Functions of operators and the classes associated with them - Labuschagne, L E; Labuschagne, Louis E (Department of Mathematics and Applied Mathematics, 2017)
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.