Functional Analysis and Economic Theory PDF
Preview Functional Analysis and Economic Theory
Functional Analysis and Economic Theory Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo Y. Abramovich . E. Avgerinos N. C. Yannelis (Eds.) Functional Analysis and Economic Theory With 14 Figures and 6 Tables Springer Professor Yuri Abramovich IUPUI Department of Mathematical Sciences Indianapolis, IN 46202 USA Dr. Evgenios Avgerinos University of the Aegean Department of Education 1 Demokratias Ave. 85100 Rhodes Greece Professor Nicholas C. Yannelis University of Illinois at Urbana Champaign Department of Economics Champaign, IL 61801 USA Library of Congress Cataloging-in-Publication Data Die Deutsche Bibliothek - CIP-Einheitsaufnahme Functional analysis and economic theory: with 6 tables I Y. Abramovich ... (ed.). - Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Santa Clara; Singapore; Tokyo: Springer, 1998 ISBN-13: 978-3-642-72224-0 e-ISBN-I3: 978-3-642-72222-6 DO I: 10.1007/ 978-3-642-72222-6 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is per mitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. @ Springer-Verlag Berlin· Heidelberg 1998 Softcover reprint ofthe hardcover 1st edition 1998 The use of general descriptive names, registered names, trademarks, etc. in this pub lication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Hardcover-Design: Erich Kirchner, Heidelberg SPIN 10679966 42/2202-5 4 3 2 1 0 - Printed on acid-free paper Preface In July of 1996, the conference Nonlinear Analysis and its Applications in Engineering and Economics took place on the Greek island of Samos, the birthplace of Pythagoras. During this conference, a special session was held on the occasion of the 50th birthday of the well known mathematician and math ematical economist Professor Charalambos Aliprantis, who, by his numerous friends, is usually called Roko. The story behind this nickname is not quite clear yet; it will be investigated further and will be made public prior to his 60th birthday. (At this moment we have already found out that it has nothing to do with the famous movie Rocco and his Brothers even though Roko does have two brothers.) Roko was born on the Greek island of Cephalonia on May 12,1946, and his elementary and secondary school education took place there. At 18 he entered the Mathematics Department at the University of Athens. Upon graduation from the University of Athens he proceeded with his graduate studies at Cal tech, where in 1973 he completed his Ph.D. degree in Mathematics under the supervision of Professor W. A. J. Luxemburg. His research career can be divided into two periods. The first one, till 1981, was devoted entirely to pure mathematics. The other one, after 1981, has been subdivided between pure mathematics and mathematical economics. The main objects of Roko's work in pure mathematics are spaces with order structure (Riesz spaces) and operators acting on them. His books (co-authored with O. Burkinshaw) Locally Solid Riesz Spaces and Positive Operators, both published by Academic Press, are highly recognized reference books used by researchers all over the world. The knowledge of Riesz spaces turned out to be rather crucial for Roko's further development in mathematical economics. In 1981 Professor Donald Brown, an economist from Yale University, was visiting Caltech. Brown needed to speak to a mathematician who had a good knowledge of Banach lattices, and Wim Luxemburg suggested to him to talk to Roko, who was also visiting Caltech that year. That is how Roko got in volved in Mathematical Economics and started his collaboration with Don. The paper of Aliprantis and Brown, Equilibria in Markets with a Riesz Space of Commodities, made it clear for the first time that the theory of Riesz spaces and Banach lattices is the natural setting for general equilibrium and infinite horizon economic models. Moreover, this was the first paper which utilized the idea of the order ideal generated by the social endowment, a concept which played a fundamental role subsequently in the work of many economists. By now Roko has written four books and over twenty papers in Economics. In vi Preface 1990 Roko's presence in this field grew even further when he launched, with the support and encouragement of Springer-Verlag, a new Journal called Eco nomic Theory, which has become a leading journal in the rigorous treatment of economic thought. This volume is coming out at a special moment for Roko. He has just moved from the Department of Mathematical Sciences at IUPUI to Purdue University, where he will hold a joint appointment in the Departments of Economics and Mathematics. Together with his numerous friends we wish him all the best for this new phase of his life. All those who are familiar with Roko know that he is a dependable, dy namic, decisive, enthusiastic, energetic, generous, honest, hard working, and extremely fair-minded individual. To make this litany of nearly perfect quali ties bearable, Roko luckily also has a great sense of humor. We were together with him on a tour when one of the guides at a historic city of Perga (Turkey) mentioned in his speech that Alexander the Great was not Greek, but rather he was Macedonian. Roko immediately retorted: You know, Ronald Reagan is not American; he is Californian! Everybody laughed and a historical crisis was avoided. We are fortunate to have known Roko for a long time, and we are happy to dedicate this volume to him. This volume contains contributions from many colleagues and friends who participated in the conference as well as from many who did not manage to attend. The papers are divided into two parts. The first part contains papers in mathematics and the second one contains papers in economics. We conclude by expressing our thanks to all those who helped organize the conference. For financial support we are very thankful to Professor David Stocum, Dean of the School of Science at IUPUI, to Professor Bart Ng, for mer Chair of the Department of Mathematical Sciences at IUPUI and to Dr. Werner Muller of Springer-Verlag. Last but not least, we thank A. Zaslavsky for his help in tackling endless 'lEX problems during-our preparation of this volume. Y. Abramovich E. Av gerinos N. Yannelis Indianapolis Rodos Champaign USA Greece USA Table of Contents Preface. v Part I. Mathematics Y. Abramovich and A. Kitover Bijective disjointness preserving operators . 1 A. Arias and A. Koldobsky A remark on positive isotropic random vectors 9 G. Barbieri and H. Weber A topological approach to the study of fuzzy measures . . . . . . . . 17 A. Basile On the ranges of additive correspondences . . . . . . . . . . . . 47 P. Enflo Extremal vectors for a class of linear operators . . . . . . . . . . . . 61 Z. Ercan and A. Wickstead Towards a theory of nonlinear orthomorphisms . . . . . . . . . . . . 65 C. Huijsmans Finitely generated vector sublattices . . . . . . . . . . . . . . . . . . 79 V. Lomonosov Duality in operator spaces ..... 97 C. Niculescu Topological transitivity and recurrence as a source of chaos . 101 A. Wickstead Order bounded operators may be far from regular .109 Part II. Economics R. Becker and C. Foias Implicit programming and the invariant manifold for Ramsey equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . 119 D. Glycopantis and A. Muir An approach to bargaining for general payoffs regions . 145 M. Kurz Social states of belief and the determinant of the equity risk premium in a rational belief equilibrium . . . . . . . . . . . . . . . . 171 viii Table of Contents K. Podczeck Quasi-equilibrium and equilibrium in a large production economy with differentiated commodities .................... 221 W. Trockel An exact implementation of the Nash bargaining solution in dominant strategies . . . . . . . . . . . . . . . . . . . . . . . . 271 D. Yannelis On the existence of a temporary unemployment equilibrium . . 281 N. Yannelis On the existence of a Bayesian Nash equilibrium . . . . . . . . 291 BIJECTIVE DISJOINTNESS PRESERVING OPERATORS YURI ABRAMOVICH and ARKADY KITOVER Abstract. A linear operator T : X -T Y between vector lattices is said to be disjointness preserving if T sends disjoint elements in X to disjoint elements in Y. Let T : X -T Y be a bijective disjointness preserving operator, and so the inverse operator T-1 exists. In this paper we discuss the most recent results regarding the following problem: when is T-l disjointness preserving? Apart from presenting several counterexamples to this problem we also formulate many sufficient conditions for the affirmative answer to it. Two elements Xl, X2 of a vector lattice are called disjoint (in notation: Xl ..L X2) if IXIIA IX21 = o. A (linear) operator T : X -+ Y between vector lattices is said to be disjointness preserving if T sends disjoint elements in X to disjoint elements in Y, that is, Xl ..L X2 in X implies that TXI ..L TX2 in Y. In some form or the other disjointness preserving operators appeared in the literature for the first time in early 30-s, but only during the last 15-20 years have they become the object of a systematic study. We mention here only several monographs [2, 8, 15, 16, 18], and a survey [10] in which these operators occupy a prominent role. (We do not even touch here any literature on the spectral properties of disjointness preserving operators.) One of the reasons for the recent interest in the disjointness preserving operators lies in the fact that it is precisely these operators that allow a multi plicative representation as weighted composition operators; thus the disjoint ness preserving operators provide an abstract framework for a very important class of operators in analysis. We refer to [1] and [2] for the results in this di rection. All our vector lattices are assumed to be Archimedean and considered either over IR or over C. We refer to [6, 17, 18] for all necessary terminology regarding vector and Banach lattices, and operators on them. 1. Main problem and related background In this paper, we will concern ourselves with the internal problems regarding the disjointness preserving operators. The following question was posed by the first named author several years ago. Problem A. Let X, Y be two arbitrary vector lattices and T : X -+ Y be a disjointness preserving bijection. Is it true that the inverse operator T-1 is also disjointness preserving? On the one hand, the question is very simple as it addresses the very basic structure of the operators preserving disjointness but, on the other hand, it has Y. Abramovich et al. (eds.), Functional Analysis and Economic Theory © Springer-Verlag Berlin · Heidelberg 1998 2 Y. A. Abramovich and A. K. Kitover turned out to be rather stimulating and fruitful, having generated a number of results devoted to it. To some extent, Problem A was motivated by the following result which, implicitly, is in [4, 5] and which solves this problem for a special class of disjointness preserving operators on Banach lattices. Recall that an operator T : X -+ X on a vector lattice is band preserving if X1.lX2 implies that Tx1.lX2' 1.1. Theorem. Let X be a Banach lattice and T be an invertible band pre serving operator from X onto X. Then the inverse operator T-1 is also band preserving. The first considerable progress regarding Problem A was made in 1990 by K. Jarosz [13] who answered it in the affirmative in the case when X and Y are the classical Banach lattices of continuous functions on compact Hausdorff spaces. A few years later a much stronger result was proved by C. Huijsmans-B. de Pagter [11] and independently by A. Koldunov [14]. To formulate their theo rem, recall that a vector lattice X is said to be relatively uniformly complete ((ru)-complete, in short) if each principal ideal in X is order isomorphic to the C(K) space for some compact Hausdorff space K. The most important examples of (ru)-complete vector lattices are Banach lattices and Dedekind complete vector lattices. 1.2. Theorem. If X is an (ru)-complete vector lattice and Y is a normed vector lattice, and T : X -+ Y is a bijective disjointness preserving operator, then T-1 is also disjointness preserving. Moreover, T is a regular operator. The list of the known facts regarding Problem A can be easily continued. We will confine ourselves to three more results. Each of them describes some special case with an affirmative solution to Problem A. In [11] an affirmative answer is obtained in the case when X is a discrete vector lattice (and Y is an arbitrary vector lattice). C. Huijsmans and A. Wickstead [12] proved that the conclusion of Theo rem 1.1 remains true if the vector lattice X either is (ru)-complete or has the principal projection property. Very recently, J. Araujo, E. Beckenstein, and L. Narici [7] proved that the answer to Problem A is affirmative if X = C(Sl) and Y = C(S2), where Sl, S2 are Tykhonoff spaces and additionally either Sl is zero-dimensional, or S2 is connected, or S2 is pseudo compact. The last case is, of course, an improvement of Jarosz's result but, on the other hand, it is just a special case of Theorem 1.2.
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