On a pattern of reflexive operator spaces

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.

show abstract

On a pattern of reflexive operator spaces

Cited by 10 publications

References 12 publications

Compressions, Graphs, and Hyperreflexivity

Compressions, Graphs, and Hyperreflexivity

Algebraic Reflexivity and Local Linear Dependence: Generic Aspects

Local linear dependence seen through duality I

Contact Info

Product

Resources

About