Table Of ContentFunctional 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
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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.
