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ON SPACES OF BANACH LATTICE VALUED FUNCTIONS AND MEASURES \\ i Φ* m О ? -O Gé Groenewegen ON SPACES OF BANACH LATTICE VALUED FUNCTIONS AND MEASURES This thesis is printed on recycled paper. PROMOTOR: PROF. DR. A.C.M. VAN ROOIJ ON SPACES OF BANACH LATTICE VALUED FUNCTIONS AND MEASURES PROEFSCHRIFT TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE WISKUNDE EN NATUURWETENSCHAPPEN AAN DE KATHOLIEKE UNIVERSITEIT TE NIJMEGEN, OP GEZAG VAN DE RECTOR MAGNIFICUS PROF. DR. P.G.A.B. WIJDEVELD, VOLGENS BESLUIT VAN HET COLLEGE VAN DEKANEN IN HET OPENBAAR TE VERDEDIGEN OP DONDERDAG 24 JUNI 1982, DES NAMIDDAGS TE 2 UUR PRECIES DOOR GELINUS LAMBERTUS MARIA GROENEWEGEN GEBOREN TE VLAARDINGEN NIJMEGEN 1982 De tekening op de omslag is gemaakt door Hans Groenewegen. CONTENTS INTRODUCTION AND SUMMARY vii SOME NOTATIONS AND CONVENTIONS xii CHAPTER I: PRELIMINARIES ON BANACH LATTICES 1 § 1. Riesz spaces 2 §2. Banach lattices 5 §3. Disjoint sequences 10 §4. Fatou and Levi norms on Banach lattices 19 §5. Operators between Banach lattices 25 CHAPTER II: KOTHE SPACES OF VECTOR-VALUED FUNCTIONS 33 § 1. Vector-valued measurable functions 34 §2. The range space is a Banach lattice 40 §3. Generalized normed Kôthe spaces 44 §4. Step functions 50 §5. Absolute continuity and properties of the integral 59 CHAPTER III: VECTOR MEASURES 71 §1. Vector measures 72 §2. Vector measures with values in a Banach lattice 81 §3. The order structure and measures of bounded variation 85 §4. The property (C) 93 55. The order structure and strongly additive measures 100 §6. Countably additive vector measures and an application to weakly compact operators 108 §7. Radon-Nikodym derivatives for Banach lattice valued measures 113 CHAPTER IV: THE DUAL OF L (E) 117 Ρ §1. The dual of L (Ε) 118 Ρ §2. p-integrals 128 §3. m-integrals and their relation to vector measures 133 §4. m-integrals and operators 139 CHAPTER V: REPRESENTABLE OPERATORS 145 §1. Introduction to representable operators 146 §2. P-representable operators 151 ІЗ. Weekly compact operators on AL-spaces 162 §4. Some compactness properties of representable operators 167 REFERENCES 172 LIST OF SYMBOLS 176 SUBJECT INDEX 178 SAMENVATTING 179 CURRICULUM VITAE 185 INTRODUCTION AND SUMMARY At the basis of the research, out of which this thesis has grown, stands the following problem. Consider the space L(EF) of operators between two Banach lattices r E and F. This space is partially ordered in a natural way. We pose the problem: how well, does this order structure fit in with various properties of operators, such as compactness and weak compactness? Somewhat more specified: if T: E -*• F has a certain property and be sides has a modulus |τ| in L(E,F), does |τ| also have this property? We were particularly interested in weak compactness. A question which is posed in such generality, can of course initiate all kinds of research. First I would like to sketch the main lines of the de velopment that this problem has undergone. After that I shall say something about the results that came out of it. We started with looking at a simple case, namely operators with domain L (μ). The simplicity of this situation is due to a classical representation theorem of Dunford, Pettis and Phillips: take a Banách space X, and T: L (Ω,Σ,ρ) -»• X weakly compact. Then there exists a norm bounded function g: Ω •*· X such that (i) Tf is the Bochner integral ƒ fg dy for all f e L (μ), (ii) g(Ω) is relatively weakly compact. Conversely, any norm bounded, strongly measurable function g: Ω -»• X that satisfies (ii), induces a weakly compact operator by means of (i). Thanks to this theorem, we can solve our problem for domain spaces L.(y). Encouraged by this first succes, we continued with various kinds of representable operators. Operators that are given by means of vector valued integrals formed a first class. A second important class are those operators vii which can be described by means of a vector valued measure. One of the most interesting theorems in this field is due to Bartle, Dunfotd and Schwartz. They describe the operators C(K) -> X (here К is a compact Hausdorff space and X a Banach space). Also this representation bore fruités. It resulted in a description of the order structure on the space W(C(K),X) of all weak ly compact operators C(K) -* X. By this approach a number of questions were raised about the ordering on spaces of vector valued functions and measures, since they are at the root of the representations we mentioned. The research was also guided into this direction by a seminar held in Nijmegen, where the recent book of J. Diestel and J.J. Uhi Jr.: On vector measures (Γπ,Ι)]) was studied. Within the context as we sketched it above, it was natural to ask: what can you say in relation to the order structure about the subjects that are treated in Γο,υ] if you consider range spaces that are Banach lattices ? In this way the Banach lattices L (E) of Ε-valued functions came up Ρ (with E a Banach lattice). They form a generalization of the well-known Ba nach function spaces. The functionals on these spaces L (E) lead again to operators. We obtain a class of operators of which the kernel operators bet ween Banach function spaces are the most important examples. All this did result in a thesis where we study the order structure on spaces of functions and measures with values in a Banach lattice. Along these lines we derive results about the ordering on spaces of operators. Let us now look at the content of this thesis in some more detail. The the sis consists of five chapters. In the first chapter we sketch some parts of the theory on Riesz spaces and especially on Banach lattices that will be used further on in the the sis. We hope that in this way the readers who are not familiar with Riesz spaces, but are acquainted with functional analysis, get a good idea of what is going on in Banach lattice theory and can grasp the central techniques and results that will be employed. Chapter II starts with a short reflection on strongly measurable func tions with values in a Banach space, before we come to our real subject, the space B(y,E) of strongly measurable functions f : Ω -*• E (where (Ω, Σ, μ) viii