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Reproducing kernels, de Branges-Rovnyak spaces, and norms of weighted composition operators. (English) Zbl 1137.47019
Given a Hilbert space \(H\) of functions defined on a set \(X\) and a function \(b:X \rightarrow X\), the composition operator \(C_b\) can be defined by \((C_b f)(x) = f(b(x))\). In the case of the Hardy space, \(C_b\) is bounded for every analytic function \(b\) mapping the unit disk to the unit disk and the standard proof uses the Littlewood subordination principle. A weighted composition operator on the Hardy space is of the form \(T_{\phi}C_b\), where \(\phi\) is the weight and \(T_{\phi}\) denotes the Toeplitz operator with symbol \(\phi\).
In this paper, the author considers the boundedness of weighted composition operators on the Hardy space from the perspective of the reproducing kernel \(k^b(z,w) = \frac{1-\overline{b(w)} b(z)}{1-\overline{w}z}\) associated with the de Branges–Rovnyak space \(H_b\). The boundedness is expressed in terms of positivity of kernels related to \(k_b\) and provides a useful alternate proof to the standard approach. The techniques can be generalized to the standard weighted Bergman spaces.

MSC:
47B33 Linear composition operators
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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