Algebraic Reflexivity and Local Linear Dependence: Generic Aspects

Abstract

Both reflexivity and the LLD (local linear dependence) property for a space of linear operators are defined in terms of (“local”) one-sided action on vectors.W e survey the main developments concerning these two areas during the last decade, and observe a major difference between them.Whe reas the LLD property is verified on generic sets of (so-called separating) vectors, reflexivity must be tested on the entire vector space.W e duly introduce a modified notion of reflexivity (generic reflexivity) which is verified on generic sets of vectors, study its properties, and show that it is closer in spirit to LLD than the usual notion of reflexivity. In passing, we simplify and complete some recent results on LLD spaces of low dimension, in the special case of matrix spaces.W e answer an open problem on matrix pairs (A,B) for which AB belongs to the reflexive hull of B and B.Our approach is based on determinant-based genericity constructions and the Kronecker-Weierstrass canonical form. The study of reflexivity and the LLD property is restricted to operator spaces of very low dimension.W e provide basic partial results which apply also in arbitrary dimension.