Reproducing Kernels, De Branges-rovnyak Spaces, and Norms of Weighted Composition Operators

SPACES, AND NORMS OF WEIGHTED COMPOSITION OPERATORS MICHAEL T. JURY Abstract. We prove that the norm of a weighted composition operator on the Hardy space H2 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 H2, 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 su cient for the boundedness of a given composition operator on the standard functions spaces on the unit ball. IfH is a vector space of functions de ned on a setX, given a function b : X ! X one can de ne a composition operator Cb by (Cbf)(x) = f(b(x)). When H is the Hardy space H2(D), the Hilbert space of functions analytic in the open unit disk D equipped with the norm kfk2 = sup 0<r<1 1 2 Z 2 0 jf(rei )j2 d ; the composition operator Cb is bounded for every analytic map b : D! D, and kCbk 1 + jb(0)j 1 jb(0)j 1=2 : The standard proof of these facts appeals to the Littlewood subordination principle in harmonic analysis; see [4]. In this note we give a proof of the boundedness of Cb on H2 which does not use the Littlewood subordination principle, only reproducing kernel methods. The idea behind the proof is to express the boundedness of certain weighted composition operators TfCb in terms of the positivity of kernels related to b and H2, in particular the kernels of the de Branges-Rovnyak spaces. The boundedness of Cb is obtained as Date: September 19, 2006. 2000 Mathematics Subject Classi cation. 47B33 (primary), 47B32, 46E22 (secondary).