Analytic reproducing kernel Hilbert spaces and their operators
A criterion for boundedness of composition operators acting on the general class of Hilbert spaces of entire Dirichlet series, namely the class $\mathcal{H}(\beta,E)$, was obtained in [15]. Varied results of properties were analysed in earlier papers [22, 13, 2]. In this thesis we extend these resul...
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| Format: | Final Year Project (FYP) |
| Language: | English |
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Nanyang Technological University
2020
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| Online Access: | https://hdl.handle.net/10356/138941 |
| _version_ | 1826125035836801024 |
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| author | Mau, Camille |
| author2 | Le Hai Khoi |
| author_facet | Le Hai Khoi Mau, Camille |
| author_sort | Mau, Camille |
| collection | NTU |
| description | A criterion for boundedness of composition operators acting on the general class of Hilbert spaces of entire Dirichlet series, namely the class $\mathcal{H}(\beta,E)$, was obtained in [15]. Varied results of properties were analysed in earlier papers [22, 13, 2]. In this thesis we extend these results to the general setting of spaces of Dirichlet series holomorphic on the half-plane. A complete characterisation of boundedness of polynomial-induced composition operators is found. We then study several properties of these operators, obtaining several characterisations in complex symmetry, compactness, etc. A proof that a system of normalised reproducing kernels $(\widetilde{k_{\lambda_n}})$ is never a frame for the Hardy space $H^2$ is also analysed. A generalisation of the method was made to determine classes of spaces and sequences $(\widetilde{k_{\lambda_n}})$ which do not constitute frames for their parent spaces. |
| first_indexed | 2024-10-01T06:30:09Z |
| format | Final Year Project (FYP) |
| id | ntu-10356/138941 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T06:30:09Z |
| publishDate | 2020 |
| publisher | Nanyang Technological University |
| record_format | dspace |
| spelling | ntu-10356/1389412023-02-28T23:16:24Z Analytic reproducing kernel Hilbert spaces and their operators Mau, Camille Le Hai Khoi School of Physical and Mathematical Sciences University of Lille 1 Emmanuel Fricain lhkhoi@ntu.edu.sg; emmanuel.fricain@univ-lille.fr Science::Mathematics::Analysis A criterion for boundedness of composition operators acting on the general class of Hilbert spaces of entire Dirichlet series, namely the class $\mathcal{H}(\beta,E)$, was obtained in [15]. Varied results of properties were analysed in earlier papers [22, 13, 2]. In this thesis we extend these results to the general setting of spaces of Dirichlet series holomorphic on the half-plane. A complete characterisation of boundedness of polynomial-induced composition operators is found. We then study several properties of these operators, obtaining several characterisations in complex symmetry, compactness, etc. A proof that a system of normalised reproducing kernels $(\widetilde{k_{\lambda_n}})$ is never a frame for the Hardy space $H^2$ is also analysed. A generalisation of the method was made to determine classes of spaces and sequences $(\widetilde{k_{\lambda_n}})$ which do not constitute frames for their parent spaces. Bachelor of Science in Mathematical Sciences 2020-05-14T04:24:14Z 2020-05-14T04:24:14Z 2020 Final Year Project (FYP) https://hdl.handle.net/10356/138941 en application/pdf Nanyang Technological University |
| spellingShingle | Science::Mathematics::Analysis Mau, Camille Analytic reproducing kernel Hilbert spaces and their operators |
| title | Analytic reproducing kernel Hilbert spaces and their operators |
| title_full | Analytic reproducing kernel Hilbert spaces and their operators |
| title_fullStr | Analytic reproducing kernel Hilbert spaces and their operators |
| title_full_unstemmed | Analytic reproducing kernel Hilbert spaces and their operators |
| title_short | Analytic reproducing kernel Hilbert spaces and their operators |
| title_sort | analytic reproducing kernel hilbert spaces and their operators |
| topic | Science::Mathematics::Analysis |
| url | https://hdl.handle.net/10356/138941 |
| work_keys_str_mv | AT maucamille analyticreproducingkernelhilbertspacesandtheiroperators |