Spaces of Analytic Functions on the Complex Half-Plane

In this thesis we present certain spaces of analytic functions on the complex half-plane, including the Hardy, the Bergman spaces, and their generalisation: Zen spaces. We use the latter to construct a new type of spaces, which include the Dirichlet and the Hardy-Sobolev spaces. We show that the Laplace transform defines an isometric map from the weighted L^2(0, ∞) spaces into these newly-constructed spaces. These spaces are reproducing kernel Hilbert spaces, and we employ their reproducing kernels to investigate their features. We compare corresponding spaces on the disk and on the half-plane. We present the notions of Carleson embeddings and Carleson measures and characterise them for the spaces introduced earlier, using the reproducing kernels, Carleson squares and Whitney decomposition of the half-plane into an abstract tree. We also study multiplication operators for these spaces. We show how the Carleson measures can be used to test the boundedness of these operators. We show that if a Hilbert space of complex valued functions is also a Banach algebra with respect to the pointwise multiplication, then it must be a reproducing kernel Hilbert space and its kernels are uniformly bounded. We provide examples of such spaces. We examine spectra and character spaces corresponding to multiplication operators. We study weighted composition operators and, using the concept of causality, we link the boundedness of such operators on Zen spaces to Bergman kernels and weighted Bergman spaces. We use this to show that a composition operator on a Zen space is bounded only if it has a finite angular derivative at infinity. We also prove that no such operator can be compact. We present an application of spaces of analytic functions on the half-plane in the study of linear evolution equations, linking the admissibility criterion for control and observation operators to the boundedness of Laplace-Carleson embeddings.