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Functions of operators and the classes associated with them
The important classes of normally solvable, ϴ₊ (ϴ₋) and strictly singular (strictly cosingular) operators have long been studied in the setting of bounded or closed operators between Banach spaces. Results by Kato, Lacey, et al (see Goldberg [16; III.1.9, III.2.1 and III.2.3] ) led to the definition...
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| Language: | English English |
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Department of Mathematics and Applied Mathematics
2017
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| _version_ | 1867614255302311936 |
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| access_status_str | Open Access |
| author | Labuschagne, L E Labuschagne, Louis E |
| author2 | Cross, Ron W |
| author_browse | Cross, Ron W Labuschagne, L E Labuschagne, Louis E |
| author_facet | Cross, Ron W Labuschagne, L E Labuschagne, Louis E |
| author_sort | Labuschagne, L E |
| collection | Thesis |
| description | The important classes of normally solvable, ϴ₊ (ϴ₋) and strictly singular (strictly cosingular) operators have long been studied in the setting of bounded or closed operators between Banach spaces. Results by Kato, Lacey, et al (see Goldberg [16; III.1.9, III.2.1 and III.2.3] ) led to the definition of certain norm related functions of operators (Γ, Δ and Γ₀) which provided a powerful new way to study the classes of ϴ₊ and strictly singular operators (see for example Gramsch[19], Lebow and Schechter[28] and Schechter[36]). Results by Brace and R.-Kneece[4] among others led to the definition of analogous functions (Γ' and Δ') which were used to study ϴ₋ and strictly cosingular operators (see for example Weis, [37] and [38]). Again this problem was considered mainly for the case of bounded operators between Banach spaces. This thesis represents a contribution to knowledge in the sense that by considering the functions Γ', Δ' and Γ'₀, as well as the minimum modulus function in the more general setting of unbounded linear operators between normed linear spaces, we obtain the classes of F₋ and Range Open operators which turn out to be closely related to the classes of ϴ₋ and normally solvable operators respectively. We also define unbounded strictly cosingular operators and find that many of the classical results on ϴ₋, normally solvable and bounded strictly cosingular operators go through for F₋, range open and unbounded strictly cosingular operators respectively. This ties up with work done by R. W. Cross and provides a workable framework within which to study ϴ₋ and ϴ₊ type operators in the much more. general setting of unbounded linear operators between normed linear spaces. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/23260 |
| institution | University of Cape Town (South Africa) |
| language | eng eng |
| last_indexed | 2026-06-10T12:49:08.234Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2017 |
| publishDateRange | 2017 |
| publishDateSort | 2017 |
| publisher | Department of Mathematics and Applied Mathematics |
| publisherStr | Department of Mathematics and Applied Mathematics |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/23260 Functions of operators and the classes associated with them Functions of operators and the classes associated with them Labuschagne, L E Labuschagne, Louis E Cross, Ron W Cross, Ron W Operator theory Operator theory Mathematics The important classes of normally solvable, ϴ₊ (ϴ₋) and strictly singular (strictly cosingular) operators have long been studied in the setting of bounded or closed operators between Banach spaces. Results by Kato, Lacey, et al (see Goldberg [16; III.1.9, III.2.1 and III.2.3] ) led to the definition of certain norm related functions of operators (Γ, Δ and Γ₀) which provided a powerful new way to study the classes of ϴ₊ and strictly singular operators (see for example Gramsch[19], Lebow and Schechter[28] and Schechter[36]). Results by Brace and R.-Kneece[4] among others led to the definition of analogous functions (Γ' and Δ') which were used to study ϴ₋ and strictly cosingular operators (see for example Weis, [37] and [38]). Again this problem was considered mainly for the case of bounded operators between Banach spaces. This thesis represents a contribution to knowledge in the sense that by considering the functions Γ', Δ' and Γ'₀, as well as the minimum modulus function in the more general setting of unbounded linear operators between normed linear spaces, we obtain the classes of F₋ and Range Open operators which turn out to be closely related to the classes of ϴ₋ and normally solvable operators respectively. We also define unbounded strictly cosingular operators and find that many of the classical results on ϴ₋, normally solvable and bounded strictly cosingular operators go through for F₋, range open and unbounded strictly cosingular operators respectively. This ties up with work done by R. W. Cross and provides a workable framework within which to study ϴ₋ and ϴ₊ type operators in the much more. general setting of unbounded linear operators between normed linear spaces. 2017-01-26T07:46:15Z 2017-01-26T07:46:15Z 1988 2016-11-22T09:56:44Z Doctoral Thesis Doctoral PhD http://hdl.handle.net/11427/23260 eng eng application/pdf Department of Mathematics and Applied Mathematics Faculty of Science University of Cape Town |
| spellingShingle | Operator theory Operator theory Mathematics Labuschagne, L E Labuschagne, Louis E Functions of operators and the classes associated with them |
| thesis_degree_str | Doctoral |
| title | Functions of operators and the classes associated with them |
| title_full | Functions of operators and the classes associated with them |
| title_fullStr | Functions of operators and the classes associated with them |
| title_full_unstemmed | Functions of operators and the classes associated with them |
| title_short | Functions of operators and the classes associated with them |
| title_sort | functions of operators and the classes associated with them |
| topic | Operator theory Operator theory Mathematics |
| url | http://hdl.handle.net/11427/23260 |
| work_keys_str_mv | AT labuschagnele functionsofoperatorsandtheclassesassociatedwiththem AT labuschagnelouise functionsofoperatorsandtheclassesassociatedwiththem |