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Compactness property associated with the quasi-normed integral operator ideals
Thesis (PhD (Mathematical Sciences))--University of Pretoria, 2024.
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University of Pretoria
2024
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| _version_ | 1867613458994823168 |
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| access_status_str | Open Access |
| author2 | Maepa, Charles |
| author_browse | Maepa, Charles |
| author_facet | Maepa, Charles |
| collection | Thesis |
| dc_rights_str_mv | © 2023 University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria. |
| description | Thesis (PhD (Mathematical Sciences))--University of Pretoria, 2024. |
| format | Thesis |
| id | oai:repository.up.ac.za:2263/99965 |
| institution | University of Pretoria (South Africa) |
| language | English |
| last_indexed | 2026-06-10T12:36:28.597Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UPSpace — University of Pretoria Institutional Repository |
| publishDate | 2024 |
| publishDateRange | 2024 |
| publishDateSort | 2024 |
| publisher | University of Pretoria |
| publisherStr | University of Pretoria |
| record_format | dspace |
| source_str | UPSpace — University of Pretoria Institutional Repository |
| spelling | oai:repository.up.ac.za:2263/99965 Compactness property associated with the quasi-normed integral operator ideals Maepa, Charles nchihinga@gmail.com Ndumba, Brian Chihinga UCTD Sustainable Development Goals (SDGs) Quasi-normed integral operator ideals (p,r)-compactness mid (p,r)-compact sets mid (p,r)-compact operators (p,r)-limited sets Thesis (PhD (Mathematical Sciences))--University of Pretoria, 2024. In this thesis, we conduct a study on the (p, r)-compactness and mid (p, r)-compactness of subsets in Banach spaces for 1 ≤ p ≤ ∞, and 1 ≤ r ≤ p∗, where p∗ is the conjugate index of p. We begin by introducing and studying a compactness property which a Banach space may or may not have. This compactness property will be denoted by C_p^r and it is the class of all Banach spaces X such that X belongs to C_p^r if for every bounded subset A of X, A is relatively (p, r)-compact if, and only if, U_A^∗ belongs to the injective hull of the (p, r∗, 1)-integral operators where U_A^∗ is the adjoint of the operator U_A : ℓ_1(A) → X. Our main interest is to investigate the relationship between the (p, r)-compactness of sets and the C_p^r Property of Banach spaces. Moreover, we will also prove a characterization that a Banach space Y has the C_p^r Property precisely when the (p, r)-compact operators from X into Y equals the surjective hull of the dual of the (p, r∗, 1)-integral operators from X into Y for every Banach space X. Other results with regard to the C_p^r Property of Banach spaces will also be proved. We also introduce and study mid (p, r)-compact sets and operators. We begin by introducing and defining the mid (p, r)-compact subsets of a Banach space X and the mid (p, r)-compact operators between Banach spaces X and Y . The set of mid (p, r)-compact operators between Banach spaces X and Y is denoted by K^mid_(p,r)(X, Y ). We prove that the ideal (K^mid_(p,r)(X, Y ), κ^mid_(p,r)(·)) is a quasi-Banach operator ideal. Finally, we introduce and study the (p, r)-limited subsets in Banach spaces. We prove that every mid (p, r)-compact subset of X is (p, r)-limited and that the set K^mid_(p,r)(X, Y ) consists of (p, r)-limited sets. Other results with regard to this ideal (K^mid_(p,r)(X, Y ), κ^mid_(p,r)(·)) and the (p, r)-limited sets will also be proved. Brian Chihinga Ndumba Mathematics and Applied Mathematics PhD (Mathematical Sciences) Unrestricted Faculty of Natural and Agricultural Sciences SDG-04: Quality education 2024-12-12T11:59:51Z 2024-12-12T11:59:51Z 2025-04-15 2024-11-14 Thesis * A2025 http://hdl.handle.net/2263/99965 NA en © 2023 University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria. application/pdf University of Pretoria |
| spellingShingle | UCTD Sustainable Development Goals (SDGs) Quasi-normed integral operator ideals (p,r)-compactness mid (p,r)-compact sets mid (p,r)-compact operators (p,r)-limited sets Compactness property associated with the quasi-normed integral operator ideals |
| title | Compactness property associated with the quasi-normed integral operator ideals |
| title_full | Compactness property associated with the quasi-normed integral operator ideals |
| title_fullStr | Compactness property associated with the quasi-normed integral operator ideals |
| title_full_unstemmed | Compactness property associated with the quasi-normed integral operator ideals |
| title_short | Compactness property associated with the quasi-normed integral operator ideals |
| title_sort | compactness property associated with the quasi normed integral operator ideals |
| topic | UCTD Sustainable Development Goals (SDGs) Quasi-normed integral operator ideals (p,r)-compactness mid (p,r)-compact sets mid (p,r)-compact operators (p,r)-limited sets |
| url | http://hdl.handle.net/2263/99965 |