Local linear dependence seen through duality I

TL;DR

This paper explores the structure of locally linearly dependent operator spaces, providing new theorems on minimal rank operators, duality-based proofs, and a classification involving novel algebraic structures called LDB division algebras.

Contribution

It introduces a duality approach to LLD spaces, improves existing theorems on minimal rank, and classifies non-reflexive operator spaces using new algebraic structures.

Findings

Reproved Bresar-Semrl theorem on minimal rank in LLD spaces.

Improved Meshulam-Semrl theorem for rank bounds.

Classified non-reflexive operator spaces with minimal rank constraints.

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 of 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 Bresar-Semrl 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 Meshulam-Semrl 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 and Semrl states that, for all fields with large enough cardinality, 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.