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

Abstract:

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.