Spaces of analytic functions are usually studied in the context of the open unit disk of the complex plane, and other domains are seldom considered. We will present the so-called Zen spaces, which are a generalisation of the Hardy and the weighted Bergman spaces of analytic functions defined on the complex half-plane. We will also show how these can be generalised even further, to include for example the weighted Dirichlet spaces and the Hardy-Sobolev spaces. Thanks to a variant of the Paley-Wiener theorem, the versions of these spaces defined on the disk and on the half-plane can be seen as discrete and continuous counterparts. We will illustrate the difference between them by studying the weighted composition operators and a related notion of a Carleson measure.

 

And finally, we will present an application of spaces of analytic functions defined on the open right complex half-plane in the study of linear evolution equations, in particular, linking the admissibility criterion for control and observation operators to the boundedness of the Laplace-Carleson embeddings.

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