Spectral Properties of Noncommuting Operators
Forming functions of operators is a basic task of many areas of linear analysis and quantum physics. Weyl’s functional calculus, initially applied to the position and momentum operators of quantum mechanics, also makes sense for finite systems of selfadjo
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Brian Jefferies
Spectral Properties of Noncommuting Operators
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Author Brian Jefferies School of Mathematics University of New South Wales Sydney, NSW, 2052 Australia e-mail: [email protected] http://www.maths.unsw.edu.au/˜brianj
Library of Congress Control Number: 2004104471
Mathematics Subject Classification (2000): 47A13, 47A60, 30G35, 42B20 ISSN 0075-8434 ISBN 3-540-21923-4 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted 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 is a part of Springer Science + Business Media http://www.springeronline.com c Springer-Verlag Berlin Heidelberg 2004 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors SPIN: 11001249
41/3142/du - 543210 - Printed on acid-free paper
Preface
The work described in these notes has had a long gestation. It grew out of my sojourn at Macquarie University, Sydney, 1986-87 and 1989-90, during which time Alan McIntosh was applying Clifford analysis techniques to the study of singular integral operators and irregular boundary value problems. His research group provided a stimulating and convivial environment over the years. I would like to thank my collaborators in this enterprise: Jerry Johnson, Alan McIntosh, Susumu Okada, James Picton-Warlow, Werner Ricker, Frank Sommen and Bernd Straub. The work was supported by two large grants from the Australian Research Council. Sydney, March 2004
Brian Jefferies
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
Weyl Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Operators of Paley-Wiener Type s . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Joint Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13 13 16 21
3
Clifford Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Clifford Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 B
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