Quasi-range-compatible affine maps on large operator spaces

Abstract

Let $U$ and $V$ be finite-dimensional vector spaces over an arbitrary field, and $\mathcal{S}$ be a subset of the space $\mathcal{L}(U,V)$ of all linear maps from $U$ to $V$. A map $F : \mathcal{S} \rightarrow V$ is called range-compatible when it satisfies $F(s) \in \mathrm{im}(s)$ for all $s \in \mathcal{S}$; 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 $s \mapsto s(x)$ for fixed $x \in U$. Recently, the range-compatible group homomorphisms on $\mathcal{S}$ were classified when $\mathcal{S}$ is a linear subspace of small codimension in $\mathcal{L}(U,V)$. In this work, we consider several variations of that problem: we investigate range-compatible affine maps on affine subspaces of linear operators; when $\mathcal{S}$ is a linear subspace, we give the optimal bound on its codimension for all quasi-range-compatible homomorphisms on $\mathcal{S}$ to be local. Finally, we give the optimal upper bound on the codimension of an affine subspace $\mathcal{S}$ of $\mathcal{L}(U,V)$ for all quasi-range-compatible affine maps on it to be local.