Normed Spaces of Continuous and Holomorphic Functions and Weighted Composition Operators
| dc.contributor.author | Bilokopytov, Ievgen | |
| dc.contributor.examiningcommittee | Schippers, Eric (Mathematics) | en_US |
| dc.contributor.examiningcommittee | Shamseddine, Khodr (Physics and Astronomy) | en_US |
| dc.contributor.examiningcommittee | Bonet, Jose (Universitat Politecnica de Valencia) | en_US |
| dc.contributor.supervisor | Zorboska, Nina (Mathematics) | en_US |
| dc.date.accessioned | 2019-01-08T16:21:09Z | |
| dc.date.available | 2019-01-08T16:21:09Z | |
| dc.date.issued | 2018-12-21 | en_US |
| dc.date.submitted | 2018-12-21T07:46:40Z | en |
| dc.degree.discipline | Mathematics | en_US |
| dc.degree.level | Doctor of Philosophy (Ph.D.) | en_US |
| dc.description.abstract | In the present work we develop a unified way of looking at Normed Spaces of Continuous and Holomorphic Functions (NSCF’s and NSHF’s) and of Weighted Composition Operators (WCO’s) between these spaces. If F is a normed space of continuous functions over a topological space X, we relate different properties of F. We show that compactness of the inclusion map JF from F into C(X) is equivalent to continuity of the operation of evaluation on BF×X, where BF is the closed unit ball of F endowed with the weak topology. Moreover, we show that these conditions follow from mere continuity of the norm of the point evaluations, under the additional assumption that F is a uniformly smooth Banach space (this was previously known for Hilbert spaces). Also, we show that F is reflexive if and only if JF is weakly compact and BF is closed in C(X) in any equivalent norm on F. On the other hand, we show that some mild conditions on F imply that X is metrizable. We also give some sufficient conditions that force WCO’s between NSCF’s to have continuous symbols (this problem was previously considered only for a specific choice of NSCF’s). We provide counterexamples that show that the problem is not trivial. An analogous problem is considered in the holomorphic setting. We also show that on a wide class of NSCF’s the only unitary multiplication operators are the multiples of identity. For a Reproducing Kernel Hilbert Space (RKHS) of holomorphic functions on a complex domain we give a formula that describes the Hermitean metrics on the domain which are pullbacks of some metric on the (dual of) the RKHS via the evaluation map. Then we consider the question when such metrics are invariant with respect to the group of automorphisms of the domain, and obtain some partial results in that direction. | en_US |
| dc.description.note | February 2019 | en_US |
| dc.identifier.uri | http://hdl.handle.net/1993/33648 | |
| dc.language.iso | eng | en_US |
| dc.rights | open access | en_US |
| dc.subject | Function Spaces | en_US |
| dc.subject | Weighted Composition Operators | en_US |
| dc.subject | Reproducing Kernel Hilbert Spaces | en_US |
| dc.title | Normed Spaces of Continuous and Holomorphic Functions and Weighted Composition Operators | en_US |
| dc.type | doctoral thesis | en_US |