In this paper we will show how the boundedness condition for the weightedcomposition operators on a class of spaces of analytic functions on the openright complex half-plane called Zen spaces (which include the Hardy spaces andweighted Bergman spaces) can be stated in terms of Carleson measures andBergman kernels. In Hilbertian setting we will also show how the norms of\emph{causal} weighted composition operators on these spaces are related toeach 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 finiteangular derivative at infinity. Finally, we will show that there is no compactcomposition operator on Zen spaces.
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