Reproducing Kernel Hilbert Spaces Supporting Nontrivial Hermitian Weighted Composition Operators |
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Authors: | Paul Bourdon Wenling Shang |
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Institution: | 1. Mathematics Department, Washington and Lee University, Lexington, VA, 24450, USA
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Abstract: | We characterize those generating functions ${k(z) = \sum_{j=0}^\infty z^j/\beta(j)^2}$ that produce weighted Hardy spaces H 2(β) of the unit disk ${\mathbb D}$ supporting nontrivial Hermitian weighted composition operators. Our characterization shows that the spaces associated with the “classical reproducing kernels” ${z \mapsto (1 - \bar{w}z)^{-\eta}}$ , where ${w \in \mathbb D}$ and η > 0, as well as certain natural extensions of these spaces, are precisely those that are hospitable to Hermitian weighted composition operators. It also leads to a refinement of a necessary condition for a weighted composition to be Hermitian, obtained recently by Cowen, Gunatillake, and Ko, into one that is both necessary and sufficient. |
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