatible affine maps on large operator spaces

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Let U and V be finite-dimensional vector spaces over an arbitrary field, and S be a subset of the space L(U,V) of all linear maps from U to V. A map F: S → V is called range-compatible when it satisfies F(s) ∈ im(s) for all s ∈ S; it is called quasi-range-compatible when the condition is only assumed to apply to the operators whose range does not include a fixed 1-dimensional linear subspace of V. Among the range-compatible maps are the so-called local maps s s(x) for fixed x ∈ U. Recently, the range-compatible group homomorphisms on S were classified when S is a linear subspace of small codimension in L(U,V). In this work, we consider several variations of that problem: we investigate range-compatible affine maps on affine subspaces of linear operators; when S is a linear subspace, we give the optimal bound on its codimension for all quasi-range-compatible homomorphisms on S to be local. Finally, we give the optimal upper bound on the codimension of an affine subspace S of L(U,V) for all quasi-range-compatible affine maps on it to be local.