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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ndltd-netd.ac.za-oai-union.ndltd.org-uct-oai-localhost-11427-232602020-07-22T05:08:04Z 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 University of Cape Town Faculty of Science Department of Mathematics and Applied Mathematics |
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English English |
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Doctoral Thesis |
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Operator theory Operator theory Mathematics |
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Operator theory Operator theory Mathematics Labuschagne, L E Labuschagne, Louis E Functions of operators and the classes associated with them |
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. |
author2 |
Cross, Ron W |
author_facet |
Cross, Ron W Labuschagne, L E Labuschagne, Louis E |
author |
Labuschagne, L E Labuschagne, Louis E |
author_sort |
Labuschagne, L E |
title |
Functions of operators and the classes associated with them |
title_short |
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_sort |
functions of operators and the classes associated with them |
publisher |
University of Cape Town |
publishDate |
2017 |
url |
http://hdl.handle.net/11427/23260 |
work_keys_str_mv |
AT labuschagnele functionsofoperatorsandtheclassesassociatedwiththem AT labuschagnelouise functionsofoperatorsandtheclassesassociatedwiththem |
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1719331444208173056 |