We propose and analyze optimal additive multilevel solvers for isogeometric discretizations of scalar elliptic problems for locally refined T-meshes. Applying the refinement strategy in Morgenstern and Peterseim, Comput. Aided Geom. Design, 34 (2015), we can guarantee that the obtained T-meshes are p-admissible, which implies that the associated T-splines are analysis suitable. Taking advantage of the multilevel structure of p-admissible T-meshes, we develop a BPX preconditioner on the basis of local smoothing only for the functions affected by a newly added edge by bisection, and prove that our method has optimal complexity. Several numerical experiments confirm our theoretical result and also show the practical performance of the proposed preconditioner.