Operator Birkha" user Carlos S. Kubrusly Elements of Operator Theory Birkhauser Boston Basel Berlin Carlos S. Kubrusly Catholic University R. Marques de S. Vicente 225 22453-900, Rio de Janeiro, Brazil e-mail:
[email protected] puc-rio. br Ubrary of Congress Cataloging-In-Publication Data Kubrusly, Carlos S., 1947- Elements of operator theory / Carlos S. Kubrusly p. cm. Includes bibliographical references and index. ISBN 0-8176-4174-2 (acid-free paper) - ISBN 3-7643-4174-2 (acid-free paper) 1. Operator theory. I Title. QA329.K79 2001 515'.724-dc2I 2001018439 AMS Subject Classifications: 47-XX, 47-01, 47A05, 47A10, 47A12, 47AI5, 47A75, 47610, 47B15, 47B20, 47637, 47005, 47L05. 46-XX, 46-01 .46A22. 46A30, 46B 10, 461315, 46820.46B45, 46B50. 46C05,46C07, 46015, 54-XX, 54-01, 54A20, 54B05, 54B10, 54B15, 54005. 54020, 54E35. 54E45, 54850, 54E52, 15-XX, 15-01, 15A03, 15A04. 03Exx. 03E10. 03820 Printed on acid-free paper. 02001 Birkhiuser Boston Birkhiiuser rt) L11 1), All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (BirkhBuser Boston, c/o Springer-Verlag New York. Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks. etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act. may accordingly be used freely by anyone. ISBN 0-8176-4174-2 SPIN 10754156 ISBN 3-7643-4174-2 Reformatted from author's files in LATEX2e by TEXniques, Inc., Cambridge, MA Printed and bound by Hamilton Pnnting, Rensselaer, NY Printed in the United States of Amenca 987654321 To the memory of my father The truth, he thought, has never been of any real value to any human being - it is a symbol for mathematicians and philosophers to pursue. In human relations kindness and lies are worth a thousand truths. He involved himself in what he always knew was a vain struggle to retain the lies. Graham Greene Preface "Elements" in the title of this book has its standard meaning, namely, basic principles and elementary theory. The main focus is operator theory, and the topics range from sets to the spectral theorem. Chapter 1 (Set-Theoretic Structures) introduces the reader to ordering, lattices and cardinality. Linear spaces are presented in Chapter 2 (Algebraic Structures). Metric (and topological) spaces in Chapter 3 (Topological Structures). The purpose of Chapter 4 (Banach Spaces) is to put algebra and topology to work together. Continuity plays a central role in the theory of topological spaces, and linear transformation plays a central role in the theory of linear spaces. When algebraic and topological structures are compatibly laid on the same underlying set, leading to the notion of topological vector spaces, then we may consider the concept of continuous linear transformations. By an operator we mean a continuous linear transformation of a normed space into itself. Chapter 5 (Hilbert Spaces) is central. There a geometric structure is properly added to the algebraic and topological structures. The spectral theorem is a cornerstone in the theory of operators on Hilbert spaces. It gives a full statement on the nature and structure of normal