A study of Dunford–Pettis–like properties with applications to polynomials and analytic functions on normed spaces
Zeekoei, Elroy Denovanne
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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∞(µ) for some finite measure µ and X is closed in some Lp(µ) for 1 ≤ p < ∞, then X is finite dimensional. The fact that the well known spaces L1(µ) and C(Ω) 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(µ) for p ≥ 1. In fact, it played an important role in proving that for each weakly compact operator T : L1(µ) → L1(µ) or T : C(Ω) → C(Ω), 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 , , , , ,  and  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  and  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  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  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) 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  (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  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  and  as well as some selected (relevant) results from other papers (for instance,  and ). 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 ) of Banach spaces with DPPp. In Section 2.2 (of Chapter 2) we continue to report on the results of the paper , 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  and , but also on the work in some articles such as . 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 . 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 , 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  (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 , 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  by using the weakly p-summable sequences in X and weakly null sequences in X*. 3. Focusing on the work in the paper , 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  in the study of a scale of DP-properties and the work in the paper , 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  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?"
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