Spaces of Analytic Functions on the Complex Half-Plane
Andrzej Kucik (Leeds University)
Tuesday 23rd May, 2017 16:00-17:00 Math seminar room
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.