Kucik, Andrzej Stanislaw (2017) Spaces of Analytic Functions on the Complex HalfPlane. PhD thesis, University of Leeds.

Text
Thesis.pdf  Final eThesis  complete (pdf) Available under License Creative Commons AttributionNoncommercialShare Alike 2.0 UK: England & Wales. Download (818Kb)  Preview 
Abstract
In this thesis we present certain spaces of analytic functions on the complex halfplane, 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 HardySobolev spaces. We show that the Laplace transform defines an isometric map from the weighted L^2(0, ∞) spaces into these newlyconstructed 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 halfplane. 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 halfplane 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 halfplane in the study of linear evolution equations, linking the admissibility criterion for control and observation operators to the boundedness of LaplaceCarleson embeddings.
Item Type:  Thesis (PhD) 

Keywords:  Hardy space; Bergman space; Zen space; Dirichlet space; HardySobolev space; Laplace transform; Lebesgue function space; Bergman kernel; reproducing kernel; reproducing kernel Hilbert space; Hilbert space; Banach space; Banach algebra; complex halfplane; analytic function; Carleson embedding; LaplaceCarleson embedding; Carleson measure; Carleson square; Whitney decomposition; multiplication operator; composition operator; weighted composition operator; multiplier; bounced operator; causal operator; causality; angular derivative; compact operator; evolution equation; LaplaceCarleson embedding; control operator; observation operator; admissibility; admissible operator; sectorial measure 
Academic Units:  The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) 
Identification Number/EthosID:  uk.bl.ethos.715055 
Depositing User:  Mr Andrzej Kucik 
Date Deposited:  26 Jun 2017 11:03 
Last Modified:  25 Jul 2018 09:55 
URI:  http://etheses.whiterose.ac.uk/id/eprint/17573 
You do not need to contact us to get a copy of this thesis. Please use the 'Download' link(s) above to get a copy.
You can contact us about this thesis. If you need to make a general enquiry, please see the Contact us page.