Quasi-range-compatible affine maps on large operator spaces

Abstract

Let U and V be finite-dimensional vector spaces over an arbitrary field K, 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 [3,4], 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.

Large spaces of bounded rank matrices revisited

“…F maps every matrix of H(V) to a vector of its range. Using the classification of range-compatible maps over large matrix spaces [14,15], it is then possible except in very specific situations -in which the structure of H(V) is already fairly simple -to find a non-zero vector that is annihilated by all the matrices in V, which yields that V is equivalent to a subspace of R(0, r).…”

“…There, we will also recall the results on range-compatible and quasi-rangecompatible maps that were proved recently [14] and which will be used in the proofs of the special lifting results.…”

“…The first one deals with affine range-compatible maps on affine subspaces of matrices. Theorem 2.6 (Theorem 3.1 of [14]). Let n and p be non-negative integers, and S be an affine subspace of M n,p (K) such that codim S ≤ n − 2.…”