Dissertations / Theses on the topic 'Normed linear spaces and Banach spaces'

"A natural process in examining properties of Banach spaces is to see if a Banach space can be decomposed into simpler Banach spaces; in other words, to see if a Banach space has complemented subspaces. This thesis concentrates on three main aspects of this problem: norm of projections of a Banach space onto its finite dimensional subspaces; a class of Banach spaces, each of which has a large number of infinite dimensional complemented subspaces; and methods of finding Banach spaces which have uncomplemented subspaces, where the subspaces and the quotient spaces are chosen as well-known classical sequence spaces (finding non-trivial twisted sums)." --Abstract.
Master of Mathematical Sciences

"A natural process in examining properties of Banach spaces is to see if a Banach space can be decomposed into simpler Banach spaces; in other words, to see if a Banach space has complemented subspaces. This thesis concentrates on three main aspects of this problem: norm of projections of a Banach space onto its finite dimensional subspaces; a class of Banach spaces, each of which has a large number of infinite dimensional complemented subspaces; and methods of finding Banach spaces which have uncomplemented subspaces, where the subspaces and the quotient spaces are chosen as well-known classical sequence spaces (finding non-trivial twisted sums)." --Abstract.
Master of Mathematical Sciences

"A natural process in examining properties of Banach spaces is to see if a Banach space can be decomposed into simpler Banach spaces; in other words, to see if a Banach space has complemented subspaces. This thesis concentrates on three main aspects of this problem: norm of projections of a Banach space onto its finite dimensional subspaces; a class of Banach spaces, each of which has a large number of infinite dimensional complemented subspaces; and methods of finding Banach spaces which have uncomplemented subspaces, where the subspaces and the quotient spaces are chosen as well-known classical sequence spaces (finding non-trivial twisted sums)." --Abstract.
Master of Mathematical Sciences

Banachverbände spielen sowohl in der Theorie als auch in der Anwendung von geordneten normierten Räume eine bedeutende Rolle. Einerseits erweisen sich viele in der Praxis relevanten Räume als Banachverbände, andererseits ermöglichen die Vektorverbandsstruktur und die enge Beziehung zwischen Ordnung und Norm ein tiefes Verständnis solcher normierter Räume. An dieser Stelle setzen folgende Überlegungen an: - Die genaue Untersuchung einiger Resultate der reichhaltigen Banachverbandstheorie ließ (zu Recht) vermuten, dass in manchen Fällen die Verbandsnormeigenschaft keine notwendige Voraussetzung ist. In der Literatur gibt es bereits einige interessante Untersuchungen allgemeiner geordneter normierter Räume mit qualifizierten positiven Kegeln und in dem Zusammenhang eine Reihe wertvoller Dualitätsaussagen. An dieser Stelle sind die Eigenschaften der Normalität, der Nichtabgeflachtheit und der Regularität eines Kegels erwähnt, welche selbst im Falle eines mit einer Norm versehenen Vektorverbandes eine schwächere Relation zwischen Ordnung und Norm ergeben als die Verbandsnormeigenschaft. - In einer neueren Arbeit wurde der aus der Theorie der Vektorverbände gut bekannte Begriff der Disjunktheit bereits auf beliebige geordnete Räume verallgemeinert, wobei viele Eigenschaften disjunkter Vektoren, des disjunkten Komplements einer Menge usw., welche aus der Verbandstheorie bekannt sind, erhalten bleiben. Auf entsprechende Weise, d.h. durch das Ersetzen exakter Infima und Suprema durch Mengen unterer bzw. oberer Schranken, können der Modul eines Vektors sowie der Begriff der Solidität einer Menge für geordnete (normierte) Räume eingeführt werden. An solchen Überlegungen knüpft die vorliegende Arbeit an. Im Kapitel m-Normen ======== werden verallgemeinerte Formen der M-Norm Eigenschaft eingeführt und untersucht. AM-Räume und (approximative) Ordnungseinheit-Räume sind Beispiele für geordnete normierte Räume mit m-Norm. Die Schwerpunkte dieses Kapitels sind zum Einen Kegel- und Normeigenschaften dieser Räume und deren Charakterisierung mit Hilfe solcher Eigenschaften und zum Anderen Dualitätsaussagen, wie sie zum Teil bereits aus der Theorie der AM- und AL-Räume bekannt sind. Minimal totale Mengen ===================== Ziel dieses Kapitels ist es, den oben erwähnten verallgemeinerten Disjunktheitsbegiff für geordnete normierte Räume zu untersuchen. Eine zentrale Rolle spielen dabei totale Mengen im Dualraum und insbesondere minimal totale Mengen sowie deren Zusammenhang mit der Disjunktheit von Elementen des Ausgangsraumes. Normierte pre-Riesz Räume ========================= Wie bereits bekannt, lässt sich jeder pre-Riesz Raum ordnungsdicht in einen (bis auf Isomorphie) eindeutigen minimalen Vektorverband einbetten, die so genannte Riesz Vervollständigung. Ist der pre-Riesz Raum normiert und sein positiver Kegel abgeschlossen, dann kann eine Verbandsnorm auf der Riesz Vervollständigung eingeführt werden, welche sich in vielen Fällen als äquivalent zur Ausgangsnorm auf dem pre-Riesz Raum erweist. Es ist allgemein bekannt, dass sich dann auch stetige lineare Funktionale fortsetzen lassen. In diesem Kapitel wird nun untersucht, inwiefern sich Ordnungsrelationen auf einer Menge stetiger linearer Funktionale beim Übergang zur Menge der Fortsetzungen erhalten lassen. Die gewonnenen Erkenntnisse kommen anschließend bei Untersuchungen zur schwachen bzw. schwach*-Topologie auf geordneten normierten Räumen zur Anwendung. Hierbei werden zwei Fragestellungen behandelt. Zum Einen gilt das Augenmerk disjunkten Folgen in geordneten normierten Räumen. Als Beispiel seien ordnungsbeschränkte disjunkte Folgen in geordneten normierten Räumen mit halbmonotoner mNorm genannt, welche stets schwach gegen Null konvergieren. Zum Anderen werden monoton fallende Folgen und Netze bzw. disjunkte Folgen von stetigen linearen Funktionalen auf einem geordneten normierten Raum betrachtet
Banach lattices play an important role in the theory of ordered normed spaces. One reason is, that many ordered normed vector spaces, that are important in practice, turn out to be Banach lattices, on the other hand, the lattice structure and strong relations between order and norm allow a deep understanding of such ordered normed spaces. At this point the following is to be considered. - The analysis of some results in the rich Banach lattice theory leads to the conjecture, that sometimes the lattice norm property is no necessary supposition. General ordered normed spaces with a convenient positive cone were already examined, where some valuable duality properties could be achieved. We point out the properties of normality, non-flatness and regularity of a cone, which are a weaker relation between order and norm than the lattice norm property in normed vector lattices. - The notion of disjointness in vector lattices has already been generalized to arbitrary ordered vector spaces. Many properties of disjoint elements, the disjoint complement of a set etc., well known from the vector lattice theory, are preserved. The modulus of a vector as well as the concept of the solidness of a set can be introduced in a similar way, namely by replacing suprema and infima by sets of upper and lower bounds, respectively. We take such ideas up in the present thesis. A generalized version of the M-norm property is introduced and examined in section m-norms. ======= AM-spaces and approximate order unit spaces are examples of ordered normed spaces with m-norm. The main points of this section are the special properties of the positive cone and the norm of such spaces and the duality properties of spaces with m-norm. Minimal total sets ================== In this section we examine the mentioned generalized disjointness in ordered normed spaces. Total sets as well as minimal total sets and their relation to disjoint elements play an inportant at this. Normed pre-Riesz spaces ======================= As already known, every pre-Riesz space can be order densely embedded into an (up to isomorphism) unique vector lattice, the so called Riesz completion. If, in addition, the pre-Riesz space is normed and its positive cone is closed, then a lattice norm can be introduced on the Riesz completion, that turns out to be equivalent to the primary norm on the pre-Riesz space in many cases. Positive linear continuous functionals on the pre-Riesz space are extendable to positive linear continuous functionals in this setting. Here we investigate, how some order relations on a set of continuous functionals can be preserved to the set of the extension. In the last paragraph of this section the obtained results are applied for investigations of some questions concerning the weak and the weak* topology on ordered normed vector spaces. On the one hand, we focus on disjoint sequences in ordered normed spaces. On the other hand, we deal with decreasing sequences and nets and disjoint sequences of linear continuous functionals on ordered normed spaces

Banachverbände spielen sowohl in der Theorie als auch in der Anwendung von geordneten normierten Räume eine bedeutende Rolle. Einerseits erweisen sich viele in der Praxis relevanten Räume als Banachverbände, andererseits ermöglichen die Vektorverbandsstruktur und die enge Beziehung zwischen Ordnung und Norm ein tiefes Verständnis solcher normierter Räume. An dieser Stelle setzen folgende Überlegungen an: - Die genaue Untersuchung einiger Resultate der reichhaltigen Banachverbandstheorie ließ (zu Recht) vermuten, dass in manchen Fällen die Verbandsnormeigenschaft keine notwendige Voraussetzung ist. In der Literatur gibt es bereits einige interessante Untersuchungen allgemeiner geordneter normierter Räume mit qualifizierten positiven Kegeln und in dem Zusammenhang eine Reihe wertvoller Dualitätsaussagen. An dieser Stelle sind die Eigenschaften der Normalität, der Nichtabgeflachtheit und der Regularität eines Kegels erwähnt, welche selbst im Falle eines mit einer Norm versehenen Vektorverbandes eine schwächere Relation zwischen Ordnung und Norm ergeben als die Verbandsnormeigenschaft. - In einer neueren Arbeit wurde der aus der Theorie der Vektorverbände gut bekannte Begriff der Disjunktheit bereits auf beliebige geordnete Räume verallgemeinert, wobei viele Eigenschaften disjunkter Vektoren, des disjunkten Komplements einer Menge usw., welche aus der Verbandstheorie bekannt sind, erhalten bleiben. Auf entsprechende Weise, d.h. durch das Ersetzen exakter Infima und Suprema durch Mengen unterer bzw. oberer Schranken, können der Modul eines Vektors sowie der Begriff der Solidität einer Menge für geordnete (normierte) Räume eingeführt werden. An solchen Überlegungen knüpft die vorliegende Arbeit an. Im Kapitel m-Normen ======== werden verallgemeinerte Formen der M-Norm Eigenschaft eingeführt und untersucht. AM-Räume und (approximative) Ordnungseinheit-Räume sind Beispiele für geordnete normierte Räume mit m-Norm. Die Schwerpunkte dieses Kapitels sind zum Einen Kegel- und Normeigenschaften dieser Räume und deren Charakterisierung mit Hilfe solcher Eigenschaften und zum Anderen Dualitätsaussagen, wie sie zum Teil bereits aus der Theorie der AM- und AL-Räume bekannt sind. Minimal totale Mengen ===================== Ziel dieses Kapitels ist es, den oben erwähnten verallgemeinerten Disjunktheitsbegiff für geordnete normierte Räume zu untersuchen. Eine zentrale Rolle spielen dabei totale Mengen im Dualraum und insbesondere minimal totale Mengen sowie deren Zusammenhang mit der Disjunktheit von Elementen des Ausgangsraumes. Normierte pre-Riesz Räume ========================= Wie bereits bekannt, lässt sich jeder pre-Riesz Raum ordnungsdicht in einen (bis auf Isomorphie) eindeutigen minimalen Vektorverband einbetten, die so genannte Riesz Vervollständigung. Ist der pre-Riesz Raum normiert und sein positiver Kegel abgeschlossen, dann kann eine Verbandsnorm auf der Riesz Vervollständigung eingeführt werden, welche sich in vielen Fällen als äquivalent zur Ausgangsnorm auf dem pre-Riesz Raum erweist. Es ist allgemein bekannt, dass sich dann auch stetige lineare Funktionale fortsetzen lassen. In diesem Kapitel wird nun untersucht, inwiefern sich Ordnungsrelationen auf einer Menge stetiger linearer Funktionale beim Übergang zur Menge der Fortsetzungen erhalten lassen. Die gewonnenen Erkenntnisse kommen anschließend bei Untersuchungen zur schwachen bzw. schwach*-Topologie auf geordneten normierten Räumen zur Anwendung. Hierbei werden zwei Fragestellungen behandelt. Zum Einen gilt das Augenmerk disjunkten Folgen in geordneten normierten Räumen. Als Beispiel seien ordnungsbeschränkte disjunkte Folgen in geordneten normierten Räumen mit halbmonotoner mNorm genannt, welche stets schwach gegen Null konvergieren. Zum Anderen werden monoton fallende Folgen und Netze bzw. disjunkte Folgen von stetigen linearen Funktionalen auf einem geordneten normierten Raum betrachtet.
Banach lattices play an important role in the theory of ordered normed spaces. One reason is, that many ordered normed vector spaces, that are important in practice, turn out to be Banach lattices, on the other hand, the lattice structure and strong relations between order and norm allow a deep understanding of such ordered normed spaces. At this point the following is to be considered. - The analysis of some results in the rich Banach lattice theory leads to the conjecture, that sometimes the lattice norm property is no necessary supposition. General ordered normed spaces with a convenient positive cone were already examined, where some valuable duality properties could be achieved. We point out the properties of normality, non-flatness and regularity of a cone, which are a weaker relation between order and norm than the lattice norm property in normed vector lattices. - The notion of disjointness in vector lattices has already been generalized to arbitrary ordered vector spaces. Many properties of disjoint elements, the disjoint complement of a set etc., well known from the vector lattice theory, are preserved. The modulus of a vector as well as the concept of the solidness of a set can be introduced in a similar way, namely by replacing suprema and infima by sets of upper and lower bounds, respectively. We take such ideas up in the present thesis. A generalized version of the M-norm property is introduced and examined in section m-norms. ======= AM-spaces and approximate order unit spaces are examples of ordered normed spaces with m-norm. The main points of this section are the special properties of the positive cone and the norm of such spaces and the duality properties of spaces with m-norm. Minimal total sets ================== In this section we examine the mentioned generalized disjointness in ordered normed spaces. Total sets as well as minimal total sets and their relation to disjoint elements play an inportant at this. Normed pre-Riesz spaces ======================= As already known, every pre-Riesz space can be order densely embedded into an (up to isomorphism) unique vector lattice, the so called Riesz completion. If, in addition, the pre-Riesz space is normed and its positive cone is closed, then a lattice norm can be introduced on the Riesz completion, that turns out to be equivalent to the primary norm on the pre-Riesz space in many cases. Positive linear continuous functionals on the pre-Riesz space are extendable to positive linear continuous functionals in this setting. Here we investigate, how some order relations on a set of continuous functionals can be preserved to the set of the extension. In the last paragraph of this section the obtained results are applied for investigations of some questions concerning the weak and the weak* topology on ordered normed vector spaces. On the one hand, we focus on disjoint sequences in ordered normed spaces. On the other hand, we deal with decreasing sequences and nets and disjoint sequences of linear continuous functionals on ordered normed spaces.

In this thesis we consider the problem of simultaneously approximating elements of a set B C X by a single element of a set K C X. This type of a problem arises when the element to be approximated is not known precisely but is known to belong to a set.Thus, best simultaneous approximation is a natural generalization of best approximation which has been studied extensively. The theory of best simultaneous approximation has been studied by many authors, see for example [4], [8], [25], [28], [26] and [12] to name but a few.

Bibliography: leaves 176-182.
Certain properties associated with these classes are stable under small perturbation, i.e. stable under additive perturbation by continuous operators whose norms are less than the minimum modulus of the relation being perturbed, and are also stable under perturbation by compact, strictly singular or strictly cosingular operators. In this work we continue the study of these classes and introduce the classes of α-Atkinson and β-Atkinson relations. These are subclasses of upper and lower semi-Fredholm relations respectively, having generalised inverses and defined in terms of the existence of continuous projections onto their ranges and nullspaces.

The traditional first-order analysis in Euclidean spaces relies on the Sobolev spaces W1,p(Ω), where Ω ⊂ Rn is open and p ∈ [1, ∞].The Sobolev norm is then defined as the sum of Lp norms of a function and its distributional gradient.We generalize the notion of Sobolev spaces in two different ways. First, the underlying function norm will be replaced by the “norm” of a quasi-Banach function lattice. Second, we will investigate functions defined on an abstract metric measure space and that is why the distributional gradients need to be substituted. The thesis consists of two papers. The first one builds up the elementary theory of Newtonian spaces based on quasi-Banach function lattices. These lattices are complete linear spaces of measurable functions with a topology given by a quasinorm satisfying the lattice property. Newtonian spaces are first-order Sobolev-type spaces on abstract metric measure spaces, where the role of weak derivatives is passed on to upper gradients. Tools such asmoduli of curve families and the Sobolev capacity are developed, which allows us to study basic properties of the Newtonian functions.We will see that Newtonian spaces can be equivalently defined using the notion of weak upper gradients, which increases the number of techniques available to study these spaces. The absolute continuity of Newtonian functions along curves and the completeness of Newtonian spaces in this general setting are also established. The second paper in the thesis then continues with investigation of properties of Newtonian spaces based on quasi-Banach function lattices. The set of all weak upper gradients of a Newtonian function is of particular interest.We will prove that minimalweak upper gradients exist in this general setting.Assuming that Lebesgue’s differentiation theoremholds for the underlyingmetricmeasure space,wewill find a family of representation formulae. Furthermore, the connection between pointwise convergence of a sequence of Newtonian functions and its convergence in norm is studied.

The purpose of this paper is to explore certain properties of Banach spaces. The first chapter begins with basic definitions, includes examples of Banach spaces, and concludes with some properties of continuous linear functionals. In the second chapter, dimension is discussed; then one version of the Hahn-Banach Theorem is presented. The third chapter focuses on dual spaces and includes an example using co, RI, and e'. The role of locally convex spaces is also explored in this chapter. In the fourth chapter, several more theorems concerning dual spaces and related topologies are presented. The final chapter focuses on reflexive spaces. In the main theorem, the relation between compactness and reflexivity is examined. The paper concludes with an example of a non-reflexive space.

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&quot
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.

This thesis consists of four papers and focuses on function spaces related to first-order analysis in abstract metric measure spaces. The classical (i.e., Sobolev) theory in Euclidean spaces makes use of summability of distributional gradients, whose definition depends on the linear structure of Rn. In metric spaces, we can replace the distributional gradients by (weak) upper gradients that control the functions’ behavior along (almost) all rectifiable curves, which gives rise to the so-called Newtonian spaces. The summability condition, considered in the thesis, is expressed using a general Banach function lattice quasi-norm and so an extensive framework is built. Sobolev-type spaces (mainly based on the Lp norm) on metric spaces, and Newtonian spaces in particular, have been under intensive study since the mid-1990s. In Paper I, the elementary theory of Newtonian spaces based on quasi-Banach function lattices is built up. Standard tools such as moduli of curve families and the Sobolev capacity are developed and applied to study the basic properties of Newtonian functions. Summability of a (weak) upper gradient of a function is shown to guarantee the function’s absolute continuity on almost all curves. Moreover, Newtonian spaces are proven complete in this general setting. Paper II investigates the set of all weak upper gradients of a Newtonian function. In particular, existence of minimal weak upper gradients is established. Validity of Lebesgue’s differentiation theorem for the underlying metric measure space ensures that a family of representation formulae for minimal weak upper gradients can be found. Furthermore, the connection between pointwise and norm convergence of a sequence of Newtonian functions is studied. Smooth functions are frequently used as an approximation of Sobolev functions in analysis of partial differential equations. In fact, Lipschitz continuity, which is (unlike -smoothness) well-defined even for functions on metric spaces, often suffices as a regularity condition. Thus, Paper III concentrates on the question when Lipschitz functions provide good approximations of Newtonian functions. As shown in the paper, it suffices that the function lattice quasi-norm is absolutely continuous and a fractional sharp maximal operator satisfies a weak norm estimate, which it does, e.g., in doubling Poincaré spaces if a non-centered maximal operator of Hardy–Littlewood type is locally weakly bounded. Therefore, such a local weak boundedness on rearrangement-invariant spaces is explored as well. Finer qualitative properties of Newtonian functions and the Sobolev capacity get into focus in Paper IV. Under certain hypotheses, Newtonian functions are proven to be quasi-continuous, which yields that the capacity is an outer capacity. Various sufficient conditions for local boundedness and continuity of Newtonian functions are established. Finally, quasi-continuity is applied to discuss density of locally Lipschitz functions in Newtonian spaces on open subsets of doubling Poincaré spaces.

We study monotone operators in general Banach spaces. Properties and characterizations of monotone linear relations are presented. We focus on the "sum problem" which is the most famous open problem in Monotone Operator Theory, and we provide a powerful sufficient condition for the sum problem. We work on classical types of maximally monotone operators and provide affirmative answers to several open problems posed by Phelps and by Simons. Borwein-Wiersma decomposition and Asplund decomposition of maximally monotone operators are also studied.

In the study of Banach spaces, the development of some key properties require studying topologies on the collection of closed convex subsets of the space. The subcollection of closed linear subspaces is studied under the relative slice topology, as well as a class of topologies similar thereto. It is shown that the collection of closed linear subspaces under the slice topology is homeomorphic to the collection of their respective intersections with the closed unit ball, under the natural mapping. It is further shown that this collection under any topology in the aforementioned class of similar topologies is a strong Choquet space. Finally, a collection of category results are developed since strong Choquet spaces are also Baire spaces.

Thesis (Ph.D.) - University of Glasgow, 2000.
Ph.D. thesis submitted to the Department of Mathematics, University of Glasgow, 2000. Includes bibliographical references. Print version also available.

In this thesis we study Floquet theory on a Banach space. We are concerned about the linear differential equation of the form: y'(t) = A(t)y(t), where t ∈ R, y(t) is a function with values in a Banach space X, and A(t) are linear, bounded operators on X. If the system is periodic, meaning A(t+ω) = A(t) for some period ω, then it is called a Floquet system. We will investigate the existence and uniqueness of the periodic solution, as well as the stability of a Floquet system. This thesis will be presented in five main chapters. In the first chapter, we review the system of linear differential equations on Rn: y'= A(t)y(t) + f(t), where A(t) is an n x n matrix-valued function, y(t) are vectors and f(t) are functions with values in Rn. We establish the general form of the all solutions by using the fundamental matrix, consisting of n independent solutions. Also, we can find the stability of solutions directly by using the eigenvalues of A. In the second chapter, we study the Floquet system on Rn, where A(t+ω) = A(t). We establish the Floquet theorem, in which we show that the Floquet system is closely related to a linear system with constant coefficients, so the properties of those systems can be applied. In particular, we can answer the questions about the stability of the Floquet system. Then we generalize the Floquet theory to a linear system on Banach spaces. So we introduce to the readers Banach spaces in chapter three and the linear operators on Banach spaces in chapter four. In the fifth chapter we study the asymptotic properties of solutions of the system: y'(t) = A(t)y(t), where y(t) is a function with values in a Banach space X and A(t) are linear, bounded operators on X with A (t+ω) = A(t). For that system, we can show the evolution family U(t,s) representing the solutions is periodic, i.e. U(t+ω, s+ω) = U(t,s). Next we study the monodromy of the system V := U(ω,0). We point out that the spectrum set of V actually determines the stability of the Floquet system. Moreover, we show that the existence and uniqueness of the periodic solution of the nonhomogeneous equation in a Floquet system is equivalent to the fact that 1 belongs to the resolvent set of V.

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.

For many years mathematicians have been interested in the problem of whether an operator ideal is complemented in the space of all bounded linear operators. In this dissertation the complementation of various classes of operators in the space of all bounded linear operators is considered. This paper begins with a preliminary discussion of linear bounded operators as well as operator ideals. Let L(X, Y ) be a Banach space of all bounded linear operator 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 denote space all operator ideals by O.

Orientador: Jorge Tulio Mujica Ascui
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação
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Resumo: Nosso primeiro objetivo é provar uma importante caracterização de conjuntos fracamente compactos em espaços de Banach, o Teorema de Eberlein-Smulian, que diz que um subconjunto K de um espaço de Banach é fracamente compacto se, e somente se, toda seqüência em K tem uma subseqüência que converge fracamente para um elemento de K. Em seguida nós provamos uma importante caracterização de operadores fracamente compactos entre espaços de Banach, o Teorema de Gantmacher, que diz que um operador linear contínuo T: E -> F entre espaços de Banach é fracamente compacto se, e somente se, o seu adjunto T': F'-> E' é fracamente compacto. Finalmente, nós provamos o resultado principal deste trabalho, o Teorema de Fatoração de Davis, Figiel, Johnson e Pelczynski, que diz que, um operador linear contínuo T: E -> F entre espaços deBanach é fracamente compacto se, e somente se, T fatora-se através de um espaço de Banach reflexivo, isto é, existem um espaço de Banach reflexivo G e operadores lineares contínuos S: E-> G and L: G -> F tais que T = L o S. U ma aplicação deste resultado é que um polinômio m- homogêneo contínuo P: E -> F entre espaços de Banach é fracamente compacto se, e somente se, existem um espaço de Banach reflexivo G, um polinômio contínuo m-homogêneo Q: E-> G e um operador linear contínuo L: G -> F tais que P = L o Q
Abstract: Our first aim is to prove an important caracterization of weakly compact sets in Banach spaces, the Eberlein-¿mulian Theorem which says that a subset K of a Banach space is weakly compact if and only if each sequence in K has a subsequence which converges weakly to an element of K. We next prove an important caracterization of weakly compact operators between Banach spaces, the Gantmacher Theorem, which says that a continuous linear operator T: E -> F between Banach spaces is weakly compact if and only if its adjoint T': F'-> E' is weakly compact. Finally, we prove the principal result of this work, the Factorization Theorem of Davis, Figiel, Johnson and Pelczynski, which says that a continuous linear operator T: E -> F between Banach spaces is weakly compact if and only if T factors through a reflexive Banach space, i.e, there are a reflexive Banach space G and continuous linear operators S: E-> G and L: G -> F such that T = L o S. An application of this result is that an m-homogeneous continuous polynomial P: E -> F between Banach spaces is weakly compact if and only if there are a reflexive Banach space G, an m-homogeneous continuous polynomial Q: E -> G and a continuous linear operator L: G -> F such that P = L o Q
Mestrado
Analise Funcional
Mestre em Matemática

No presente trabalho apresentamos dois teoremas obtidos por Gorak em 2011, que são generalizações para o Teorema de Banach-Stone, envolvendo uma classe de funções não-necessariamente lineares, denominadas quasi-isometrias.
In this work we present two theorems proved by Gorak in 2011. These results are generalizations of the Banach-Stone Theorem envolving a class of not-necessarily linear functions, called quasi-isometries.

Orientador: Mario Carvalho de Matos
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Neste trabalho, damos uma generalização do conceito e da teoria das aplicações lineares p-fatoráveis para o caso multilinear. Fornecemos duas definições; baseadas na definição 2.2 chegamos a obter alguns resultados. Seguindo a ideas do Pietsch, e baseada na definição 3.9 previa generalização de algumas definições e teoremas dos ideais lineares para o caso multilinear tentamos provar a equivalência das duas definições
Abstract: In this work, we give one generalization of the concept and the linear theory of applications p - factories for the multilinear case. We supply two definitions; based in definition 2.2 we arrive to get some results. Following the ideas of the Pietsch, and based in definition 3.9 it foresaw generalization of some definitions and theorems of the linear ideals for the multilinear case we try to prove the equivalence of the two definitions
Doutorado
Matematica
Doutor em Matemática

by Chi-keung Ng.
Thesis (M.Phil.)--Chinese University of Hong Kong, 1991.
Includes bibliographical references.
Introduction --- p.1
Chapter Chapter 0. --- Preliminary --- p.4
Chapter 0.1 --- Topological vector spaces
Chapter 0.2 --- Ordered vector spaces
Chapter 0.3 --- Ordered normed spaces
Chapter 0.4 --- Ordered topological vector spaces
Chapter 0.5 --- Ordered bornological vector spaces
Chapter Chapter 1. --- Results on Ordered Normed Spaces --- p.23
Chapter 1.1 --- Results on e∞-spaces and e1-spaces
Chapter 1.2 --- Complemented subspaces of ordered normed spaces
Chapter 1.3 --- Half injections and Half surjections
Chapter 1.4 --- Strict quotients and strict subspaces
Chapter Chapter 2. --- Helley's Selection Theorem and Local Reflexivity Theorem of order type --- p.55
Chapter 2.1 --- Helley's selection theorem of order type
Chapter 2.2 --- Local reflexivity theorem of order type
Chapter Chapter 3. --- Operator Modules and Ideal Cones --- p.68
Chapter 3.1 --- Operator modules and ideal cones
Chapter 3.2 --- Space cones and space modules
Chapter 3.3 --- Injectivity and surjectivity
Chapter 3. 4 --- Dual and pre-dual
Chapter Chapter 4. --- Topologies and Bornologies Defined by Operator Modules and Ideal Cones --- p.95
Chapter 4.1 --- Generalized polars
Chapter 4.2 --- Topologies and bornologies defined by β and ε
Chapter 4. 3 --- Injectivity and generating topologies
Chapter 4.4 --- Surjectivity and generating bornologies
Chapter 4.5 --- The solid property and the generating topologies
Chapter 4.6 --- The solid property and the generating bornologies
Chapter Chapter 5. --- Semi-norms and Bounded disks defined by Operator Modules and Ideal Cones --- p.129
Chapter 5.1 --- Results on semi-norms
Chapter 5.2 --- Results on bounded disks
References --- p.146
Notations --- p.149

by Zang Rui.
"Nov 2004."
Thesis (Ph.D.)--Chinese University of Hong Kong, 2004.
Includes bibliographical references (p. 78-82)
Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Mode of access: World Wide Web.
Abstracts in English and Chinese.

We consider the Cauchy problem(UNFORMATTED TABLE OR EQUATION FOLLOWS)$$\eqalign{u\sp\prime(t)&= \int\sbsp{0}{t}\ K(t - s)Au(s)ds,\quad t\geq 0\cr u(0)&= f,\cr}\leqno(P)$$(TABLE/EQUATION ENDS)and we are interested in continuous dependence of solutions $u(t) = U(t)f$ on A and K, where $\{U(t)\}\sb{t\geq 0}$ is the resolvent family for (P) Given a family of operators $\{A\sb{n}\}$ and a family of scalar kernels $\{ K\sb{n}\}$, we study the family of Cauchy problems(UNFORMATTED TABLE OR EQUATION FOLLOWS)$$\eqalign{u\sbsp{n}{\prime}(t)&= \int\sbsp{0}{t}\ K\sb{n}(t - s)A\sb{n}u\sb{n}(s)ds,\quad t\geq 0\cr u\sb{n}(0)&= f\sb{n}.\cr}\leqno(Pn)$$(TABLE/EQUATION ENDS)We show that under certain stability conditions for $\{ A\sb{n}\}$ and $\{ K\sb{n}\},$ if $A\sb{n} \to A\sb{o}$ and if $K\sb{n} \to K\sb{o},$ in a certain sense, then $u\sb{n}(t) \to u\sb{o}(t).$ Our result is a partial extension of the Trotter-Neveu-Kato theorem to integro-differential equations
acase@tulane.edu

Tesis (Lic. en Matemática)--Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física, 2010.
En el capítulo I presentamos el 'Teorema de convexidad de Riesz-Thorin' y diferentes aplicaciones. Y concluimos con el 'Teorema de interpolación de Riesz-Stein'. En el capítulo II, hacemos un breve estudio de 'operadores de tipo débil y función distribución', definimos la clase de Marcinkievicz y demostramos los 'Teoremas de interpolación de Marcinkievicz. Caso diagonal y caso general'. Damos algunas aplicaciones. Concluimos este capítulo con las 'Condiciones de Kolmogoroff y Zygmund'. Este trabajo contiene un Apéndice donde destacamos variados resultados matemáticos, fundamentales para el entendimiento de los dos capítulos.

We introduce a new framework of generalized operators to handle vector valued distributions, intertwined evolution operators of B-evolution equations and Fokker Planck type evolution equations. Generalized operators capture these operators. The framework is a marriage between vector valued distribution theory and abstract harmonic analysis: a new convolution algebra is the offspring. The new algebra shows that convolution is more fundamental than operator composition. The framework is complete with a Hille-Yosida theorem for implicit evolution equations for generalized operators. Feller semigroups and processes fit perfectly into the framework of generalized operators. Feller semigroups are intertwined by the Chapman Kolmogorov equation. Our framework handles more complex intertwinements which naturally arise from a dynamic boundary approach to an absorbing barrier of a fly trap model: we construct an entwined pseudo Poisson process which is a pair of stochastic processes entwined by the extended Chapman Kolmogorov equation. Similarly, we introduce the idea of an entwined Brownian motion. We show that the diffusion equation of an entwined Brownian motion involves an implicit evolution equation on a suitable scalar test space. We end off by constructing a new convolution of operator valued measures which generalizes the convolution of Feller convolution semigroups.
Thesis (PhD)--University of Pretoria, 2015.
Mathematics and Applied Mathematics
Unrestricted

Скрипник К. В. Дослідження властивостей компактних операторів у банахових просторах : кваліфікаційна робота магістра спеціальності 111 "Математика" / наук. керівник І. В. Красікова. Запоріжжя : ЗНУ, 2020. 50 с.
UA : Робота викладена на 50 сторінках друкованого тексту, містить 12 джерел. Об’єкт дослідження: компактні оператори у банахових просторах. Мета роботи: дослідити властивості компактних операторів у банахових просторах. Метод дослідження: аналітичний. У кваліфікаційній роботі досліджуються властивості компактних операторів, заданих у банахових просторах. Весь теоретичний матеріал проілюстровано прикладами та задачами. Розглянуто застосування компактних операторів до розв’язання інтегральних рівнянь.
EN : The work is presented on 50 pages of printed text, 12 references. The object of the study is compact operators in Banach space. The aim of the study is to study the properties of the compact operators in the Banach space. The method of research is analytical. In the qualification paper the properties of compact operators are investigated, given in Banach spaces. All theoretical material is illustrated with examples and tasks. The application of compact operators to the solution of integral equations is considered.