Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 60-61). Also available on the Internet.

In this paper, exhaustivity, continuity, and strong additivity are studied in the setting of topological Riesz spaces. Of particular interest is the link between strong additivity and exhaustive elements of Dedekind s-complete Banach lattices. There is a strong connection between the Diestel-Faires Theorem and the Meyer-Nieberg Lemma in this setting. Also, embedding properties of Banach lattices are linked to the notion of strong additivity. The Meyer-Nieberg Lemma is extended to the setting of topological Riesz spaces and uniform absolute continuity and uniformly exhaustive elements are studied in this setting. Counterexamples are provided to show that the Vitali-Hahn-Saks Theorem and the Brooks-Jewett Theorem cannot be extended to submeasures or to the setting of Banach lattices.

Ph.D. (Mathematics) / Our objective in this thesis is to investigate two fundamental questions concerning Riesz operators de ned on a Banach space. Recall that Riesz operators are generalizations of compact operators in the sense that Riesz operators have the same spectral properties as compact operators. However, they do not possess the same algebraic properties as compact operators. Our rst question that we investigate is: When is a Riesz operator a nite rank operator? This question is motivated from the fact that if a compact operator de ned on a Banach space has closed range, then it is a nite rank operator. Also, Ghahramani proved that a compact homomorphism de ned on a C -algebra is a nite rank operator, see . Martin Mathieu, in his paper, generalized the result of Ghahramani by proving that a weakly compact homomorphism de ned on a C -algebra is a nite rank operator...

Student Number : 0413210T - MSc dissertation - School of Mathematics - Faculty of Science / The aim of this work is to do a literature study on spaces of martingales on Riesz spaces and Banach lattices, using [16, 19, 20, 17, 18, 2, 30] as a point of departure. Convergence of martingales in the classical theory of stochastic processes has many applications in mathematics and related areas. Operator theoretic approaches to the classical theory of stochastic processes and martingale theory in particular, can be found in, for example, [4, 5, 6, 7, 13, 15, 26, 27]. The classical theory of stochastic processes for scalar-valued measurable functions on a probability space ( ,#6;, μ) utilizes the measure space ( ,#6;, μ), the norm structure of the associated Lp(μ)-spaces as well as the order structure of these spaces. Motivated by the existing operator theoretic approaches to classical stochastic processes, a theory of discrete-time stochastic processes has been developed in [16, 19, 20, 17, 18] on Dedekind complete Riesz spaces with weak order units. This approach is measure-free and utilizes only the order structure of the given Riesz space. Martingale convergence in the Riesz space setting is considered in [18]. It was shown there that the spaces of order bounded martingales and order convergent martingales, on a Dedekind complete Riesz space with a weak order unit, coincide. A measure-free approach to martingale theory on Banach lattices with quasi-interior points has been given in [2]. Here, the groundwork was done to generalize the notion of a filtration on a vector-valued Lp-space to the M-tensor product of a Banach space and a Banach lattice (see [1]). In [30], a measure-free approaches to martingale theory on Banach lattices is given. The main results in [30] show that the space of regular norm bounded martingales and the space of norm bounded martingales on a Banach lattice E are Banach lattices in a natural way provided that, for the former, E is an order continuous Banach lattice, and for the latter, E is a KB-space. The definition of a ”martingale” defined on a particular space depends on the type of space under consideration and on the ”filtration,” which is a sequence of operators defined on the space. Throughout this dissertation, we shall consider Riesz spaces, Riesz spaces with order units, Banach spaces, Banach lattices and Banach lattices with quasi-interior points. Our definition of a ”filtration” will, therefore, be determined by the type of space under consideration and will be adapted to suit the case at hand. In Chapter 2, we consider convergent martingale theory on Riesz spaces. This chapter is based on the theory of martingales and their properties on Dedekind complete Riesz spaces with weak order units, as can be found in [19, 20, 17, 18]. The notion of a ”filtration” in this setting is generalized to Riesz spaces. The space of martingales with respect to a given filtration on a Riesz space is introduced and an ordering defined on this space. The spaces of regular, order bounded, order convergent and generated martingales are introduced and properties of these spaces are considered. In particular, we show that the space of regular martingales defined on a Dedekind complete Riesz space is again a Riesz space. This result, in this context, we believe is new. The contents of Chapter 3 is convergent martingale theory on Banach lattices. We consider the spaces of norm bounded, norm convergent and regular norm bounded martingales on Banach lattices. In [30], filtrations (Tn) on the Banach lattice E which satisfy the condition 1[n=1 R(Tn) = E, where R(Tn) denotes the range of the filtration, are considered. We do not make this assumption in our definition of a filtration (Tn) on a Banach lattice. Our definition yields equality (in fact, a Riesz and isometric isomorphism) between the space of norm convergent martingales and 1Sn=1R(Tn). The aforementioned main results in [30] are also considered in this chapter. All the results pertaining to martingales on Banach spaces in subsections 3.1.1, 3.1.2 and 3.1.3 we believe are new. Chapter 4 is based on the theory of martingales on vector-valued Lp-spaces (cf. [4]), on its extension to the M-tensor product of a Banach space and a Banach lattice as introduced by Chaney in [1] (see also [29]) and on [2]. We consider filtrations on tensor products of Banach lattices and Banach spaces as can be found in [2]. We show that if (Sn) is a filtration on a Banach lattice F and (Tn) is a filtration on a Banach space X, then 1[n=1 R(Tn Sn) = 1[n=1 R(Tn) e M 1[n=1 R(Sn). This yields a distributive property for the space of convergent martingales on the M-tensor product of X and F. We consider the continuous dual of the space of martingales and apply our results to characterize dual Banach spaces with the Radon- Nikod´ym property. We use standard notation and terminology as can be found in standard works on Riesz spaces, Banach spaces and vector-valued Lp-spaces (see [4, 23, 29, 31]). However, for the convenience of the reader, notation and terminology used are included in the Appendix at the end of this work. We hope that this will enhance the pace of readability for those familiar with these standard notions.

This study revolves around operators defined by conditional expectations and operators generated by random measures. Studies of operators in function spaces defined by conditional expectations first appeared in the mid 1950's by S-T.C. Moy [22] and S. Sidak [26]. N. Kalton studied them in the setting of Lp-spaces 0 < p < 1 in [15, 131 and in L1-spaces, [14], while W. Arveson [5] studied them in L2-spaces. Their averaging properties were studied by P.G. Dodds and C.B. Huijsmans and B. de Pagter in [7] and C.B. Huijsmans and B. de Pagter in [lo]. A. Lambert [17] studied their relationship with multiplication operators in C*-modules. It was shown by J.J. Grobler and B. de Pagter [8] that partial integral operators that were studied A.S. Kalitvin et a1 in [2, 4, 3, 11, 121 and the special cases of kernel operators that were, inter alia, studied by A.R. Schep in [25] were special cases of conditional expectation operators. On the other hand, operators generated by random measures or pseudo-integral operators were studied by A. Sourour [28, 271 and L.W. Weis [29,30], building on the studies of W. Arveson [5] and N. Kalton [14, 151, in the late 1970's and early 1980's. In this thesis we extend the work of J.J. Grobler and B. de Pagter [8] on Multiplication Conditional Expectation-representable (MCE-representable) operators. We also generalize the result of A. Sourour [27] and show that order continuous linear maps between ideals of almost everywhere finite measurable functions on u-finite measure spaces are MCE-representable. This fact enables us to easily deduce that sums and compositions of MCE-representable operators are again MCE-representable operators. We also show that operators generated by random measures are MCE-representable. The first chapter gathers the definitions and introduces notions and concepts that are used throughout. In particular, we introduce Riesz spaces and operators therein, Riesz and Boolean homomorphisms, conditional expectation operators, kernel and absolute T-kernel operators. In Chapter 2 we look at MCE-operators where we give a definition different from that given by J.J. Grobler and B. de Pagter in [8], but which we show to be equivalent. Chapter 3 involves random measures and operators generated by random measures. We solve the problem (positively) that was posed by A. Sourour in [28] about the relationship of the lattice properties of operators generated by random measures and the lattice properties of their generating random measures. We show that the total variation of a random signed measure representing an order bounded operator T, it being the difference of two random measures, is again a random measure and represents ITI. We also show that the set of all operators generated by a random measure is a band in the Riesz space of all order bounded operators. In Chapter 4 we investigate the relationship between operators generated by random measures and MCE-representable operators. It was shown by A. Sourour in [28, 271 that every order bounded order continuous linear operator acting between ideals of almost everywhere measurable functions is generated by a random measure, provided that the measure spaces involved are standard measure spaces. We prove an analogue of this theorem for the general case where the underlying measure spaces are a-finite. We also, in this general setting, prove that every order continuous linear operator is MCE-representable. This rather surprising result enables us to easily show that sums, products and compositions of MCE-representable operator are again MCE-representable. Key words: Riesz spaces, conditional expectations, multiplication conditional expectation-representable operators, random measures. / Thesis (Ph.D. (Mathematics))--North-West University, Potchefstroom Campus, 2004.

This study revolves around operators defined by conditional expectations and operators generated by random measures. Studies of operators in function spaces defined by conditional expectations first appeared in the mid 1950's by S-T.C. Moy [22] and S. Sidak [26]. N. Kalton studied them in the setting of Lp-spaces 0 < p < 1 in [15, 131 and in L1-spaces, [14], while W. Arveson [5] studied them in L2-spaces. Their averaging properties were studied by P.G. Dodds and C.B. Huijsmans and B. de Pagter in [7] and C.B. Huijsmans and B. de Pagter in [lo]. A. Lambert [17] studied their relationship with multiplication operators in C*-modules. It was shown by J.J. Grobler and B. de Pagter [8] that partial integral operators that were studied A.S. Kalitvin et a1 in [2, 4, 3, 11, 121 and the special cases of kernel operators that were, inter alia, studied by A.R. Schep in [25] were special cases of conditional expectation operators. On the other hand, operators generated by random measures or pseudo-integral operators were studied by A. Sourour [28, 271 and L.W. Weis [29,30], building on the studies of W. Arveson [5] and N. Kalton [14, 151, in the late 1970's and early 1980's. In this thesis we extend the work of J.J. Grobler and B. de Pagter [8] on Multiplication Conditional Expectation-representable (MCE-representable) operators. We also generalize the result of A. Sourour [27] and show that order continuous linear maps between ideals of almost everywhere finite measurable functions on u-finite measure spaces are MCE-representable. This fact enables us to easily deduce that sums and compositions of MCE-representable operators are again MCE-representable operators. We also show that operators generated by random measures are MCE-representable. The first chapter gathers the definitions and introduces notions and concepts that are used throughout. In particular, we introduce Riesz spaces and operators therein, Riesz and Boolean homomorphisms, conditional expectation operators, kernel and absolute T-kernel operators. In Chapter 2 we look at MCE-operators where we give a definition different from that given by J.J. Grobler and B. de Pagter in [8], but which we show to be equivalent. Chapter 3 involves random measures and operators generated by random measures. We solve the problem (positively) that was posed by A. Sourour in [28] about the relationship of the lattice properties of operators generated by random measures and the lattice properties of their generating random measures. We show that the total variation of a random signed measure representing an order bounded operator T, it being the difference of two random measures, is again a random measure and represents ITI. We also show that the set of all operators generated by a random measure is a band in the Riesz space of all order bounded operators. In Chapter 4 we investigate the relationship between operators generated by random measures and MCE-representable operators. It was shown by A. Sourour in [28, 271 that every order bounded order continuous linear operator acting between ideals of almost everywhere measurable functions is generated by a random measure, provided that the measure spaces involved are standard measure spaces. We prove an analogue of this theorem for the general case where the underlying measure spaces are a-finite. We also, in this general setting, prove that every order continuous linear operator is MCE-representable. This rather surprising result enables us to easily show that sums, products and compositions of MCE-representable operator are again MCE-representable. Key words: Riesz spaces, conditional expectations, multiplication conditional expectation-representable operators, random measures. / Thesis (Ph.D. (Mathematics))--North-West University, Potchefstroom Campus, 2004.

Thesis (Ph. D. in Mathematics)--Vanderbilt University, Aug. 2006. / Title from title screen. Includes bibliographical references.

The payoff functions of contingent claims (options) of one variable are prominent in Financial Economics and thus assume a fundamental role in option pricing theory. Some of these payoff functions are continuous, piecewise-defined and linear or affine. Such option payoff functions can be analysed in a useful way when they are represented in additive, Boolean normal, graphical and linear form. The issue of converting such payoff functions expressed in the additive, linear or graphical form into an equivalent Boolean normal form, has been considered by several authors for more than half-a-century to better-understand the role of such functions. One aspect of our study is to unify the foregoing different forms of representation, by creating algorithms that convert a payoff function expressed in graphical form into Boolean normal form and then into the additive form and vice versa. Applications of these algorithms are considered in a general theoretical sense and also in the context of specific option contracts wherever relevant. The use of these algorithms have yielded easy computation of the area enclosed by the graph of various functions using min and max operators in several ways, which, in our opinion, are important in option pricing. To summarise, this study effectively dealt with maximum and minimum operators from several perspectives