Reproducing kernels, de Branges-Rovnyak spaces, and norms of weighted composition operators

We prove that the norm of a weighted composition operator on the Hardy space H-2 of the disk is controlled by the norm of the weight function in the de Branges- Rovnyak space associated to the symbol of the composition operator. As a corollary we obtain a new proof of the boundedness of composition operators on H-2 and recover the standard upper bound for the norm. Similar arguments apply to weighted Bergman spaces. We also show that the positivity of a generalized de Branges- Rovnyak kernel is sufficient for the boundedness of a given composition operator on the standard function spaces on the unit ball.