Conjugate operators that preserve the nonconvergence of bounded martingales

We show that the conjugate T∗ of an operator , 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 such that the kernel N(T∗+K∗) fails the Radon–Nikodým property.