Abstract: The Gross-Pitaevskii equation admits in two dimensions stationary solutions, among them the Ginzburg-Landau vortex of degree one. Its orbital stability was proved by Gravejat-Pacherie-Smets (Proc. London Math. Soc. 2022). We obtain sharp estimates for its linear asymptotic stability, via a detailed spectral study of the linearized operator, through the description of its spectrum, of its generalized eigenfunctions, and of the properties of the associated distorted Fourier transform. The presence of a resonance at zero energy weakens the decay in dispersive estimates, and makes the L2 norm grow logarithmically in time. Under a suitable orthogonality condition with respect to the resonance, improved dispersive estimates are obtained and the L2 norm remains bounded. This is joint work with P. Germain (Imperial College London) and E. Pacherie (CNRS & CY Cergy Paris Université).