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We prove that the norm of a weighted composition operator on the Hardy spaceH^2 of the disk is controlled by the norm of the weight function in the deBranges-Rovnyak space associated to the symbol of the composition operator. Asa corollary we obtain a new proof of the boundedness of composition operatorson H^2, and recover the standard upper bound for the norm. Similar argumentsapply to weighted Bergman spaces. We also show that the positivity of ageneralized de Branges-Rovnyak kernel is sufficient for the boundedness of agiven composition operator on the standard functions spaces on the unit ball.

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