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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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