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