Local linear dependence seen through duality I

Abstract

A vector space S of linear operators between vector spaces U and V is called locally linearly dependent (in abbreviated form: LLD) when every vector x ∈ U is annihilated by a non-zero operator in S. A duality argument bridges the theory of LLD spaces to the one of vector spaces of non-injective operators. This new insight yields a unified approach to rediscover basic LLD theorems and obtain many additional ones thanks to the power of formal matrix computations.In this article, we focus on the minimal rank for a non-zero operator in an LLD space. Among other things, we reprove the Brešar-Šemrl theorem, which states that an n-dimensional LLD operator space always contains a non-zero operator with rank less than n, and we improve the MeshulamSemrl theorem that examines the case when no non-zero operator has rank less than n − 1.We also tackle the minimal rank problem for a non-zero operator in an n-dimensional operator space that is not algebraically reflexive. A theorem of Meshulam andŠemrl states that, for all fields with cardinality large enough, a non-reflexive operator space with dimension n must contain a non-zero operator with rank at most 2n − 2. We show that there are infinitely many integers n for which this bound is optimal for general infinite fields. Moreover, under mild cardinality assumptions, we obtain a complete classification of the non-reflexive n-dimensional operator spaces in which no non-zero operator has rank less than 2n − 2. This classification involves a new algebraic structure called left-division-bilinearizable (in abbreviated form: LDB) division algebras, which generalize a situation that is encountered with quaternions and octonions and whose systematic study occupies a large part of the present article.

On the minimal rank in non-reflexive operator spaces over finite fields

“…This follows from Theorem 6.1 of [8] and from the fact, over a finite field, a quadratic form whose dimension is greater than 2 is always isotropic.…”

“…In [8], we examined whether the upper-bound 2 dim S − 2 from Meshulam anď Semrl's result was optimal or if one could improve it in the case when dim S ≥ 3. First, it was proved that this upper-bound still held under the milder cardinality assumption #K > dim S, and then, under that provision, a classification of the non-reflexive n-dimensional operator spaces S such that mrk S = 2n − 2 was achieved (see Theorem 6.1 of [8]): it was shown in particular that the existence of such spaces is connected to the existence of exotic division algebra structures over the field K, called left-division-bilinearizable (LDB) division algebras.…”

“…First, it was proved that this upper-bound still held under the milder cardinality assumption #K > dim S, and then, under that provision, a classification of the non-reflexive n-dimensional operator spaces S such that mrk S = 2n − 2 was achieved (see Theorem 6.1 of [8]): it was shown in particular that the existence of such spaces is connected to the existence of exotic division algebra structures over the field K, called left-division-bilinearizable (LDB) division algebras. The existence of LDB division algebras over K is deeply connected to the quadratic structure of K. LDB division algebras were entirely classified in [7], and as a consequence the following result was obtained 1 : Theorem 1.1.…”

“…Until now, the best known result over such fields was the following one: Proposition 1.3 (See Theorem 4.5 in [8]). Let S be an n-dimensional nonreflexive operator space.…”

“…To obtain the general case, it suffices to extend the former to all vector spaces U and V , which can be done by noticing that Meshulam anď Semrl's Corollary 2.5 of [5] states that if mrk(S) = 2 dim S − 2 then all the operators in S have finite-rank (provided that #K > n + 2, but it has been shown in [8] that it suffices to assume that #K > n) and one can then simply apply the above results to the reduced space associated with S, whose source and target spaces are finite-dimensional (see Section 3 for the definition of that reduced space).…”