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Polynomially Riesz elements: a study via Fredholm and Ruston theory
Thesis (MSc)--Stellenbosch University, 2025.
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Stellenbosch : Stellenbosch University
2026
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| _version_ | 1867614134876504064 |
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| access_status_str | Open Access |
| author | Sander, Nicholas Jan |
| author2 | Mouton, Sonja |
| author_browse | Mouton, Sonja Sander, Nicholas Jan |
| author_facet | Mouton, Sonja Sander, Nicholas Jan |
| author_sort | Sander, Nicholas Jan |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (MSc)--Stellenbosch University, 2025. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/134804 |
| institution | Stellenbosch University (South Africa) |
| last_indexed | 2026-06-10T12:47:13.037Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2026 |
| publishDateRange | 2026 |
| publishDateSort | 2026 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/134804 Polynomially Riesz elements: a study via Fredholm and Ruston theory Sander, Nicholas Jan Mouton, Sonja Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Fredholm operators Operator theory Banach algebras Homomorphisms (Mathematics) Riesz spaces Thesis (MSc)--Stellenbosch University, 2025. Sander, N. J. 2025. Polynomially Riesz elements: a study via Fredholm and Ruston theory. Unpublished masters thesis. Stellenbosch: Stellenbosch University [online]. Available: https://scholar.sun.ac.za/items/f8d0ccf7-bba5-4fa1-a8fd-22d27f8f7354 ENGLISH ABSTRACT: The Fredholm theory of Banach algebras studies different concepts relative to Banach algebra homomorphisms that are related to invertibility. Riesz elements explore the notion of being quasinilpotent relative to a Banach algebra homomorphism. As a result, Riesz elements play a vital role in the Fredholm theory of Banach algebras. In this thesis we discuss even more general notions of quasinilpotent and Riesz elements: polynomially quasinilpotent and polynomially Riesz elements that were introduced by Živković-Zlatanović, Djordjević and Harte. Let A and B be Banach algebras, T : A → B a (Banach algebra) homomorphism and a ∈ A. Then a is said to be polynomially quasinilpotent if there exists a non-zero polynomial p such that p(a) is quasinilpotent, i.e. σ(p(a)) = {0}. Furthermore, a is said to be polynomially Riesz if there exists a non-zero polynomial p such that p(a) is Riesz, i.e. σ_T(p(a)) = {0}. We show that quasinilpotent and Riesz elements form an important part of the Fredholm theory of Banach algebras. Then we investigate which of their properties have analogues where we replace quasinilpotent elements with polynomially quasinilpotent elements and Riesz elements with polynomially Riesz elements. We focus on the perturbation and spectral properties involving Fredholm, Weyl and Browder elements and spectra. AFRIKAANSE OPSOMMING: Die Fredholmteorie van Banach-algebras bestudeer verskillende konsepte relatief tot ’n Banach-algebra homomorfisme wat verwant aan inverteerbaarheid is. Rieszelemente ondersoek wat dit beteken om kwasi-nulpotent relatief tot ’n Banachalgebra homomorfisme te wees. As gevolg hiervan speel Riesz-elemente ’n noodsaaklike rol in die Fredholmteorie van Banach-algebras. In hierdie tesis bespreek ons meer algemene weergawes van kwasi-nulpotente en Riesz-elemente wat deur Zivkovi´c-Zlatanovi´c, Djordjevi´c en Harte bekend gestel is, naamlik polinomiese ˇ kwasi-nulpotente en polinomiese Riesz-elemente. Laat A en B Banach-algebras wees, T : A → B ’n (Banach-algebra) homomorfisme en a ∈ A. Dan is a polinomies kwasi-nulpotent indien daar ’n polinoom p is wat nie nul is nie sodat p(a) kwasi-nulpotent is, dit wil sˆe, σ(p(a)) = {0}. Verder is a polinomies Riesz indien daar ’n polinoom p is wat nie nul is nie sodat p(a) Riesz is, dit wil sˆe, σ T p(a) = {0}. Ons wys dat kwasi-nulpotente en Riesz-elemente ’n belangrike gedeelte van die Fredholmteorie van Banach-algebras vorm. Daarna ondersoek ons watter van hul eienskappe analo¨e het waar ons kwasi-nulpotente elemente met polinomiese kwasi-nulpotente elemente en Riesz-elemente met polinomiese Riesz-elemente kan vervang. Ons fokus op die steuring- en spektrale eienskappe wat verband hou met Fredholm, Weyl en Browder elemente en spektra. Masters 2026-01-08T10:31:56Z 2026-01-08T10:31:56Z 2025-12 Thesis https://scholar.sun.ac.za/handle/10019.1/134804 Stellenbosch University xv, 105 pages application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Fredholm operators Operator theory Banach algebras Homomorphisms (Mathematics) Riesz spaces Sander, Nicholas Jan Polynomially Riesz elements: a study via Fredholm and Ruston theory |
| title | Polynomially Riesz elements: a study via Fredholm and Ruston theory |
| title_full | Polynomially Riesz elements: a study via Fredholm and Ruston theory |
| title_fullStr | Polynomially Riesz elements: a study via Fredholm and Ruston theory |
| title_full_unstemmed | Polynomially Riesz elements: a study via Fredholm and Ruston theory |
| title_short | Polynomially Riesz elements: a study via Fredholm and Ruston theory |
| title_sort | polynomially riesz elements a study via fredholm and ruston theory |
| topic | Fredholm operators Operator theory Banach algebras Homomorphisms (Mathematics) Riesz spaces |
| url | https://scholar.sun.ac.za/handle/10019.1/134804 |
| work_keys_str_mv | AT sandernicholasjan polynomiallyrieszelementsastudyviafredholmandrustontheory |