Asymptotic reflexivity Original Research Article

We introduce a new version of reflexivity, akin to approximate reflexivity, called Asymptotic Reflexivity. We prove that the unital algebra generated by any operator in image, where image is a Hilbert space, is asymptotically reflexive. We also show that a linear subspace image of image is asymptotically reflexive if and only if image is asymptotically, where image is the set of finite rank operators in image. This result, in particular, implies that the space image is asymptotically reflexive. An analogous version of Loginov–Shulman Theorem will be also proved for this notion of reflexivity. This result, in particular, implies that any linear subspace of normal operators is asymptotically reflexive. The relation between this notion of reflexivity and completely rank-nonincreasing maps will be studied as well.