Treatise on the Shift Operator Spectral Function Theory
This book is an elementary introduction to non-classical spectral theory. Mter the basic definitions and a reduction to the study of the functional model the discussion will be centered around the simplest variant of such a model which, formally speaking,
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Editors M. Artin S. S. Chern 1. M. Frohlich E. Heinz H. Hironaka F. Hirzebruch L. Hormander S. MacLane W. Magnus C. C. Moore 1. K Moser M. Nagata W. Schmidt D. S. Scott Va. G. Sinai 1. Tits B. L. van der Waerden M. Waldschmidt S. Watanabe Managing Editors
M. Berger B. Eckmann S. R. S. Varadhan
N. K Nikol'skii
Treatise on the Shift Operator Spectral Function Theory With an Appendix by S. V. Hruscev and V. V. Peller Translated from the Russian by Jaak Peetre
Springer-Verlag Berlin Heidelberg New York Tokyo
NikolaI K. Nikol'skil Leningrad Branch of the Steklov Mathematical Institute Fontanka 27,191011 Leningrad, USSR Jaak Peetre Lund Institute of Technology, Department of Mathematics Box 118, S-22100 Lund, Sweden
Title of the Russian original edition: Lektsii ob operatore sdviga Publisher Nauka, Moscow 1980 This volume is part of the Springer Series in Soviet Mathematics Advisers: L.D. Faddeev (Leningrad), R. V. Gamkrelidze (Moscow)
Mathematics Subject Classification (1980): 47A45, 47 A20, 47A15, 47A60, 47A70, 47B35, 47B40, 47C05, 46J15, 30B30, 30B50, 30D50, 30D55, 30E05, 46J15, 41A20, 41A50, 60G25. ISBN-13: 978-3-642-70153-5 e-ISBN-13: 978-3-642-70151-1 DOl: 10.1007/978-3-642-70151-1
Library of Congress Cataloging in Publication Data Nikol'skil, N. K. (NikolaI Kapitonovich) Treatise on the shift operator. (Grundlehren der mathematischen Wissenschaften ; 273) Translation of: Lektsii ob operatore sdviga. Bibliography: p. Includes indexes. I. Shift operators (Operator theory) 2. Invariant subspaces. 3. Spectral theory (Mathematics) I. Title. II. Series: Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Beriicksichtigung der Anwendungsgebiete ; Bd. 273. QA329.2.N5513 1985 515.7'24 84-26869 ISBN-13: 978-3-642-70153-5 (U.S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-nse of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort," Munich. © by Springer-Verlag Berlin Heidelberg 1986 Softcover reprint of the hardcover 1st edition 1986 Typesetting: Asco Trade Typesetting Ltd., Hong Kong 214113140-543210
Preface
This book is an elementary introduction to non-classical spectral theory. Mter the basic definitions and a reduction to the study of the functional model the discussion will be centered around the simplest variant of such a model which, formally speaking, comprises only the class of contraction operators with a onedimensional rank of non-unitarity (rank(I - T*T) = rank(I - TT*) = 1). The main emphasis is on the technical side of the subject, the book being mostly devoted to a development of the analytical machinery of spectral theory rather than to this discipline itself. The functional model of Sz.-Nagy and Foia§ reduces the study of general operators to an inv
