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Positive weighted koopman semigroups on banach lattice modules
Thesis (MSc)--Stellenbosch University, 2023.
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| Format: | Thesis |
| Language: | en_ZA |
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Stellenbosch : Stellenbosch University
2023
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| _version_ | 1867614089376694272 |
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| access_status_str | Open Access |
| author | Olabiyi, Tobi David |
| author2 | Heymann, Retha |
| author_browse | Heymann, Retha Olabiyi, Tobi David |
| author_facet | Heymann, Retha Olabiyi, Tobi David |
| author_sort | Olabiyi, Tobi David |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (MSc)--Stellenbosch University, 2023. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/127258 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:46:29.473Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2023 |
| publishDateRange | 2023 |
| publishDateSort | 2023 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/127258 Positive weighted koopman semigroups on banach lattice modules Olabiyi, Tobi David Heymann, Retha Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Banach modules (Algebra) Topological algebras Banach lattices Commutative algebra UCTD Thesis (MSc)--Stellenbosch University, 2023. ENGLISH SUMMARY: In this thesis, we introduce the notion of a positive weighted semigroup representation on a Banach lattice module over a group representation on a commutative Banach lattice algebra. One main theme of this work is the following: for topological dynamics, we obtain the abstract representation of the lattice of continuous sections vanishing at infinity of a topological Banach lattice bundle (over a locally compact space Ω) as a structure which we call an AM m-lattice module over C0(Ω) on which every positive weighted semigroup representation over the Koopman group representation on C0(Ω) is isomorphic to a positive weighted Koopman semigroup representation induced by a unique positive semiflow on the underlying topological Banach lattice bundle (over the continuous flow on the base space Ω). And as a result, every positive dynamical Banach lattice bundle can be assigned uniquely to a certain positive dynamical m-lattice module and vice versa, which is the Gelfand-type theorem that we proved. In order to do this, we, in particular, establish the following two categories of (i) Banach lattice modules and their dynamics; and (ii) Banach lattice bundles and their dynamics. We pay special attention to the case of a topological positive R+-dynamical Banach lattice bundle by which we obtain the corresponding C0-semigroup of positive weighted Koopman operators, and using the theory of strongly continuous semigroup of positive operators, we obtain results pertaining to properties of the generator, and spectral theory of this positive semigroup. AFRIKAANSE OPSOMMING: In hierdie tesis stel ons die idee van ’n positiewe geweegde halfgroepvoorstelling op ’n Banach-roostermodule oor die groepvoorstelling op ’n kommutatiewe Banach-roosteralgebra bekend. Een hooftema van hierdie tesis is die volgende: vir topologiese dinamika verkry ons die abstrakte voorstelling van die rooster van kontinue snitte wat verdwyn by oneindig van ’n topologiese Banach roosterbundel (oor ’n lokaal-kompakte ruimte Ω ) as ’n struktuur wat ons ’n AM-m-roostermodule oor C0(Ω) noem, waarop elke positiewe geweegde halfgroepvoorstelling oor die Koopman-groepvoorstelling op C0(Ω) isomorfies i s a an ’ n positiewe geweegde Koopmanhalfgroepvoorstelling geinduseer deur ’n unieke positiewe halfvloei op die onderliggende topologiese Banach-roosterbundel (oor die kontinue vloei op die basisruimte Ω). Gevolglik kan elke positiewe dinamiese Banach-roosterbundel uniek aan ’n sekere positiewe dinamiese m-roostermodule toegeken word en omgekeerd, wat die Gelfand-tipe stelling is wat ons bewys het. Om dit te doen, stel ons veral die volgende twee kategoriee van (i) Banachroostermodules en hul dinamika; en (ii) Banach-roosterbundels en hul dinamika bekend. Ons gee besondere aandag aan die geval van ’n topolo giese positiewe R+-dinamiese Banach-roosterbundel waardeur ons die ooreenstemmende C0-halfgroep van positiewe geweegde Koopman-operatore verkry en deur die teorie van sterk-kontinue halfgroepe van positiewe operatore te gebruik, verkry ons resultate wat betrekking het op eienskappe van die generator, en spektraalteorie van hierdie positiewe halfgroep. Masters 2023-03-07T07:29:16Z 2023-05-18T07:12:27Z 2023-03-07T07:29:16Z 2023-05-18T07:12:27Z 2023-03 Thesis http://hdl.handle.net/10019.1/127258 en_ZA Stellenbosch University xvii, 220 pages ; includes annexures application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Banach modules (Algebra) Topological algebras Banach lattices Commutative algebra UCTD Olabiyi, Tobi David Positive weighted koopman semigroups on banach lattice modules |
| title | Positive weighted koopman semigroups on banach lattice modules |
| title_full | Positive weighted koopman semigroups on banach lattice modules |
| title_fullStr | Positive weighted koopman semigroups on banach lattice modules |
| title_full_unstemmed | Positive weighted koopman semigroups on banach lattice modules |
| title_short | Positive weighted koopman semigroups on banach lattice modules |
| title_sort | positive weighted koopman semigroups on banach lattice modules |
| topic | Banach modules (Algebra) Topological algebras Banach lattices Commutative algebra UCTD |
| url | http://hdl.handle.net/10019.1/127258 |
| work_keys_str_mv | AT olabiyitobidavid positiveweightedkoopmansemigroupsonbanachlatticemodules |