  We present the construction of additive multilevel preconditioners, also
known as BPX preconditioners, for the solution of the linear system arising in
isogeometric adaptive schemes with (truncated) hierarchical B-splines. We show
that the locality of hierarchical spline functions, naturally defined on a
multilevel structure, can be suitably exploited to design and analyze efficient
multilevel decompositions. By obtaining smaller subspaces with respect to
standard tensor-product B-splines, the computational effort on each level is
reduced. We prove that, for suitably graded hierarchical meshes, the condition
number of the preconditioned system is bounded independently of the number of
levels. A selection of numerical examples validates the theoretical results and
the performance of the preconditioner.
