Reflexive sets of operators

Abstract

For a set $\mathcal{M}$ of operators on a complex Banach space $\mathcal{X}$, the reflexive cover of $\mathcal{M}$ is the set $Ref\left(\mathcal{M}\right)$ of all those operators $T$ satisfying $Tx\in \overline{\mathcal{M}x}$ for every $x\in \mathcal{X}$. Set $\mathcal{M}$ is reflexive if $Ref\left(\mathcal{M}\right)=\mathcal{M}$. The notion is well known, especially for Banach algebras or closed spaces of operators, because it is related to the problem of invariant subspaces. We study reflexivity for general sets of operators. We are interested in how the reflexive cover behaves towards basic operations between sets of operators. It is easily seen that the intersection of an arbitrary family of reflexive sets is reflexive, as well. However this does not hold for unions, since the union of two reflexive sets of operators is not necessarily a reflexive set. We give some sufficient conditions under which the union of reflexive sets is reflexive. We explore how the reflexive cover of the sum (resp., the product) of two sets is related to the reflexive covers of summands (resp., factors). We also study the relation between reflexivity and convexity, with special interest in the question: under which conditions is the convex hull of a reflexive set reflexive?