Compact-Like Operators in Lattice-Normed Spaces
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Lifting Convex Approximation Properties and Cyclic Operators with Vector Lattices رﻓﻊ ﺧﺼﺎﺋﺺ اﻟﺘﻘﺮﯾﺐ اﻟﻤﺤﺪب واﻟﻤﺆﺛﺮات اﻟﺪورﯾﺔ ﻣﻊ ﺷﺒﻜﺎت اﻟﻤﺘﺠﮫ
University of Sudan for Science and Technology College of graduate Studies Lifting Convex Approximation Properties and Cyclic Operators with Vector Lattices رﻓﻊ ﺧﺼﺎﺋﺺ اﻟﺘﻘﺮﯾﺐ اﻟﻤﺤﺪب واﻟﻤﺆﺛﺮات اﻟﺪورﯾﺔ ﻣﻊ ﺷﺒﻜﺎت اﻟﻤﺘﺠﮫ A thesis Submitted in Partial Fulfillment of the Requirement of the Master Degree in Mathematics By: Marwa Alyas Ahmed Supervisor: Prof. Shawgy Hussein Abdalla November 2016 1 Dedication I dedicate my dissertation work to my family, a special feeling of gratitude to my loving parents, my sisters and brothers. I also dedicate this dissertation to my friends and colleagues. I ACKNOWLEDGEMENTS First of all I thank Allah.. I wish to thank my supervisor Prof. Shawgi huSSien who was more than generous with his expertise and precious time. A special thanks to him for his countless hours of reflecting, reading, encouraging, and most of all patience throughout the entire process. I would like to acknowledge and thank Sudan University for Sciences & Technology for allowing me to conduct my research and providing any assistance requested. Special thanks goes to the all members of the university for their continued support. II Abstract We demonstrate that rather weak forms of the extendable local reflexivity and of the principle of local reflexivity are needed for the lifting of bounded convex approximation properties from Banach spaces to their dual spaces. We show that certain adjoint multiplication operators are convex- cyclic and show that some are convex- cyclic but no convex polynomial of the operator is hypercyclic. Also some adjoint multi- plication operators are convex- cyclic but not 1-weakly hypercyclic. We deal with two weaker forms of injectivity which turn out to have a rich structure behind: separable injectivity and universal separable injectivity. -
ORDER EXTREME POINTS and SOLID CONVEX HULLS 1. Introduction at the Very Heart of Banach Space Geometry Lies the Study of Three I
ORDER EXTREME POINTS AND SOLID CONVEX HULLS T. OIKHBERG AND M.A. TURSI To the memory of Victor Lomonosov Abstract. We consider the \order" analogues of some classical notions of Banach space geometry: extreme points and convex hulls. A Hahn- Banach type separation result is obtained, which allows us to establish an \order" Krein-Milman Theorem. We show that the unit ball of any infinite dimensional reflexive space contains uncountably many order extreme points, and investigate the set of positive norm-attaining func- tionals. Finally, we introduce the \solid" version of the Krein-Milman Property, and show it is equivalent to the Radon-Nikod´ym Property. 1. Introduction At the very heart of Banach space geometry lies the study of three in- terrelated subjects: (i) separation results (starting from the Hahn-Banach Theorem), (ii) the structure of extreme points, and (iii) convex hulls (for instance, the Krein-Milman Theorem on convex hulls of extreme points). Certain counterparts of these notions exist in the theory of Banach lattices as well. For instance, there are positive separation/extension results; see e.g. [1, Section 1.2]. One can view solid convex hulls as lattice analogues of convex hulls; these objects have been studied, and we mention some of their properties in the paper. However, no unified treatment of all three phenomena listed above has been attempted. In the present paper, we endeavor to investigate the lattice versions of (i), (ii), and (iii) above. We introduce the order version of the classical notion of an extreme point: if A is a subset of a Banach lattice X, then a 2 A is called an order extreme point of A if for all x0; x1 2 A and t 2 (0; 1) the inequality a ≤ (1 − t)x0 + tx1 implies x0 = a = x1. -
Locally Solid Riesz Spaces with Applications to Economics / Charalambos D
http://dx.doi.org/10.1090/surv/105 alambos D. Alipr Lie University \ Burkinshaw na University-Purdue EDITORIAL COMMITTEE Jerry L. Bona Michael P. Loss Peter S. Landweber, Chair Tudor Stefan Ratiu J. T. Stafford 2000 Mathematics Subject Classification. Primary 46A40, 46B40, 47B60, 47B65, 91B50; Secondary 28A33. Selected excerpts in this Second Edition are reprinted with the permissions of Cambridge University Press, the Canadian Mathematical Bulletin, Elsevier Science/Academic Press, and the Illinois Journal of Mathematics. For additional information and updates on this book, visit www.ams.org/bookpages/surv-105 Library of Congress Cataloging-in-Publication Data Aliprantis, Charalambos D. Locally solid Riesz spaces with applications to economics / Charalambos D. Aliprantis, Owen Burkinshaw.—2nd ed. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 105) Rev. ed. of: Locally solid Riesz spaces. 1978. Includes bibliographical references and index. ISBN 0-8218-3408-8 (alk. paper) 1. Riesz spaces. 2. Economics, Mathematical. I. Burkinshaw, Owen. II. Aliprantis, Char alambos D. III. Locally solid Riesz spaces. IV. Title. V. Mathematical surveys and mono graphs ; no. 105. QA322 .A39 2003 bib'.73—dc22 2003057948 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. -
When Is the Core Equivalence Theorem Valid?
DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CALIFORNIA 91125 WHEN IS THE CORE EQUIVALENCE THEOREM VALID? Charalambos D. Aliprantis California Institute of Technology Indiana University - Purdue Univeristy at Indianapolis Owen Burkinshaw Indiana University - Purdue University at Indianapolis SOCIAL SCIENCE WORKING PAPER 707 September 1989 WHEN IS THE CORE EQUIVALENCE THEOREM VALID? Charalambos D. Aliprantis and Owen Burkinshaw ABSTRACT In 1983 L. E. Jones exhibited a surprising example of a weakly Pareto optimal al location in a two consumer pure exchange economy that failed to be supported by prices. In this example the price space is not a vector lattice (Riesz space). Inspired by Jones' example, A. Mas-Colell and S. F. Richard proved that this pathological phenomenon cannot happen when the price space is a vector lattice. In particu lar, they established that (under certain conditions) in a pure exchange economy the lattice structure of the price space is sufficient to guarantee the supportability of weakly Pareto optimal allocations by prices-i.e., they showed that the second welfare theorem holds true in an exchange economy whose price space is a vector lattice. In addition, C. D. Aliprantis, D. J. Brown and 0. Burkinshaw have shown that when the price space of an exchange economy is a certain vector lattice, the Debreu-Scarf core equivalence theorem holds true, i.e., the sets of Walrasian equilib ria and Edgeworth equilibria coincide. (An Edgeworth equilibrium is an allocation that belongs to the core of every replica economy of the original economy.) In other words, the lattice structure of the price space is a sufficient condition for avoiding the pathological situation occuring in Jones' example. -
Riesz Vector Spaces and Riesz Algebras Séminaire Dubreil
Séminaire Dubreil. Algèbre et théorie des nombres LÁSSLÓ FUCHS Riesz vector spaces and Riesz algebras Séminaire Dubreil. Algèbre et théorie des nombres, tome 19, no 2 (1965-1966), exp. no 23- 24, p. 1-9 <http://www.numdam.org/item?id=SD_1965-1966__19_2_A9_0> © Séminaire Dubreil. Algèbre et théorie des nombres (Secrétariat mathématique, Paris), 1965-1966, tous droits réservés. L’accès aux archives de la collection « Séminaire Dubreil. Algèbre et théorie des nombres » im- plique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Seminaire DUBREIL-PISOT 23-01 (Algèbre et Theorie des Nombres) 19e annee, 1965/66, nO 23-24 20 et 23 mai 1966 RIESZ VECTOR SPACES AND RIESZ ALGEBRAS by Lássló FUCHS 1. Introduction. In 1940, F. RIESZ investigated the bounded linear functionals on real function spaces S, and showed that they form a vector lattice whenever S is assumed to possess the following interpolation property. (A) Riesz interpolation property. -If f , g~ are functions in S such that g j for i = 1 , 2 and j = 1 , 2 , then there is some h E S such that Clearly, if S is a lattice then it has the Riesz interpolation property (choose e. g. h = f 1 v f~ A g 2 ~, but there exist a number of important function spaces which are not lattice-ordered and have the Riesz interpolation property. -
Order-Quasiultrabarrelled Vector Lattices and Spaces
Periodica Mathematica Hungarica YoL 6 (4), (1975), pp. 363--371. ~ ORDER-QUASIULTRABARRELLED VECTOR LATTICES AND SPACES by T. HUSAIN and S. M. KHALEELULLA (Hamilton) 1. Introduetion In this paper, we introduce and study a class of topological vector lattices (more generally, ordered topological vector spaces) which we call the class of order-quasiultrabarreUed vector lattices abbreviated to O. Q. U. vector lattices (respectively, O. Q. U. spaces). This class replaces that of order-infrabarrened Riesz spaces (respectively, order-infrabarrelled spaces) [8] in situations where local convexity is not assumed. We obtain an analogue of the Banach--Stein- haus theorem for lattice homom0rphisms on O. Q. U. vector lattices (respec- tively positive linear maps on O. Q. U. spaces) and the one for O. Q. U. spaces is s used successfully to obtain an analogue of the isomorphism theorem concerning O. Q.U. spaces and similar Schauder bases. Finally, we prove a closed graph theorem for O. Q. U, spaces, analogous to that for ultrabarrelled spa~s [6]. 2. Notations and preliminaries We abbreviate locally convex space, locally convex vector lattice, tope- logical vector space, and topological vector lattice to 1.c.s., 1.c.v.l., t.v.s., and t.v.l., respectively. We write (E, C) to denote a vector lattice (or ordered vector space) over the field of reals, with positive cone (or simply, cone) C. A subset B of a vector lattice (E, G) is solid if Ixl ~ ]y], yEB implies xEB; a vector subspace M of (~, C) which is also solid is called a lattice ideal. -
Version of 4.9.09 Chapter 35 Riesz Spaces the Next Three Chapters Are
Version of 4.9.09 Chapter 35 Riesz spaces The next three chapters are devoted to an abstract description of the ‘function spaces’ described in Chapter 24, this time concentrating on their internal structure and relationships with their associated measure algebras. I find that any convincing account of these must involve a substantial amount of general theory concerning partially ordered linear spaces, and in particular various types of Riesz space or vector lattice. I therefore provide an introduction to this theory, a kind of appendix built into the middle of the volume. The relation of this chapter to the next two is very like the relation of Chapter 31 to Chapter 32. As with Chapter 31, it is not really meant to be read for its own sake; those with a particular interest in Riesz spaces might be better served by Luxemburg & Zaanen 71, Schaefer 74, Zaanen 83 or my own book Fremlin 74a. I begin with three sections in an easy gradation towards the particular class of spaces which we need to understand: partially ordered linear spaces (§351), general Riesz spaces (§352) and Archimedean Riesz spaces (§353); the last includes notes on Dedekind (σ-)complete spaces. These sections cover the fragments of the algebraic theory of Riesz spaces which I will use. In the second half of the chapter, I deal with normed Riesz spaces (in particular, L- and M-spaces)(§354), spaces of linear operators (§355) and dual Riesz spaces (§356). Version of 16.10.07 351 Partially ordered linear spaces I begin with an account of the most basic structures which involve an order relation on a linear space, partially ordered linear spaces. -
Operators and the Space of Integrable Scalar Functions with Respect to a Frechet-Valued Measure
J. Austral. Math. Soc. (Series A) 65 (1998), 176-193 OPERATORS AND THE SPACE OF INTEGRABLE SCALAR FUNCTIONS WITH RESPECT TO A FRECHET-VALUED MEASURE ANTONIO FERNANDEZ and FRANCISCO NARANJO (Received 21 May 1997; revised 10 October 1997) Communicated by P. G. Dodds Abstract We consider the space L' (v, X) of all real functions that are integrable with respect to a measure v with values in a real Frechet space X. We study L-weak compactness in this space. We consider the problem of the relationship between the existence of copies of t°° in the space of all linear continuous operators from a complete DF-space Y to a Frechet lattice E with the Lebesgue property and the coincidence of this space with some ideal of compact operators. We give sufficient conditions on the measure v and the space X that imply that L' (v, X) has the Dunford-Pettis property. Applications of these results to Frechet AL-spaces and Kothe sequence spaces are also given. 1991 Mathematics subject classification (Amer. Math. Soc): primary 46A04, 46A40,46G10,47B07. Keywords and phrases: Frechet lattice, L-weak compactness, Dunford-Pettis property, L-weakly compact operators, generalized AL-spaces, Kothe sequences spaces. 1. Introduction In this paper we study operators with values in, or defined on, spaces of scalar-valued integrable functions with respect to a vector measure with values in a real Frechet space. This kind of integration was introduced by Lewis in [19] and developed essentially by Kluvanek and Knowles in [18] for locally convex spaces. Let us recall briefly the basic definitions (see [19] and [18]). -
Inapril 3, 1970
AN ABSTRACT OF THE THESIS OF Ralph Leland James for theDoctor of Philosophy (Name) (Degree) inAprilMathematics 3,presented 1970on (Major) (Date) Title: CONVERGENCE OF POSITIVE OPERATORS Abstract approved: Redacted for Privacy P. M. Anselone The extension and convergence of positiveoperators is investi- gated by means of a monotone approximation technique.Some gener- alizations and extensions of Korovkin's monotoneoperator theorem on C[0, 1] are given. The concept of a regular set is introduced and it is shownthat pointwise convergence is uniform on regular sets.Regular sets are investigated in various spaces andsome characterizations are obtained. These concepts are applied to the approximate solutionof a large class of integral equations. Convergence of Positive Operators by Ralph Leland James A THESIS submitted to Oregon State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy June 1970 APPROVED: Redacted for Privacy P nf eS ::33r of _Department of Ma,thernat ic s in charge of major Redacted for Privacy ActingChairman o Department of Mathematics Redacted for Privacy Dean of Graduate School Date thesis is presented April 3, 1970 Typed by Barbara Eby for Ralph Leland James ACKNOWLEDGEMENT I wish to express my appreciation to Professor P. M.Anselone for his guidance and encouragement during the preparation ofthis thesis. CONVERGENCE OF POSITIVE OPERATORS I.INTRODUCTION §1.Historical Remarks The ordinary Riemann integral can be regarded as an extension of the integral of a continuous function to a larger space in the follow- ing way.Let (1,03,C denote respectively the linear spaces of all, bounded, and continuous real valued functions on [0, 1] .For x in define 1 P0 x = x(t)dt is posi- thenP0is a linear functional defined on C. -
Applications of Convergence Spaces to Vector Lattice Theory
Volume 41, 2013 Pages 311{331 http://topology.auburn.edu/tp/ Applications of Convergence Spaces to Vector Lattice Theory by Jan Harm van der Walt Electronically published on October 3, 2012 Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: [email protected] ISSN: 0146-4124 COPYRIGHT ⃝c by Topology Proceedings. All rights reserved. http://topology.auburn.edu/tp/ TOPOLOGY PROCEEDINGS Volume 41 (2013) Pages 311-331 E-Published on October 3, 2012 APPLICATIONS OF CONVERGENCE SPACES TO VECTOR LATTICE THEORY JAN HARM VAN DER WALT Abstract. We introduce the concept of a locally solid convergence vector lattice as a generalization of locally solid Riesz spaces. This notion provides an appropriate context for a number of natural modes of convergence that cannot be described in terms of the usual Hausdorff-Kuratowski-Bourbaki (HKB) notion of topology. We discuss the dual and completion of a locally solid convergence vector lattice. As applications of this new concept we present a Closed Graph Theorem for linear operators on a class of vector lattices, and a duality result for locally convex, locally solid Riesz spaces. 1. Introduction Many of the most important spaces that arise in analysis are vector lattices. Recall [1, page 3] that a subset A of a vector lattice L is called solid whenever 8 f 2 A; g 2 L : (1.1) jgj ≤ jfj ) g 2 A: A vector space topology on L is called locally solid if it has a basis at 0 consisting of solid sets [1, Definition 5.1]. -
An Algebraic Theory of Infinite Classical Lattices I
An algebraic theory of infinite classical lattices I: General theory Don Ridgeway Department of Statistics, North Carolina State University, Raleigh, NC 27695 [email protected] Abstract We present an algebraic theory of the states of the infinite classical lattices. The construction follows the Haag-Kastler axioms from quantum field theory. By com- parison, the *-algebras of the quantum theory are replaced here with the Banach lattices (MI-spaces) to have real-valued measurements, and the Gelfand-Naimark- Segal construction with the structure theorem for MI-spaces to represent the Segal algebra as C(X). The theory represents any compact convex set of states as a decom- arXiv:math-ph/0501041v3 8 Oct 2005 position problem of states on an abstract Segal algebra C(X), where X is isomorphic with the space of extremal states of the set. Three examples are treated, the study of groups of symmetries and symmetry breakdown, the Gibbs states, and the set of all stationary states on the lattice. For relating the theory to standard problems of statistical mechanics, it is shown that every thermodynamic-limit state is uniquely identified by expectation values with an algebraic state. MSC 46A13 (primary) 46M40 (secondary) 1 2 THEORY OF MEASUREMENT I Introduction It is now generally recognized in statistical mechanics that in order to well- define even such basic thermodynamic concepts as temperature and phase transition, one must deal with systems of infinite extent [12]. Two approaches to the study of infinite systems have emerged since the 1950s, Segal’s algebraic approach in quantum field theory (QFT) ([3], [8], [13], [27]) and the theory of thermodynamic-limit (TL) states ([5],[17],[16]). -
Nonstandard Analysis and Vector Lattices Managing Editor
Nonstandard Analysis and Vector Lattices Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Volume 525 Nonstandard Analysis and Vector Lattices Edited by S.S. Kutateladze Sobolev Institute of Mathematics. Siberian Division of the Russian Academy of Sciences. Novosibirsk. Russia SPRINGER-SCIENCE+BUSINESS MEDIA, B. V. A C.LP. Catalogue record for this book is available from the Library of Congress. ISBN 978-94-010-5863-6 ISBN 978-94-011-4305-9 (eBook) DOI 10.1007/978-94-011-4305-9 This is an updated translation of the original Russian work. Nonstandard Analysis and Vector Lattices, A.E. Gutman, \'E.Yu. Emelyanov, A.G. Kusraev and S.S. Kutateladze. Novosibirsk, Sobolev Institute Press, 1999. The book was typeset using AMS-TeX. Printed an acid-free paper AII Rights Reserved ©2000 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2000 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Contents Foreword ix Chapter 1. Nonstandard Methods and Kantorovich Spaces (A. G. Kusraev and S. S. Kutateladze) 1 § 1.l. Zermelo-Fraenkel Set·Theory 5 § l.2. Boolean Valued Set Theory 7 § l.3. Internal and External Set Theories 12 § 1.4. Relative Internal Set Theory 18 § l.5. Kantorovich Spaces 23 § l.6. Reals Inside Boolean Valued Models 26 § l.7. Functional Calculus in Kantorovich Spaces 30 § l.8.