On the $J$-reflexive operators

Abstract

A bounded linear operator $T$ on a Banach space $X$ is $J$-reflexive if every bounded operator on $X$ that leaves invariant the sets $J(T,x)$ for all $x$ is contained in the closure of $orb(T)$ in the strong operator topology. We discuss some properties of $J$-reflexive operators. We also give and prove some necessary and sufficient conditions under which an operator is $J$-reflexive. We show that isomorphisms preserve $J$-reflexivity and some examples are considered. Finally, we extend the $J$-reflexive property in terms of subsets.