On Tauberian and co-Tauberian operators

Abstract

We show that a Banach space X has an infinite dimensional reflexive subspace (quotient) if and only if there exist a Banach space Z and a non- isomorphic one-to-one (dense range) Tauberian (co-Tauberian) operator form X to Z (Z to Z). We also give necessary and sufficient condition for the existence of a Tauberian operator from a separable Banach space to c0 which in turn generalizes a result of Johnson and Rosenthal. Another application of our result shows that if X** is separable, then there exists a renorming of X for which, X is essentially the only subspace contained in the set of norm attaining functionals on X*.