Browsing by Author "Cross, Ron W"

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    Functions of operators and the classes associated with them
    (1988) Labuschagne, L E; Labuschagne, Louis E; Cross, Ron W; Cross, Ron W
    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.
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    Multivalued semi-Fredholm operators in normed linear spaces
    (2002) Wilcox, Diane; Cross, Ron W
    Certain properties associated with these classes are stable under small perturbation, i.e. stable under additive perturbation by continuous operators whose norms are less than the minimum modulus of the relation being perturbed, and are also stable under perturbation by compact, strictly singular or strictly cosingular operators. In this work we continue the study of these classes and introduce the classes of α-Atkinson and β-Atkinson relations. These are subclasses of upper and lower semi-Fredholm relations respectively, having generalised inverses and defined in terms of the existence of continuous projections onto their ranges and nullspaces.
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    State diagrams for bounded and unbounded linear operators
    (1990) O'Connor, B J; Cross, Ron W
    The theme of this thesis is the construction of state diagrams and their implications. The author generalises most of the theorems in Chapter II of Goldberg [Gl] by dropping the assumption that the doin.ain of the operator is dense in X . The author also presents the standard Taylor-Halberg-Goldberg state diagrams [Gl, 61, 66]. Chapters II and III deal with F₊- and F₋-operators, which are generalisations of the ф₊- and ф₋-operators in Banach spaces of Gokhberg-Krein [GK]. Examples are given of F₊- and F₋-operators. Also, in Chapter III, the main theorems needed to construct the state diagrams of Chapter IV are discussed. The state diagrams of Chapter IV are based on states corresponding to F₊- and F₋-operators; in addition state diagrams relating T and T˝ under the assumptions ϒ(T) > 0 and ϒ(T΄) > 0 are derived. Second adjoints are important in Tauberian Theory (see Cross [Cl]). Chapters I and IV are the main chapters. In Chapter I of this thesis the author modifies many of the proofs appearing in Goldberg [Gl), to take account of the new definition of the adjoint.
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    Unbounded linear operators in seminormed spaces
    (1989) Gouveia, A I; Cross, Ron W
    Linear operator theory is usually studied in the setting of normed or Banach spaces. However, careful examination of proofs shows that in many cases the Hausdorff property of normed spaces is not used. Even in those cases where explicit use of the Hausdorff property is made, one can often get around this (should one wish to work in seminormed spaces) by suitable identification of elements and then working in the resulting normed space. Working in seminormed spaces rather than normed spaces is especially advantageous when dealing with quotients (which occur in linear operator theory when one considers the factorisation of an operator through its domain space quotiented by its null space): when taking the quotient of a normed space by a subspace, one requires the subspace to be closed in order for the quotient to be a normed space; however, in the seminormed space case the requirement that the subspace be closed is no longer necessary. Seminorms are also important in the study of certain properties of the second adjoint of an operator (for example, seminorms occur in the study of operators of the Tauberian type (see [C2]) and operators analagous to weakly compact operators (see Chapter VI). It is the aim of this work to generalise as much of the basic theory of unbounded linear operators as possible to seminormed spaces. In Chapter I, some aspects of topological vector spaces (which will be used throughout this work) are presented, the most important parts being the Hahn-Banach theorem and the section on weak topologies. In Chapter II, we restrict our attention to seminormed spaces, the setting in which the remainder of this work takes place. The basic theory of unbounded linear operators, their adjoints and the relationship between operators and their adjoints is covered in Chapter III. Chapter IV concentrates on characterising unbounded strictly singular operators while in Chapter V operators with closed range are studied. Finally, in Chapter VI, a property corresponding · to one of the equivalent conditions for a bounded operator to be weakly compact is studied for unbounded operators.