Functions of operators and the classes associated with them
| dc.contributor.advisor | Cross, Ron W | |
| dc.contributor.advisor | Cross, Ron W | |
| dc.contributor.author | Labuschagne, L E | |
| dc.contributor.author | Labuschagne, Louis E | |
| dc.date.accessioned | 2017-01-26T07:46:15Z | |
| dc.date.available | 2017-01-26T07:46:15Z | |
| dc.date.issued | 1988 | |
| dc.date.updated | 2016-11-22T09:56:44Z | |
| dc.description.abstract | 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. | |
| dc.identifier.apacitation | Labuschagne, L. E., & Labuschagne, L. E. (1988). <i>Functions of operators and the classes associated with them</i>. (Thesis). University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics. Retrieved from http://hdl.handle.net/11427/23260 | en_ZA |
| dc.identifier.chicagocitation | Labuschagne, L E, and Louis E Labuschagne. <i>"Functions of operators and the classes associated with them."</i> Thesis., University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 1988. http://hdl.handle.net/11427/23260 | en_ZA |
| dc.identifier.citation | Labuschagne, L., Labuschagne, L. 1988. Functions of operators and the classes associated with them. University of Cape Town. | en_ZA |
| dc.identifier.ris | TY - Thesis / Dissertation AU - Labuschagne, L E AU - Labuschagne, Louis E AB - 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. DA - 1988 DB - OpenUCT DP - University of Cape Town LK - https://open.uct.ac.za PB - University of Cape Town PY - 1988 T1 - Functions of operators and the classes associated with them TI - Functions of operators and the classes associated with them UR - http://hdl.handle.net/11427/23260 ER - | en_ZA |
| dc.identifier.uri | http://hdl.handle.net/11427/23260 | |
| dc.identifier.vancouvercitation | Labuschagne LE, Labuschagne LE. Functions of operators and the classes associated with them. [Thesis]. University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 1988 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/23260 | en_ZA |
| dc.language.iso | eng | |
| dc.language.iso | eng | |
| dc.publisher.department | Department of Mathematics and Applied Mathematics | en_ZA |
| dc.publisher.faculty | Faculty of Science | en_ZA |
| dc.publisher.institution | University of Cape Town | |
| dc.subject.other | Operator theory | |
| dc.subject.other | Operator theory | |
| dc.subject.other | Mathematics | |
| dc.title | Functions of operators and the classes associated with them | |
| dc.title | Functions of operators and the classes associated with them | |
| dc.type | Doctoral Thesis | |
| dc.type.qualificationlevel | Doctoral | |
| dc.type.qualificationname | PhD | |
| uct.type.publication | Research | |
| uct.type.resource | Thesis |