Local linear dependence seen through duality I

• Clémentde de Seguins Pazzis ^[1]
  1. [1] Université de Versailles Saint-Quentin-en-Yvelines, France
• Localización: Journal of pure and applied algebra, ISSN 0022-4049, Vol. 219, Nº 6 (June 2015), 2015, págs. 2144-2188
• Idioma: inglés
• DOI: 10.1016/j.jpaa.2014.07.029
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• Resumen

  • A vector space SS of linear operators between vector spaces U and V is called locally linearly dependent (in abbreviated form: LLD) when every vector x∈Ux∈U is annihilated by a non-zero operator in SS. 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 Meshulam–Šemrl 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 Šemrl 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.