Reproducing kernel Hilbert spaces supporting nontrivial Hermitian weighted composition operators

TL;DR

This paper characterizes the generating functions for weighted Hardy spaces on the unit disk that support nontrivial Hermitian weighted composition operators, refining existing necessary conditions to be both necessary and sufficient.

Contribution

It provides a complete characterization of generating functions supporting Hermitian weighted composition operators in weighted Hardy spaces, extending classical results.

Findings

Spaces with classical reproducing kernels support Hermitian operators.

Certain natural extensions of these spaces also support such operators.

A refined necessary and sufficient condition for Hermitian weighted composition operators.

Abstract

We characterize those generating functions k that produce weighted Hardy spaces of the unit disk D supporting nontrivial Hermitian weighted composition operators. Our characterization shows that the spaces associated with the "classical reproducing kernels," 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.