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 from S to V is called range-compatible when F(s) belongs to the range of s for all s in S. Obvious examples of such maps are the evaluation maps s -> s(x), with x in U. In this article, we classify all the range-compatible group homomorphisms on S provided that the codimension of S in L(U,V) is less than or equal to 2 dim(V)-3, unless this codimension equals 2 dim(V)-3 and K has only two elements. 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.Comment: 60 page

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).…”

“…The notion of a range-compatible map was introduced very recently [15]. It is motivated by its connection to the topic of our article (see Section 2 of [18]), by its connection to linear invertibility preservers (see [20]), and finally it is closely connected to the fashionable notion of algebraic reflexivity (see Section 1.1 of [16] for a thorough discussion).…”

Range-compatible homomorphisms on spaces of symmetric or alternating matrices

“…In this article, we will follow the main method from [5], which consists in performing inductive proofs over the dimension of the target space V , by using quotient spaces. Here, a major problem appears: if S is a subspace of S n (K) or of A n (K), "moding out" a non-zero subspace V 0 of V yields an operator space S mod V 0 which cannot be represented by a space of symmetric of alternating matrices, simply because the source and target spaces of the operators in S no longer have the same dimension!…”

“…The determination of the range-compatible homomorphisms on S n (K) was achieved in [5]. For the characteristic 2 case, we need some additional terminology before we can state the result.…”

Quasi-range-compatible affine maps on large operator spaces

“…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.…”