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POSITIVE OPERATORS Positive Operators POSITIVE OPERATORS Positive Operators by CHARALAMBOS D. ALIPRANTIS Purdue University, West Lafayette, U.S.A. and OWEN BURKINSHAW IUPUI, Indianapolis, U.S.A. A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN-10 1-4020-5007-0 (HB) ISBN-13 978-1-4020-5007-7 (HB) ISBN-10 1-4020-5008-9 ( e-book) ISBN-13 978-1-4020-5008-4 (e-book) Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com Printed on acid-free paper All Rights Reserved © 2006 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands. Dedication To my high school mathematics teachers, Aγγλoν Mατζαβ´ινoν και ∆ηµητριoν´ Kατσωνην´ Charalambos D. Aliprantis To: Alex, Emily, Joshua, and Andrew Grandchildren bring new joys to your life. Owen Burkinshaw Contents Foreword ix Historical Foreword xi Preface xiii Acknowledgments xv List of Special Symbols xvii Chapter 1. The Order Structure of Positive Operators 1 §1.1. Basic Properties of Positive Operators 1 §1.2. Extensions of Positive Operators 23 §1.3. Order Projections 31 §1.4. Order Continuous Operators 45 §1.5. Positive Linear Functionals 58 Chapter 2. Components, Homomorphisms, and Orthomorphisms 79 §2.1. The Components of a Positive Operator 80 §2.2. Lattice Homomorphisms 93 §2.3. Orthomorphisms 111 Chapter 3. Topological Considerations 133 §3.1. Topological Vector Spaces 133 §3.2. Weak Topologies on Banach Spaces 156 §3.3. Locally Convex-Solid Riesz Spaces 169 vii viii Contents Chapter 4. Banach Lattices 181 §4.1. Banach Lattices with Order Continuous Norms 181 §4.2. Weak Compactness in Banach Lattices 207 §4.3. Embedding Banach Spaces 221 §4.4. Banach Lattices of Operators 254 Chapter 5. Compactness Properties of Positive Operators 273 §5.1. Compact Operators 273 §5.2. Weakly Compact Operators 290 §5.3. L-and M-weakly Compact Operators 316 §5.4. Dunford–Pettis Operators 340 Bibliography 359 Monographs 369 Index 371 Foreword This monograph is a reprint of our original book Positive Operators pub- lished in 1985 as volume # 119 in the Pure and Applied Mathematics series of Academic Press. With the exception of correcting several misprints and a few mathematical errors, this edition of the book is identical to the original one. At the end of the book we have listed a collection of monographs that were published after its 1985 publication that contain material and new developments related to the subject matter of this book. West Lafayette and Indianapolis CHARALAMBOS D. ALIPRANTIS May, 2006 OWEN BURKINSHAW ix Historical Foreword Oλβιos oστιs´ ιστoρ´ιηs σχ´ µαθησιν´ . Eυριπ´ιδηs Wise is the person who knows history. Euripides Positive operators made their debut at the beginning of the nineteenth century. They were tied to integral operators (whose study triggered the birth of functional analysis) and to matrices with nonnegative entries. However, positive operators were investigated in a systematic manner much later. Their study followed closely the development of Riesz spaces. An address of F. Riesz in 1928 On the decom- position of linear functionals [166] (into their positive and negative parts), at the International Congress of Mathematicians in Bologna, Italy, marked the beginnings of the study of Riesz spaces and positive operators. The theory of Riesz spaces was developed axiomatically in the mid-1930s by H. Freudenthal [67] and L. V. Kan- torovich [85, 86, 87, 88, 89, 90, 91]. Positive operators were also introduced and studied in the mid-1930s by L. V. Kantorovich, and they made their first textbook appearance in the 1940 edition of G. Birkhoff’s book Lattice Theory [37]. Un- doubtedly, the systematic study of positive operators was originated in the 1930s by F. Riesz, L. V. Kantorovich and G. Birkhoff. In the 1940s and early 1950s one finds very few papers on positive operators. In this period, the main contributions came from the Soviet school (L. V. Kan- torovich, M. G. Krein, A. G. Pinsker, M. A. Rutman, B. Z. Vulikh) and the Japanese school (H. Nakano, K. Yosida, T. Ogasawara, and their students). In the 1950 the book Functional Analysis in Partially Ordered Spaces [92] by L. V. Kantorovich, B. Z. Vulikh, and A. G. Pinsker appeared in the Soviet literature. This book (that xi xii Historical Foreword has not been translated into English) contained an excellent treatment, up to that date, of positive operators and their applications. Since the mid-1950s, research on positive operators has gained considerable mo- mentum. From 1955 to 1970 important contributions came from T. Andˆo[20]–[24], C. Goffman [70], S. Kaplan [93], S. Karlin [95], P. P. Korovkin [98], M. A. Kras- noselskii, P. P. Zabreiko, E. I. Pustylnik, and P. E. Sobolevskii [100], U. Kren- gel [105, 106], G. Ya. Lozanovsky [119]–[123], W. A. J. Luxemburg and A. C. Za- anen [125, 130], H. Nakano [148]–[152], I. Namioka [153], A. Peressini [162], H. H. Schaefer [171], and B. Z. Vulikh [189]. Thus, by the end of the 1960s the “ground work” for the theory of positive operators was well established. The 1970s can be characterized as the “maturity period” for the theory of positive operators. In 1974, the first monograph devoted entirely to the subject appeared in the literature. This was H. H. Schaefer’s book Banach Lattices and Positive Operators [174], whose influence on the development of positive operators was enormous. By this time, the number of mathematicians working in the field had increased, and research was carried out in a more systematic manner. The growth of the subject has been very fast; and, in addition, applications to other disciplines have started to flourish. Another milestone for the 1970s was the break- through paper of P. G. Dodds and D. H. Fremlin [54] on positive compact operators. The list of contributors to the theory of positive operators in the 1970s includes Y. A. Abramovich, C. D. Aliprantis, S. J. Bernau, A. V. Buhvalov, O. Burkin- shaw, D. I. Cartwright, P. G. Dodds, M. Duhoux, P. van Eldik, J. J. Grobler, D. H. Fremlin, H. P. Lotz, W. A. J. Luxemburg, M. Meyer, P. Meyer-Nieberg, R. J. Nagel, U. Schlotterbeck, H. H. Schaefer, A. R. Schep, C. T. Tucker, A. I. Vek- sler, A. W. Wickstead, M. Wolff, and A. C. Zaanen. The 1980s have started very well for the theory of positive operators. A series of important papers on the subject have been written by the authors [9, 10, 11, 12, 13, 14, 15, 16, 17]. Another excellent book on positive operators was added to the literature of positive operators. This was A. C. Zaanen’s book Riesz Spaces II [197] dealing primarily with research done up to 1980. In addition, a pool of talented young mathematicians had joined the ranks of researchers of positive operators; many have already made important contributions. Also, more and more mathematicians of the theory of Banach spaces are studying positive operators, and this was given an extra boost to the subject. In addition, the theory of positive operators has found some impressive applications in a variety of disciplines, ranging from mathematical physics to economics. Thus, as we are progressing into the 1980s the future of positive operators looks bright. The field is alive and grows both in theory and applications. It is our hope that you, the reader, will also contribute towards this growth. Indianapolis CHARALAMBOS D. ALIPRANTIS May, 1984 OWEN BURKINSHAW Preface Linear operators have been studied in various contexts and settings in the past. Their study is a subject of great importance both to mathematics and to its applications. The present book deals mainly with the special class of linear operators known as positive operators. A linear operator between two ordered vector spaces that carries positive elements to posi- tive elements is referred to as a positive operator. For instance, the linear operator T : C[0, 1] → C[0, 1], defined by x Tf(x)= f(t) dt , 0 carries positive functions of C[0, 1] to positive functions, and is thus an ex- ample of a positive operator. The material covered in the book has some overlap with the material in the books by H. H. Schaefer [174] and A. C. Zaa- nen [197]. However, our attention is mostly focused on recent developments of the subject and the overlap with the above-mentioned books is kept to a minimum. On the other hand, we have intentionally covered those topics where the ingredient of positivity allowed us to obtain beautiful and elegant results. It is well known that many linear operators between Banach spaces aris- ing in classical analysis are in fact positive operators. For this reason, in this book positive operators are studied in the setting of Riesz spaces and Banach lattices. In order to make the book as self-sufficient as possible, some basic results from the theory of Riesz spaces and Banach lattices are included with proofs as needed. On the other hand, we assume that the reader is familiar with the elementary concepts of real analysis and functional analysis. xiii xiv Preface The material has been spread out into five chapters. Chapter 1 deals mainly with the elementary properties of positive operators. This chapter covers extension properties of positive operators, order projections, order continuous operators, and positive linear functionals. Chapter 2 studies three basic classes of operators: the components of a positive operator, the lattice homomorphisms, and the orthomorphisms.
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