  We show that the derivative nonlinear Schr\"odinger (DNLS) equation is
globally well-posed in the weighted Sobolev space $H^{2,2}(\mathbb{R})$. Our
result exploits the complete integrability of DNLS and removes certain spectral
conditions on the initial data, thanks to Xin Zhou's analysis on spectral
singularities in the context of inverse scattering.
