Thesis (Ph. D.)--Dept. of Political Science and Dept. of History. University of Missouri--Kansas City, 2004. / "A dissertation in political science and history." Advisor: Robert K. Evanson. Typescript. Vita. Title from "catalog record" of the print edition Description based on contents viewed Feb. 23, 2006. Includes bibliographical references (leaves 249-264). Online version of the print edition.

Este trabalho tem como objetivo estudar a propriedade Dunford-Pettis alternativa (propriedade DP1), como introduzida por Freedman, e algumas de suas caracterizações e relações com outras propriedades. Estudamos caracterizações para alguns espaços de operadores com a propriedade DP1, dadas por Acosta e Peralta. Vimos que um subespaço fechado do espaço dos operadores compactos em um espaço de Banach reflexivo com base de Schauder tem a propriedade DP1 se, e somente se, os operadores avaliação são operadores DP1. Estudamos um resultado análogo para espaços de Hilbert. Como consequência desses resultados, vimos uma caracterização de certas subálgebras fechadas da álgebra dos operadores compactos que possuem a propriedade DP1, supondo que os operadores composição à direita e à esquerda são operadores DP1. Finalmente, estudamos a demonstração feita por Bunce e Peralta de que as propriedades Dunford-Pettis e Duford-Pettis alternativa são equivalentes em C*-álgebras. / The main purpose of this work is to study the alternative Dunford-Pettis property (DP1 property), as introduced by Freedman, and some characterizations of the DP1 property and relations of this to other properties. We studied a characterization of certain operator subspaces which have the DP1 property, as given by Acosta and Peralta in \\cite. We saw that a closed subspace of the compact operators space in a reflexive Banach space with Schauder basis has the DP1 property if, and only if, the evaluation operators are DP1 operators. We studied a similar result for Hilbert spaces. Consequently, we also saw a characterization of certain closed subalgebras of the compact operators algebra, in which the DP1 property is held by assuming that the right and left composition operators are DP1. Finally, we studied the proof given by Bunce and Peralta that the Dunford-Pettis property and the alternative Duford-Pettis property are equivalent for C*-algebras.

Este trabalho tem como objetivo estudar a propriedade Dunford-Pettis alternativa (propriedade DP1), como introduzida por Freedman, e algumas de suas caracterizações e relações com outras propriedades. Estudamos caracterizações para alguns espaços de operadores com a propriedade DP1, dadas por Acosta e Peralta. Vimos que um subespaço fechado do espaço dos operadores compactos em um espaço de Banach reflexivo com base de Schauder tem a propriedade DP1 se, e somente se, os operadores avaliação são operadores DP1. Estudamos um resultado análogo para espaços de Hilbert. Como consequência desses resultados, vimos uma caracterização de certas subálgebras fechadas da álgebra dos operadores compactos que possuem a propriedade DP1, supondo que os operadores composição à direita e à esquerda são operadores DP1. Finalmente, estudamos a demonstração feita por Bunce e Peralta de que as propriedades Dunford-Pettis e Duford-Pettis alternativa são equivalentes em C*-álgebras. / The main purpose of this work is to study the alternative Dunford-Pettis property (DP1 property), as introduced by Freedman, and some characterizations of the DP1 property and relations of this to other properties. We studied a characterization of certain operator subspaces which have the DP1 property, as given by Acosta and Peralta in \\cite. We saw that a closed subspace of the compact operators space in a reflexive Banach space with Schauder basis has the DP1 property if, and only if, the evaluation operators are DP1 operators. We studied a similar result for Hilbert spaces. Consequently, we also saw a characterization of certain closed subalgebras of the compact operators algebra, in which the DP1 property is held by assuming that the right and left composition operators are DP1. Finally, we studied the proof given by Bunce and Peralta that the Dunford-Pettis property and the alternative Duford-Pettis property are equivalent for C*-algebras.

In this dissertation we study the structure of spaces of operators, especially the space of all compact operators between two Banach spaces X and Y. Work by Kalton, Emmanuele, Bator and Lewis on the space of compact and weakly compact operators motivates much of this paper. Let L(X,Y) be the Banach space of all bounded linear operators between Banach spaces X and Y, K(X,Y) be the space of all compact operators, and W(X,Y) be the space of all weakly compact operators. We study problems related to the complementability of different operator ideals (the Banach space of all compact, weakly compact, completely continuous, resp. unconditionally converging) operators in the space of all bounded linear operators. The structure of Dunford-Pettis sets, strong Dunford-Pettis sets, and certain spaces of operators is studied in the context of the injective and projective tensor products of Banach spaces. Bibasic sequences are used to study relative norm compactness of strong Dunford-Pettis sets. Next, we use Dunford-Pettis sets to give sufficient conditions for K(X,Y) to contain c0.

In this paper we give a characterization theorem for the reciprocal Dunford-Pettis property as defined by Grothendieck. The relationship of this property to Pelczynski's property V is examined. In particular it is shown that every Banach space with property V has the reciprocal Dunford-Pettis property and an example is given to show that the converse fails to hold. Moreover the characterizations of property V and the reciprocal Dunford-Pettis property lead to the definitions of property V* and property RDP* respectively. Me compare and contrast results for the reciprocal Dunford-Pettis property and property RDP* with those for properties V and V*. In the final chapter we use a result of Brooks to obtain a characterization for the Radon-Nikodým property.

Let (Ω, Σ, µ) be a finite measure space and X, a Banach space with continuous dual X*. A scalarly measurable function f: Ω→X is Dunford integrable if for each x* X*, x*f L1(µ). Define the operator Tf. X* → L1(µ) by T(x*) = x*f. Then f is Pettis integrable if and only if this operator is weak*-to-weak continuous. This paper begins with an overview of this function. Work by Robert Huff and Gunnar Stefansson on the operator Tf motivates much of this paper. Conditions that make Tf weak*-to-weak continuous are generalized to weak*-to­weak continuous operators on dual spaces. For instance, if Tf is weakly compact and if there exists a separable subspace D X such that for each x* X*, x*f = x*fχDµ-a.e, then f is Pettis integrable. This nation is generalized to bounded operators T: X* → Y. To say that T is determined by D means that if x*| D = 0, then T (x*) = 0. Determining subspaces are used to help prove certain facts about operators on dual spaces. Attention is given to finding determining subspaces far a given T: X* → Y. The kernel of T and the adjoint T* of T are used to construct determining subspaces for T. For example, if T*(Y*) ∩ X is weak* dense in T*(Y*), then T is determined by T*(Y*) ∩ X. Also if ker(T) is weak* closed in X*, then the annihilator of ker(T) (in X) is the unique minimal determining subspace for T.

Tensor products of Banach Spaces are studied. An introduction to tensor products is given. Some results concerning the reciprocal Dunford-Pettis Property due to Emmanuele are presented. Pelczyriski's property (V) and (V)-sets are studied. It will be shown that if X and Y are Banach spaces with property (V) and every integral operator from X into Y* is compact, then the (V)-subsets of (X⊗F)* are weak* sequentially compact. This in turn will be used to prove some stronger convergence results for (V)-subsets of C(Ω,X)*.

Esta tesis doctoral se enmarca dentro de la teoría de integración de funciones con valores en espacios de Banach. Analizamos con detalle la integral de Birkhoff de funciones vectoriales, así como sus correspondientes versiones dentro de los contextos de la integración respecto de medidas vectoriales y la integración de multi-funciones. Comparamos estos métodos de integración con otros bien conocidos (integrales de Bochner, Pettis, McShane, Debreu, etc.). Caracterizamos, en términos de integración vectorial, algunas propiedades de los espacios de Banach donde las (multi-) funciones toman valores. / The general framework of this memoir is the theory of integration of functions with values in Banach spaces. We analyze in detail the Birkhoff integral of vector-valued functions, as well as its corresponding versions within the settings of integration with respect to vector measures and integration of multi-valued functions. We compare these methods of integration with others which are well known (Bochner, Pettis, McShane, Debreu, etc.). We characterize, in terms of vector integration, some properties of the Banach spaces where the (multi-) functions take their values.

With the increase in the elderly, American's caregiver roles and responsibilities are also on the rise. Developing with this increase is the growing problem of caregiver burden. The Loma Linda V. A. Hospital has developed a service known as the Home-Based Primary Care program designed to combat caregiver burden. Studies were conducted among caregiver participants in this program in an effort to evaluate overall levels of burden and the program's efficiency at meeting caregiver needs. Overall this program provides a useful model for future programs aimed at reducing caregiver burden. Limitations that can be improved, however, include consistency in scheduling, reliable staffing, and continuity in care.

Recall that a Banach space X has the Dunford-Pettis property if every weakly compact operator defined on X takes weakly compact sets into norm compact sets. Some valuable characterisations of Banach spaces with the Dunford-Pettis property are: X has the DPP if and only if for all Banach spaces Y, every weakly compact operator from X to Y sends weakly convergent sequences onto norm convergent sequences (i.e. it requires that weakly compact operators on X are completely continuous) and this is equivalent to “if (xn) and (x*n) are sequences in X and X* respectively and limn xn = 0 weakly and limn x*n = 0 weakly then limn x*n xn = 0". A striking application of the Dunford-Pettis property (as was observed by Grothendieck) is to prove that if X is a linear subspace of L() for some finite measure  and X is closed in some Lp() for 1 ≤ p < , then X is finite dimensional. The fact that the well known spaces L1() and C() have this property (as was proved by Dunford and Pettis) was a remarkable achievement in the early history of Banach spaces and was motivated by the study of integral equations and the hope to develop an understanding of linear operators on Lp() for p ≥ 1. In fact, it played an important role in proving that for each weakly compact operator T : L1()  L1() or T : C()  C(), the operator T2 is compact, a fact which is important from the point of view that there is a nice spectral theory for compact operators and operators whose squares are compact. There is an extensive literature involving the Dunford-Pettis property. Almost all the articles and books in our list of references contain some information about this property, but there are plenty more that could have been listed. The reader is for instance referred to [4], [5], [7], [8], [10], [17] and [24] for information on the role of the DPP in different areas of Banach space theory. In this dissertation, however, we are motivated by the two papers [7] and [8] to study alternative Dunford-Pettis properties, to introduce a scale of (new) alternative Dunford-Pettis properties, which we call DP*-properties of order p (briefly denoted by DP*P), and to consider characterisations of Banach spaces with these properties as well as applications thereof to polynomials and holomorphic functions on Banach spaces. In the paper [8] the class Cp(X, Y) of p-convergent operators from a Banach space X to a Banach space Y is introduced. Replacing the requirement that weakly compact operators on X should be completely continuous in the case of the DPP for X (as is mentioned above) by “weakly compact operators on X should be p-convergent", an alternative Dunford-Pettis property (called the Dunford-Pettis property of order p) is introduced. More precisely, if 1 ≤ p ≤ , a Banach space X is said to have DPPp if the inclusion W(X, Y)  Cp(X, Y) holds for all Banach spaces Y . Here W(X, Y) denotes the family of all weakly compact operators from X to Y. We now have a scale of “Dunford-Pettis like properties" in the sense that all Banach spaces have the DPP1, if p < q, then each Banach space with the DPPq also has the DPPp and the strongest property, namely the DPP1 coincides with the DPP. In the paper [7] the authors study a property on Banach spaces (called the DP*-property, or briey the DP*P) which is stronger than the DPP, in the sense that if a Banach space has this property then it also has DPP. We say X has the DP*P, when all weakly compact sets in X are limited, i.e. each sequence (x*n)  X * in the dual space of X which converges weak* to 0, also converges uniformly (to 0) on all weakly compact sets in X. It turns out that this property is equivalent to another property on Banach spaces which is introduced in [17] (and which is called the *-Dunford-Pettis property) as follows: We say a Banach space X has the *-Dunford-Pettis property if for all weakly null sequences (xn) in X and all weak* null sequences (x*n) in X*, we have x*n(xn) n 0. After a thorough study of the DP*P, including characterisations and examples of Banach spaces with the DP*P, the authors in [7] consider some applications to polynomials and analytic functions on Banach spaces. Following an extensive literature study and in depth research into the techniques of proof relevant to this research field, we are able to present a thorough discussion of the results in [7] and [8] as well as some selected (relevant) results from other papers (for instance, [2] and [17]). This we do in Chapter 2 of the dissertation. The starting point (in Section 2.1 of Chapter 2) is the introduction of the so called p-convergent operators, being those bounded linear operators T : X  Y which transform weakly p-summable sequences into norm-null sequences, as well as the so called weakly p-convergent sequences in Banach spaces, being those sequences (xn) in a Banach space X for which there exists an x  X such that the sequence (xn - x) is weakly p-summable. Using these concepts, we state and prove an important characterisation (from the paper [8]) of Banach spaces with DPPp. In Section 2.2 (of Chapter 2) we continue to report on the results of the paper [7], where the DP*P on Banach spaces is introduced. We focus on the characterisation of Banach spaces with DP*P, obtaining among others that a Banach space X has DP*P if and only if for all weakly null sequences (xn) in X and all weak* null sequences (x*n) in X*, we have x*n(xn) n 0. An important characterisation of the DP*P considered in this section is the fact that X has DP*P if and only if every T  L(X, c0) is completely continuous. This result proves to be of fundamental importance in the study of the DP*P and its application to results on polynomials and holomorphic functions on Banach spaces. To be able to report on the applications of the DP*P in the context of homogeneous polynomials and analytic functions on Banach spaces, we embark on a study of “Complex Analysis in Banach spaces" (mostly with the focus on homogeneous polynomials and analytic functions on Banach spaces). This we do in Chapter 3; the content of the chapter is mostly based on work in the books [23] and [14], but also on the work in some articles such as [15]. After we have discussed the relevant theory of complex analysis in Banach spaces in Chapter 3, we devote Chapter 4 to considering properties of polynomials and analytic functions on Banach spaces with DP*P. The discussion in Chapter 4 is based on the applications of DP*P in the paper [7]. Finally, in Chapter 5 of the dissertation, we contribute to the study of “Dunford-Pettis like properties" by introducing the Banach space property “DP*P of order p", or briefly the DP*Pp for Banach spaces. Using the concept “weakly p-convergent sequence in Banach spaces" as is defined in [8], we define weakly-p-compact sets in Banach spaces. Then a Banach space X is said to have the DP*-property of order p (for 1 ≤ p ≤ ) if all weakly-p-compact sets in X are limited. In short, we say X has DP*Pp. As in [8] (where the DPPp is introduced), we now have a scale of DP*P-like properties, in the sense that all Banach spaces have DP*P1 and if p < q and X has DP*Pq then it has DP*Pp. The strongest property DP*P coincides with DP*P. We prove characterisations of Banach spaces with DP*Pp, discuss some examples and then consider applications to polynomials and analytic functions on Banach spaces. Our results and techniques in this chapter depend very much on the results obtained in the previous three chapters, but now we have to find our own correct definitions and formulations of results within this new context. We do this with some success in Sections 5.1 and 5.2 of Chapter 5. Chapter 1 of this dissertation provides a wide range of concepts and results in Banach spaces and the theory of vector sequence spaces (some of them very deep results from books listed in the bibliography). These results are mostly well known, but they are scattered in the literature - they are discussed in Chapter 1 (some with proof, others without proof, depending on the importance of the arguments in the proofs for later use and depending on the detail with which the results are discussed elsewhere in the literature) with the intention to provide an exposition which is mostly self contained and which will be comfortably accessible for graduate students. The dissertation reflects the outcome of our investigation in which we set ourselves the following goals: 1. Obtain a thorough understanding of the Dunford-Pettis property and some related (both weaker and stronger) properties that have been studied in the literature. 2. Focusing on the work in the paper [8], understand the role played in the study of difierent classes of operators by a scale of properties on Banach spaces, called the DPPp, which are weaker than the DP-property and which are introduced in [8] by using the weakly p-summable sequences in X and weakly null sequences in X*. 3. Focusing on the work in the paper [7], investigate the DP*P for Banach spaces, which is the exact property to answer a question of Pelczynsky's regarding when every symmetric bilinear separately compact map X x X  c0 is completely continuous. 4. Based on the ideas intertwined in the work of the paper [8] in the study of a scale of DP-properties and the work in the paper [7], introduce the DP*Pp on Banach spaces and investigate their applications to spaces of operators and in the theory of polynomials and analytic mappings on Banach spaces. Thereby, not only extending the results in [7] to a larger family of Banach spaces, but also to find an answer to the question: “When will every symmetric bilinear separately compact map X x X  c0 be p-convergent?" / Thesis (M.Sc. (Mathematics))--North-West University, Potchefstroom Campus, 2012.