Weighted composition operators on spaces of analytic functions on the complex half-plane

TL;DR

This paper characterizes the boundedness of weighted composition operators on Zen spaces, including Hardy and Bergman spaces, using Carleson measures and Bergman kernels, and explores properties of composition operators on these spaces.

Contribution

It provides new boundedness criteria for weighted composition operators on Zen spaces and links the boundedness of unweighted composition operators to angular derivatives at infinity.

Findings

Boundedness characterized by Carleson measures and Bergman kernels.

Unweighted composition operator boundedness linked to finite angular derivative.

No compact composition operators exist on Zen spaces.

Abstract

In this paper we will show how the boundedness condition for the weighted composition operators on a class of spaces of analytic functions on the open right complex half-plane called Zen spaces (which include the Hardy spaces and weighted Bergman spaces) can be stated in terms of Carleson measures and Bergman kernels. In Hilbertian setting we will also show how the norms of \emph{causal} weighted composition operators on these spaces are related to each other and use it to show that an \emph{(unweighted) composition operator} $C_\varphi$ is bounded on a Zen space if and only if $\varphi$ has a finite angular derivative at infinity. Finally, we will show that there is no compact composition operator on Zen spaces.