In this talk, we propose optimal additive multilevel solvers for isogeometric discretizations of scalar elliptic problems for locally refined T-meshes. Applying the refinement strategy in Morgenstern & Peterseim (2015, Analysis-suitable adaptive T-mesh refinement with linear complexity. Comput. Aided Geom. Design, 34, 50–66) we can guarantee that the obtained T-meshes have a multilevel structure and that the associated T-splines are analysis suitable, for which we can define a dual basis and a stable projector. Taking advantage of the multilevel structure, we prove that BPX preconditioners have optimal complexity and present several numerical experiments to confirm our theoretical results.