Conjugate operators that preserve the nonconvergence of bounded martingales

We show that the conjugate T ∗ of an operator T :X →Y , with X and Y Banach spaces, satisfies the following dichotomy: either T ∗ preserves the nonconvergence of bounded martingales in Y∗, or there exists a compact operator K :X→Y such that the kernel N(T ∗ +K∗) fails the Radon–Nikodým property. © 2007 Elsevier Inc. All rights reserved