Student Number : 0108851W - MSc Dissertation - School of Mathematics - Faculty of Science / A brief search on www.ams.org with the keyword “Markov operator” produces some 684 papers, the earliest of which dates back to 1959. This suggests that the term “Markov operator” emerged around the 1950’s, clearly in the wake of Andrey Markov’s seminal work in the area of stochastic processes and Markov chains. Indeed, [17] and [6], the two earliest papers produced by the ams.org search, study Markov processes in a statistical setting and “Markov operators” are only referred to obliquely, with no explicit definition being provided. By 1965, in [7], the situation has progressed to the point where Markov operators are given a concrete definition and studied more directly. However, the way in which Markov operators originally entered mathematical discourse, emerging from Statistics as various attempts to generalize Markov processes and Markov chains, seems to have left its mark on the theory, with a notable lack of cohesion amongst its propagators. The study of Markov operators in the Lp setting has assumed a place of importance in a variety of fields. Markov operators figure prominently in the study of densities, and thus in the study of dynamical and deterministic systems, noise and other probabilistic notions of uncertainty. They are thus of keen interest to physicists, biologists and economists alike. They are also a worthy topic to a statistician, not least of all since Markov chains are nothing more than discrete examples of Markov operators (indeed, Markov operators earned their name by virtue of this connection) and, more recently, in consideration of the connection between copulas and Markov operators. In the realm of pure mathematics, in particular functional analysis, Markov operators have proven a critical tool in ergodic theory and a useful generalization of the notion of a conditional expectation. Considering the origin of Markov operators, and the diverse contexts in which they are introduced, it is perhaps unsurprising that, to the uninitiated observer at least, the theory of Markov operators appears to lack an overall unity. In the literature there are many different definitions of Markov operators defined on L1(μ) and/or L1(μ) spaces. See, for example, [13, 14, 26, 2], all of which manage to provide different definitions. Even at a casual glance, although they do retain the same overall flavour, it is apparent that there are substantial differences in these definitions. The situation is not much better when it comes to the various discussions surrounding ergodic Markov operators: we again see a variety of definitions for an ergodic operator (for example, see [14, 26, 32]), and again the connections between these definitions are not immediately apparent. In truth, the situation is not as haphazard as it may at first appear. All the definitions provided for Markov operator may be seen as describing one or other subclass of a larger class of operators known as the positive contractions. Indeed, the theory of Markov operators is concerned with either establishing results for the positive contractions in general, or specifically for one of the aforementioned subclasses. The confusion concerning the definition of an ergodic operator can also be rectified in a fairly natural way, by simply viewing the various definitions as different possible generalizations of the central notion of a ergodic point-set transformation (such a transformation representing one of the most fundamental concepts in ergodic theory). The first, and indeed chief, aim of this dissertation is to provide a coherent and reasonably comprehensive literature study of the theory of Markov operators. This theory appears to be uniquely in need of such an effort. To this end, we shall present a wealth of material, ranging from the classical theory of positive contractions; to a variety of interesting results arising from the study of Markov operators in relation to densities and point-set transformations; to more recent material concerning the connection between copulas, a breed of bivariate function from statistics, and Markov operators. Our goals here are two-fold: to weave various sources into a integrated whole and, where necessary, render opaque material readable to the non-specialist. Indeed, all that is required to access this dissertation is a rudimentary knowledge of the fundamentals of measure theory, functional analysis and Riesz space theory. A command of measure and integration theory will be assumed. For those unfamiliar with the basic tenets of Riesz space theory and functional analysis, we have included an introductory overview in the appendix. The second of our overall aims is to give a suitable definition of a Markov operator on Banach lattices and provide a survey of some results achieved in the Banach lattice setting, in particular those due to [5, 44]. The advantage of this approach is that the theory is order theoretic rather than measure theoretic. As we proceed through the dissertation, definitions will be provided for a Markov operator, a conservative operator and an ergodic operator on a Banach lattice. Our guide in this matter will chiefly be [44], where a number of interesting results concerning the spectral theory of conservative, ergodic, so-called “stochastic” operators is studied in the Banach lattice setting. We will also, and to a lesser extent, tentatively suggest a possible definition for a Markov operator on a Riesz space. In fact, we shall suggest, as a topic for further research, two possible approaches to the study of such objects in the Riesz space setting. We now offer a more detailed breakdown of each chapter. In Chapter 2 we will settle on a definition for a Markov operator on an L1 space, prove some elementary properties and introduce several other important concepts. We will also put forward a definition for a Markov operator on a Banach lattice. In Chapter 3 we will examine the notion of a conservative positive contraction. Conservative operators will be shown to demonstrate a number of interesting properties, not least of all the fact that a conservative positive contraction is automatically a Markov operator. The notion of conservative operator will follow from the Hopf decomposition, a fundmental result in the classical theory of positive contractions and one we will prove via [13]. We will conclude the chapter with a Banach lattice/Riesz space definition for a conservative operator, and a generalization of an important property of such operators in the L1 case. In Chapter 4 we will discuss another well-known result from the classical theory of positive contractions: the Chacon-Ornstein Theorem. Not only is this a powerful convergence result, but it also provides a connection between Markov operators and conditional expectations (the latter, in fact, being a subclass of theMarkov operators). To be precise, we will prove the result for conservative operators, following [32]. In Chapter 5 we will tie the study of Markov operators into classical ergodic theory, with the introduction of the Frobenius-Perron operator, a specific type of Markov operator which is generated from a given nonsingular point-set transformation. The Frobenius-Perron operator will provide a bridge to the general notion of an ergodic operator, as the definition of an ergodic Frobenius-Perron operator follows naturally from that of an ergodic transformation. In Chapter 6 will discuss two approaches to defining an ergodic operator, and establish some connections between the various definitions of ergodicity. The second definition, a generalization of the ergodic Frobenius-Perron operator, will prove particularly useful, and we will be able to tie it, following [26], to several interesting results concerning the asymptotic properties of Markov operators, including the asymptotic periodicity result of [26, 27]. We will then suggest a definition of ergodicity in the Banach lattice setting and conclude the chapter with a version, due to [5], of the aforementioned asymptotic periodicity result, in this case for positive contractions on a Banach lattice. In Chapter 7 we will move into more modern territory with the introduction of the copulas of [39, 40, 41, 42, 16]. After surveying the basic theory of copulas, including introducing a multiplication on the set of copulas, we will establish a one-to-one correspondence between the set of copulas and a subclass of Markov operators. In Chapter 8 we will carry our study of copulas further by identifying them as a Markov algebra under their aforementioned multiplication. We will establish several interesting properties of this Markov algebra, in parallel to a second Markov algebra, the set of doubly stochastic matrices. This chapter is chiefly for the sake of interest and, as such, diverges slightly from our main investigation of Markov operators. In Chapter 9, we will present the results of [44], in slightly more detail than the original source. As has been mentioned previously, these concern the spectral properties of ergodic, conservative, stochastic operators on a Banach lattice, a subclass of the Markov operators on a Banach lattice. Finally, as a conclusion to the dissertation, we present in Chapter 10 two possible routes to the study of Markov operators in a Riesz space setting. The first definition will be directly analogous to the Banach lattice case; the second will act as an analogue to the submarkovian operators to be introduced in Chapter 2. We will not attempt to develop any results from these definitions: we consider them a possible starting point for further research on this topic. In the interests of both completeness, and in order to aid those in need of more background theory, the reader may find at the back of this dissertation an appendix which catalogues all relevant results from Riesz space theory and operator theory.

Probability theory plays a crucial role in the study of the geometry of Banach spaces. In the literature, notions from probability theory have been formulated and studied in the measure free setting of vector lattices. However, there is little evidence of these vector lattice techniques being used in the study of geometry of Banach spaces. In this thesis, we fill this niche. Using the l-tensor product of Chaney-Shaefer, we are able to extend the available vector lattice techniques and apply them to the Lebesgue-Bochner spaces. As a consequence, we obtain new characterizations of the Radon Nikod´ym property and the UMD property.

El espacio de funciones integrables con respecto a una medida vectorial, amén de interesante en si mismo, sirve de herramienta para aplicaciones en problemas importantes como la representación integral y el estudio del dominio óptimo de operadores lineales o la representación de retículos de Banach abstractos como espacios de funciones. Las medidas vectoriales clásicas se definen sobre -álgebras y con valores en un espacio de Banach, y los espacios correspondientes L1( ) y L1w( ) de funciones integrables y débilmente integrables respectivamente, han sido estudiados en profundidad por numerosos autores, siendo su comportamiento bien conocido. Sin embargo, este contexto no es suficiente, por ejemplo, para aplicaciones a operadores definidos en espacios que no contienen a las funciones características de conjuntos o retículos de Banach sin unidad débil. Estos casos requieren que la medida vectorial esté definida en una estructura más débil que la de -álgebra, a saber, en un -anillo. Más aún, la integración con respecto a medidas vectoriales definidas en -anillos es la generalización vectorial natural de la integración con respecto a medidas -finitas positivas µ, que no está incluida en el contexto de las medidas vectoriales en -álgebras si µ no es finita. En consecuencia, las medidas vectoriales definidas en un -anillo también juegan un rol importante y merecen ser estudiadas así como sus espacios de funciones integrables. La teoría de integración con respecto a estas medidas se debe a Lewis y Masani y Niemi. En este trabajo estamos interesados principalmente en encontrar las propiedades que garanticen la representación de un retículo de Banach a través de un espacio de funciones integrables. El Capítulo 4 se dedica a este objetivo y contiene nuestro resultado principal. Algunas cuestiones interesantes aparecen de forma natural al intentar resolver este problema de representación abstracto. / Juan Blanco, MA. (2011). Vector measures on delta-rings and representation theorems of banach lattices [Tesis doctoral]. Editorial Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/11300 / Palancia

Para um espaço localmente compacto K e um espaço de Banach X, seja C_0(K,X) o espaço das funções continuas que se anulam no infinito munido da norma do supremo. Nesta tese se provam resultados relacionados com a geometria destes espaços. / For a locally compact Hausdorff space K and a Banach spaces X, let C_0(K,X) be the Banach space of continuous functions which vanish at infinity endowed with the supremum norm. We prove some results about geometry of these spaces.

Para um espaço localmente compacto K e um espaço de Banach X, seja C_0(K,X) o espaço das funções continuas que se anulam no infinito munido da norma do supremo. Nesta tese se provam resultados relacionados com a geometria destes espaços. / For a locally compact Hausdorff space K and a Banach spaces X, let C_0(K,X) be the Banach space of continuous functions which vanish at infinity endowed with the supremum norm. We prove some results about geometry of these spaces.

M-matrices are extensively employed in numerical analysis. These matrices can be generalized by corresponding operators on a partially ordered normed space. We extend results which are well-known for M-matrices to this more general setting. We investigate two different notions of an M-operator, where we focus on two questions: 1. For which types of partially ordered normed spaces do the both notions coincide? This leads to the study of positive-off-diagonal operators. 2. Which conditions on an M-operator ensure that its (positive) inverse satisfies certain maximum principles? We deal with generalizations of the "maximum principle for inverse column entries". / M-Matrizen werden in der numerischen Mathematik vielfältig angewandt. Eine Verallgemeinerung dieser Matrizen sind entsprechende Operatoren auf halbgeordneten normierten Räumen. Bekannte Aussagen aus der Theorie der M-Matrizen werden auf diese Situation übertragen. Für zwei verschiedene Typen von M-Operatoren werden die folgenden Fragen behandelt: 1. Für welche geordneten normierten Räume sind die beiden Typen gleich? Dies führt zur Untersuchung außerdiagonal-positiver Operatoren. 2. Welche Bedingungen an einen M-Operator sichern, dass seine (positive) Inverse gewissen Maximumprinzipien genügt? Es werden Verallgemeinerungen des "Maximumprinzips für inverse Spalteneinträge" angegeben und untersucht.

M-matrices are extensively employed in numerical analysis. These matrices can be generalized by corresponding operators on a partially ordered normed space. We extend results which are well-known for M-matrices to this more general setting. We investigate two different notions of an M-operator, where we focus on two questions: 1. For which types of partially ordered normed spaces do the both notions coincide? This leads to the study of positive-off-diagonal operators. 2. Which conditions on an M-operator ensure that its (positive) inverse satisfies certain maximum principles? We deal with generalizations of the "maximum principle for inverse column entries". / M-Matrizen werden in der numerischen Mathematik vielfältig angewandt. Eine Verallgemeinerung dieser Matrizen sind entsprechende Operatoren auf halbgeordneten normierten Räumen. Bekannte Aussagen aus der Theorie der M-Matrizen werden auf diese Situation übertragen. Für zwei verschiedene Typen von M-Operatoren werden die folgenden Fragen behandelt: 1. Für welche geordneten normierten Räume sind die beiden Typen gleich? Dies führt zur Untersuchung außerdiagonal-positiver Operatoren. 2. Welche Bedingungen an einen M-Operator sichern, dass seine (positive) Inverse gewissen Maximumprinzipien genügt? Es werden Verallgemeinerungen des "Maximumprinzips für inverse Spalteneinträge" angegeben und untersucht.