Reproducing Kernel Hilbert Spaces Supporting Nontrivial Hermitian Weighted Composition Operators

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.