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(Generalized) B-Fredholm theory relative to a Banach algebra homomorphism
Jamiu, A. 2025. (Generalized) B-Fredholm theory relative to a Banach algebra homomorphism. Unpublished masters thesis. Stellenbosch: Stellenbosch Univeristy [online]. Available: https://scholar.sun.ac.za/items/2688538a-aa3f-4cac-829b-e8b070491524
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Stellenbosch : Stellenbosch University
2025
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| _version_ | 1867613789408460800 |
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| access_status_str | Open Access |
| author | Jamiu, Aliyu |
| author2 | Benjamin, Ronalds |
| author_browse | Benjamin, Ronalds Jamiu, Aliyu |
| author_facet | Benjamin, Ronalds Jamiu, Aliyu |
| author_sort | Jamiu, Aliyu |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Jamiu, A. 2025. (Generalized) B-Fredholm theory relative to
a Banach algebra homomorphism. Unpublished masters thesis. Stellenbosch: Stellenbosch Univeristy [online]. Available: https://scholar.sun.ac.za/items/2688538a-aa3f-4cac-829b-e8b070491524 |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/132043 |
| institution | Stellenbosch University (South Africa) |
| last_indexed | 2026-06-10T12:41:43.824Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2025 |
| publishDateRange | 2025 |
| publishDateSort | 2025 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/132043 (Generalized) B-Fredholm theory relative to a Banach algebra homomorphism Jamiu, Aliyu Benjamin, Ronalds Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Banach algebras Homomorphisms (Mathematics) Fredholm operators UCTD Jamiu, A. 2025. (Generalized) B-Fredholm theory relative to a Banach algebra homomorphism. Unpublished masters thesis. Stellenbosch: Stellenbosch Univeristy [online]. Available: https://scholar.sun.ac.za/items/2688538a-aa3f-4cac-829b-e8b070491524 Thesis (MSc)--Stellenbosch University, 2025. ENGLISH ABSTRACT: In this thesis, following [10], we study two extensions of Harte’s Fredholm theory relative to a fixed Banach algebra homomorphism, namely B-Fredholm theory and generalized B-Fredholm theory (or, in short, GB-Fredholm theory). Let T : A → B be a Banach algebra homomorphism. Following [14], an element a ∈ A (w.r.t T) is called Fredholm if T a is invertible in B, and Weyl (resp. Browder) if it is the (resp. commuting) sum of an invertible element in A and an element belonging to the null space of T. Replacing "invertible" by "Drazin invertible" (resp. "Koliha-Drazin invertible") above, gives rise to the notions of B-Fredholm, B-Weyl and B-Browder elements (resp. GB-Fredholm, GB-Weyl and GB-Browder elements), which are studied in B-Fredholm theory (resp.GB-Fredholm theory) in general Banach algebras (see [10]). We recall from [12] (resp. [16]) that an element a ∈ A is said to be Drazin invertible (resp. Koliha-Drazin invertible) if there exists an element b ∈ A such that ab = ba, b = bab and a − aba is nilpotent (resp. quasinilpotent). Clearly, invertible elements are Drazin invertible, which in turn are Koliha-Drazin invertible. In this thesis, we present several interesting properties of elements from (G)B-Fredholm theory as well as the spectra derived from them. AFRIKAANSE OPSOMMING: In hierdie tesis, deur [22] te volg, bestudeer ons twee uitbreidings van Harte se Fredholmteorie relatief tot ’n vaste Banach algebra homomorfisme, naamlik, B-Fredholmteorie en veralgemeende B-Fredholmteorie (of, in kort, GBFredholmteorie). Laat T : A → B ’n Banach algebra homomorfisme wees. Vanuit [14] word ’n element a ∈ A (m.b.t. T) Fredholm genoem indien T a inverteerbaar is in B en Weyl (ondersk. Browder) indien dit die (ondersk. kommuterende) som van ’n inverteerbare element in A en ’n element in die nulruimte van T is. Vervanging van "inverteerbaar" met "Drazin inverteerbaar" (ondersk. "KolihaDrazin inverteerbaar") hierbo, gee aanleiding tot die konsepte van B-Fredholm, B-Weyl en B-Browder elemente (ondersk. GB-Fredholm, GB-Weyl en GBBrowder elemente), wat in B-Fredholmteorie (ondersk. GB-Fredholmteorie) in algemene Banach algebras bestudeer word (sien [10]). Ons herroep vanuit [12] (ondersk. [16]) dat ’n element Drazin inverteerbaar (ondersk. Koliha-Drazin inverteerbaar ) genoem word indien daar ’n element b ∈ A bestaan sodanig dat ab = ba, b = bab en a−aba nilpotent (ondersk. quasinilpotent) is. Dit is duidelik dat inverteerbare elemente Drazin inverteerbaar is, wat weer Koliha-Drazin inverteerbaar is. In hierdie tesis gee ons verskeie interessante eienskappe van elemente vanuit (G)B-Fredholmteorie, asook die spektra wat daaruit verkry word. Masters 2025-05-20T14:17:23Z 2025-05-20T14:17:23Z 2025-03 Thesis https://scholar.sun.ac.za/handle/10019.1/132043 Stellenbosch University xix, 104 pages application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Banach algebras Homomorphisms (Mathematics) Fredholm operators UCTD Jamiu, Aliyu (Generalized) B-Fredholm theory relative to a Banach algebra homomorphism |
| title | (Generalized) B-Fredholm theory relative to a Banach algebra homomorphism |
| title_full | (Generalized) B-Fredholm theory relative to a Banach algebra homomorphism |
| title_fullStr | (Generalized) B-Fredholm theory relative to a Banach algebra homomorphism |
| title_full_unstemmed | (Generalized) B-Fredholm theory relative to a Banach algebra homomorphism |
| title_short | (Generalized) B-Fredholm theory relative to a Banach algebra homomorphism |
| title_sort | generalized b fredholm theory relative to a banach algebra homomorphism |
| topic | Banach algebras Homomorphisms (Mathematics) Fredholm operators UCTD |
| url | https://scholar.sun.ac.za/handle/10019.1/132043 |
| work_keys_str_mv | AT jamiualiyu generalizedbfredholmtheoryrelativetoabanachalgebrahomomorphism |