Harmonic analysis of operators on Hilbert space

Universitext

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Béla Sz.-Nagy • Ciprian Foias Hari Bercovici • László Kérchy

Harmonic Analysis of Operators on Hilbert Space Revised and Enlarged Edition

Béla Sz.-Nagy (Deceased)

Ciprian Foias Mathematics Department Texas A & M University College Station, TX 77843-3368 USA [email protected]

Hari Bercovici Mathematics Department Indiana University Bloomington, IN 47405 USA [email protected]

László Kérchy Bolyai Institute Szeged University H-6720 Szeged Hungary [email protected]

Editorial Board: Sheldon Axler, San Francisco State University Vincenzo Capasso, Università degli Studi di Milano Carles Casacuberta, Universitat de Barcelona Angus MacIntyre, Queen Mary, University of London Kenneth Ribet, University of California, Berkeley Claude Sabbah, CNRS, École Polytechnique Endre Süli, University of Oxford Wojbor Woyczyński, Case Western Reserve University

ISBN 978-1-4419-6093-1 e-ISBN 978-1-4419-6094-8 DOI 10.1007/978-1-4419-6094-8 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010934634 Mathematics Subject Classification (2010): 47A45  Springer Science+Business Media, LLC 1970, 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, 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 in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Foreword

Sz.-Nagy and Foias had been planning for several years to issue an updated edition of their book Harmonic Analysis of Operators on Hilbert Space (North-Holland and Akad´emiai Kiad´o, Amsterdam–Budapest, 1970). This plan was not realized due to Sz.-Nagy’s death in 1998. Sz.-Nagy’s idea was to include all developments related to dilation theory and commutant lifting. Because there are several other volumes dedicated to some of these developments, we have decided to include in this volume only those subjects that are organically related to the original contents of the book. Thus, the study of C1· -contractions and their invariant subspaces in Chap. IX has its origins in Sec. VII.5, while the theory presented in Chap. X completes the study started in Secs. III.4 and IX.4 of the English edition. The material in the English edition has been reorganized to some extent. The material in the original Chaps. I–VIII was mostly preserved, but the results in the original Chap. IX were dispersed throughout the book. We have added to several chapters a section titled Further results, where we discuss some developments related to the material of the corresponding chapter. The selection of topics was dictated by the authors’ knowledge, and by space limitations. Many significant results are certainly omitted, and only some of these are listed in the bibliography. We apologize to those authors whose work did not receive proper mention. Part of the work on this volume was performed during a semester visit by L. K´erchy to Texas A&M University. He wishes to express his gratitude to the Mathematics Department for its hospitality, and to acknowledge additional support from Hungarian research grant OTKA no. K75488. A first version of Chapters I–VIII was expertly typeset by Mrs. Robin Campbell. The authors extend their gratitude to her, as well as the Mathematics Department at Texas A&M University, for their support throughout this project. Jen˝o Heged˝us kindly translated the foreword to the Russian edition. B´ela Sz˝okefalvi-Nagy served as a mentor to all three authors. He influenced us through the clarity of his mathematical insight, and through his insistence that published results should answer the highest standards of originality, beauty, and exposition. We dedicate this edition to his memory. College Station, Bloomington, and Szeged, June, 2009 C. Foias, H. Bercovici, and L. K´erchy v

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Foreword to the French Edition In the theory of operators on Hilbert space, definitive results have long been known for self-adjoint, unitary, and normal operators—special types of operators, but types which are especially important in different branches of mathematics and theoretical physics. The theory of nonnormal operators, although also initiated a long time ago, using different methods, has not yet attained any such definitive form. The recent rapid progress in this field was stimulated largely by work of mathematicians in the USSR (M. G. Kre˘ın, M. S. Liv˘sic, M. S. Brodski˘ı, etc.) and in the United States (N. Wiener, H. Helson, D. Lowdenslager, P. Masani, etc.). The central concern of the first group was with characteristic functions of operators and the triangular models of operators obtained from them whereas the work of the second group was inspired primarily by prediction theory for stationary stochastic processes. But there is also a third research direction which started from the theorem on unitary dilations of contractions on Hilbert space (Sz.-Nagy, 1953) and was pursued by the authors of the present monograph and others (M. Schreiber, I. Halperin, H. Langer, W. Mlak, etc.). This last research direction has led, for instance, to an effective functional calculus for Hilbert space contractions. It also unifies, in a certain sense, the other two research directions. Thus the characteristic function of a contraction T appears in this study in an altogether natural way, namely by the “harmonic analysis” (or “Fourier analysis”) of the unitary dilation of T , and this in turn was inspired by prediction theory. The purpose of the present monograph is to give a detailed exposition of the information about a contraction that can be obtained from consideration of its unitary dilation. Chapter I develops the fundamentals of the theory of isometric and unitary dilations, deriving these by several different methods. Most important are the dilations of semigroups with one generator, either discrete ({T n }, n = 0, 1, . . .) or continuous ({T (s)}, 0 ≤ s < ∞). These are used throughout what follows. We also treat dilations of discrete commutative semigroups with several generators; here there are some beautiful and definitive results, but also some difficult unsolved problems. These results (Secs. 6 and 9) are not essential for the reader of the rest of the book. In Chap. II we establish some geometric and spectral properties of the unitary dilation of a contraction T (or equivalently, of the discrete contraction semigroup {T n }). Contractions are classified in terms of the asymptotic behavior of the powers of T and its adjoint T ∗ . The important notions of quasi-affinity and quasi-similarity are introduced. In Sec. 5 we prove the existence of an abundance of invariant subspaces for certain types of operators, a subject to which we return, with more powerful methods, in Chap. VII. In Chaps. III and IV we develop a functional calculus for contractions T , based on applying spectral theory to the unitary dilation of T . The relevant functions are analytic on the unit disc, in particular the class of bounded analytic functions. A. Beurling’s arithmetic of inner functions plays an essential role in connection with the “minimal functions” of contractions belonging to what we call the class C0 . Outer functions also play a key part in this calculus, especially in extending it to certain

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classes of analytic functions unbounded on the unit disc. Important applications are to continuous semigroups of contractions (considered as functions of their “cogenerators”), and to functions of accretive and dissipative operators, bounded or not (studied by use of their Cayley transforms). We define and analyze fractional powers of accretive operators, providing an illustration of the methods in a special case that has importance in its own right. Chapter V, which is independent of the preceding chapters, sets forth the ideas and general theorems of the theory of operator-valued analytic functions. This material (except for Secs. 5 and 8) is used throughout the rest of the book. In particular, we establish the existence and properties of factorizations of these functions. Fundamental in this whole development are two lemmas (Sec. 3) on Fourier representations of Hilbert spaces and certain operators on them, with respect to bilateral or unilateral shifts on the spaces. The characteristic function of a contraction T makes its appearance in Chap. VI, as the operator-valued analytic function corresponding to a certain orthogonal projection in the space of the unitary dilation of T , when this space is given its Fourier representation according to the lemmas of Chap. V. This yields at once a functional model for T . The functional model affords a tool for analyzing the structure of contractions and the relations among spectrum, minimal function, and characteristic function. In Chap. VII we establish a one-to-one correspondence between the invariant subspaces of a contraction T and certain factorizations, called the “regular” factorizations, of the characteristic function of T . This correspondence allows us to demonstrate the existence and spectral properties of invariant subspaces for certain types of contractions (class C11 ), thereby strengthening the results obtained by a more elementary method in Chap. II (Sec. 4). Chapter VIII deals with contractions T that are “weak”, that is, such that the spectrum of T is not the whole unit disc and I − T ∗ T has finite trace. For these we find a variety of invariant subspaces that furnish a spectral decomposition, in much the same sense as in the theory of normal operators. Chapter IX contains various further applications of the methods in the book: a criterion for a contraction to be similar to a unitary operator; relations of quasisimilarity for unicellular contractions; criteria for an operator to be unicellular; and finally, extension of these results, by use of a Cayley transformation, to accretive and dissipative operators and to continuous contraction semi-groups. The Notes at the end of each chapter mention additional results, sketch the history of the subject, and give references to the literature. The chapters are divided into sections and the sections into subsections. Results are designated as theorems, propositions, lemmas, and corollaries, and these are numbered separately within each section, as are the subsections and the formulas. The form of citations is the following: the second section in a chapter is called Sec. 2. Within that section, the third subsection is denoted by Sec. 2.3; the third formula by (2.3); the third theorem (or proposition, etc.) by 2.3. In references to other chapters, the appropriate roman numeral is prefixed; thus in referring to Chap. I we would write Sec. I.2.3, or (I.2.3), or Theorem I.2.3.

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We have presupposed familiarity with the elements of the theory of Hilbert space (in particular with the spectral theory for unitary, self-adjoint, and normal operators). Indeed this monograph may be regarded as a sequel to the book Lec¸ons d’analyse fonctionnelle1 by F. Riesz and B. Sz.-Nagy and to the appendix added to it in 1955 by Sz.-Nagy. An additional prerequisite is familiarity with the fundamental facts about the Hardy classes of analytic functions on the unit disc or a half-plane; these may be found in Hoffman’s book [1]. We should mention also that Chaps. V and VI of our book have points of contact with the recent book by Helson [1]; but the two books overlap only slightly in the material covered. Our thanks are due to our colleague Istv´an Kov´acs for his remarks offered in the course of reading the manuscript, and to the Publishing House of the Hungarian Academy of Sciences, and the Szeged Printing Shop for the care they showed in the technical preparation of this book. Szeged and Bucharest, October 1966 Sz.-N.—F.

Foreword to the English Edition Since this book was written in French three years ago, further progress has been made in several parts of the theory. We have made use of the opportunity of the translation into English to include some of the new results, and we have revised, improved, and completed many parts of the original. We mention in particular the following changes. It was known (Theorem I.6.4) that every commuting pair of contractions has a (commuting) unitary dilation, but it was an open question whether this holds, without further restrictions, for commuting families of more than two contractions as well. Now we know by an example due to S. Parrott that the answer to this question is negative (Sec. I.6.3). (1) For the interesting subclass of power-bounded operators, consisting of the operators that admit ρ -unitary dilations, it is proved that all of them are similar to contractions (Sec. II.8). (2) A general dilation theorem is proved in Sec. II.2 for the commutants of contractions, and this theorem is applied later to the functional model of contractions of class C00 (Sec. VI.3.8). (3) The functional calculus for contractions is slightly extended so as to include certain meromorphic functions on the unit disc also (Sec. IV.1); this generalization is immediate, and proves to be natural and even necessary in the light of some recent research on the contractions of class C0 (N); these are sketched in part 2 of the Notes to Chap. IX. (4) The important norm relation between the inverse of the characteristic function ΘT (λ ) of a contraction T and the resolvent of T , due to Gohberg and Kre˘ın, is added as Proposition VI.4.2. (5) Factorizations of a simple example of contractive analytic function are studied in Sec. V.4.5, thus providing useful information in a problem raised by Theorem VII.6.2 (see the last part of the Notes to Chap. VII). 1

References are to the English translation and are indicated by [Funct. Anal.].

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There are still other places that underwent smaller or greater changes, and we benefited from a number of remarks made by colleagues, in particular by Ju. L. ˇ Smuljan in Odessa, as well as by R. G. Douglas in Ann Arbor and Chandler Davis in Toronto, who kindly revised parts of the manuscript of the present English edition. Our sincere thanks are due to all of them. Szeged and Bucharest, May 1969 Sz.-N.—F.

Foreword to the Russian Edition The history of this book, whose Russian translation is recommended to the readers’ attention, can be easily traced. In 1953, the famous Hungarian mathematician B. Sz˝okefalvi-Nagy published in the journal Acta Scientiarum Mathematicarum (Szeged) a theorem, now widely known, on the unitary dilation of contractions. This work was soon continued by the author and other researchers. In 1958, the young Romanian mathematician C. Foias¸ joined in the elaboration of the theory of contractions. Since then a series of articles by B. Sz.-Nagy and C. Foias, under the common title On the Contractions of Hilbert Space, has appeared regularly in Acta Szeged. This research has evolved into a well-developed theory, which plays an important role in modern functional analysis. We are glad to mention that this theory has numerous, sometimes unexpected connections with works of Soviet experts on operator theory. To begin with, B. Sz.-Nagy’s original theorem was based on M. A. Na˘ımark’s result about generalized spectral functions. Later results of the authors of this book yielded explicit connections with the prediction theory of stationary processes, as well as with Beurling’s theorem on the invariant subspaces of shifts. At first it seemed that these topics were far from the spectral theory of nonnormal operators developed by Soviet authors, even when they paid special attention to contractions. The years 1963–1964 were very important in the theory of Hilbert space operators. During that time, B. Sz.-Nagy and C. Foias¸ elaborated the functional calculus of contractions, and introduced the basic concept of the minimal function for a certain class of contractive operators. It was very impressive, and in our opinion quite unexpected, when in their work B. Sz.-Nagy and C. Foias¸ arrived naturally at the concept of the characteristic function of a contraction, a concept that arose in the research of M. S. Livsiˇc (in connection with operators close to unitaries). The characteristic function played a fundamental role in the research of many Soviet mathematicians for two decades. The authors of this book obtained an essentially new functional model for arbitrary contractive operators, and in this model the characteristic function appeared in a very explicit form. From this point on, the interaction between the research carried out by B. Sz.-Nagy and C. Foias¸, and that of the Soviet school of operator theory in Hilbert space, became clear. This interaction resulted in the solution of a series of hard and important problems in numerous chapters of the theory (operators similar to unitaries; unicellular contractions and dissipative operators; multiplication theorems for characteristic functions; methods connected with

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minimal functions; and others). For this reason it is not coincidental that the book contains many references to the works of Soviet mathematicians. Another important event of the period around 1963 is connected with the success achieved by P. Lax and R. Phillips in the scattering theory of acoustic waves. These authors proposed an abstract scheme for scattering problems, and this led to a new interpretation of the S-matrix. Thus this concept, originally introduced in the quantum theory of scattering, has acquired a new life in classical mathematical physics. It turned out that the Lax–Phillips scheme is nothing else than a continuous analogue of the situation considered by B. Sz.-Nagy and C. Foias¸ in their study of the special class of C00 -contractions. It became clear that the characteristic function of a contraction can also be regarded as the S-matrix of an appropriately formulated scattering problem. We now witness the creation of a new important branch in the theory of Hilbert space operators. This involves a wide area of research including the theory of characteristic functions of various classes of operators, the calculus of triangular and multiplicative integrals, problems in the similarity theory of linear operators, several chapters of the theory of operators acting on spaces with an indefinite metric, certain aspects of the scattering theory of self-adjoint and non-self-adjoint operators, along with various applications to classical and quantum physics, and to constructive function theory. This research direction can hardly be presented within the framework of a sole monograph. Several books have appeared reflecting different facets of the aforementioned circle of problems. (Cf. for example, L. DE B RANGES [2], M. S. B RODSKI˘I [9], I. C. G OHBERG AND M. G. K RE˘I N [4], [7], P. D. L AX ˇ [4], and H. H ELSON [1].) A prominent AND R. S. P HILLIPS [2], M. S. L IVSI C place is now taken on this list by the monograph of B. Sz.-Nagy and C. Foias¸, summarizing their investigations. We are not sure that the title Harmonic Analysis of Operators on Hilbert Space fully reflects the content and the aims of the book, but it is in perfect harmony with the inner beauty of the theory, with its well-proportioned composition, and with its elegant style. It is worth mentioning that the research topics discussed in the book are supplemented by historical comments and important remarks at the end of each chapter. The book has been translated into Russian in close collaboration with the authors. Thanks to this cooperation, several small inaccuracies have been corrected, and numerous supplements have been inserted, bringing the contents of the present translation close to that of the English edition. We have no doubt that the appearance of the Russian translation of this excellent book will be well received by researchers in functional analysis. M. G. Kre˘ın

Contents

I

Contractions and Their Dilations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Unilateral shifts. Wold decomposition . . . . . . . . . . . . . . . . . . . . . . . . 2 Bilateral shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Contractions. Canonical decomposition . . . . . . . . . . . . . . . . . . . . . . . 4 Isometric and unitary dilations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Matrix construction of the unitary dilation . . . . . . . . . . . . . . . . . . . . 6 Commutative systems of contractions . . . . . . . . . . . . . . . . . . . . . . . . 7 Positive definite functions on a group . . . . . . . . . . . . . . . . . . . . . . . . . 8 Some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Regular unitary dilations of commutative systems . . . . . . . . . . . . . . 10 Another method to construct isometric dilations . . . . . . . . . . . . . . . . 11 Unitary ρ -dilations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

II

Geometrical and Spectral Properties of Dilations . . . . . . . . . . . . . . . . 59 1 Structure of the minimal unitary dilations . . . . . . . . . . . . . . . . . . . . . 59 2 Isometric dilations. Dilation of commutants . . . . . . . . . . . . . . . . . . . 63 3 The residual parts and quasi-similarities . . . . . . . . . . . . . . . . . . . . . . 70 4 A classification of contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5 Invariant subspaces and quasi-similarity . . . . . . . . . . . . . . . . . . . . . . 80 6 Spectral relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7 Spectral multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 8 Similarity of operators in Cρ to contractions . . . . . . . . . . . . . . . . . . . 95 9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 10 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

III

Functional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 1 Hardy classes. Inner and outer functions . . . . . . . . . . . . . . . . . . . . . . 103 2 Functional calculus: The classes H ∞ and HT∞ . . . . . . . . . . . . . . . . . . 112 3 The role of outer functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

1 1 4 6 9 15 19 24 27 31 38 43 49 53

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4 5 6 7 8 9 10 11

Contractions of class C0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Minimal function and spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Minimal function and invariant subspaces . . . . . . . . . . . . . . . . . . . . . 131 Characteristic vectors and unicellularity . . . . . . . . . . . . . . . . . . . . . . 136 One parameter semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Unitary dilation of semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

IV

Extended Functional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 1 Calculation rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 2 Representation of ϕ (T ) as a limit of ϕr (T ) . . . . . . . . . . . . . . . . . . . . 165 3 Functions limited by a sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 4 Accretive and dissipative operators . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5 Fractional powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 7 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

V

Operator-Valued Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 1 The spaces L2 (A) and H 2 (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 2 Inner and outer functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 3 Lemmas on Fourier representations . . . . . . . . . . . . . . . . . . . . . . . . . . 198 4 Factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 5 Nontrivial factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 6 Scalar multiples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 7 Factorization of functions with scalar multiple . . . . . . . . . . . . . . . . . 231 8 Analytic kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 10 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

VI

Functional Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 1 Characteristic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 2 Functional models for a given contraction . . . . . . . . . . . . . . . . . . . . . 247 3 Functional models for analytic functions . . . . . . . . . . . . . . . . . . . . . . 254 4 The characteristic function and the spectrum . . . . . . . . . . . . . . . . . . 264 5 The characteristic and the minimal functions . . . . . . . . . . . . . . . . . . 271 6 Spectral type of the minimal unitary dilation . . . . . . . . . . . . . . . . . . . 277 7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 8 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

VII

Regular Factorizations and Invariant Subspaces . . . . . . . . . . . . . . . . . 289 1 The fundamental theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 2 Some additional propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 3 Regular factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 4 Arithmetic of regular divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 5 Invariant subspaces for contractions of class C11 . . . . . . . . . . . . . . . 320

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Spectral decomposition and scalar multiples . . . . . . . . . . . . . . . . . . . 325 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

VIII

Weak Contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 1 Scalar multiples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 2 Decomposition C0 –C11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 3 Spectral decomposition of weak contractions . . . . . . . . . . . . . . . . . . 342 4 Dissipative operators. Class (Ω0+ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 5 Dissipative operators similar to self-adjoint ones . . . . . . . . . . . . . . . 354 6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

IX

The Structure of C1·· -Contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 1 Unitary and isometric asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 2 The spectra of C1·· -contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 3 Intertwining with unilateral shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 4 Hyperinvariant subspaces of C11 -contractions . . . . . . . . . . . . . . . . . . 387 5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

X

The Structure of Operators of Class C0 . . . . . . . . . . . . . . . . . . . . . . . . . 397 1 Local maximal functions and maximal vectors . . . . . . . . . . . . . . . . . 397 2 Jordan blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 3 Quasi-affine transforms and multiplicity . . . . . . . . . . . . . . . . . . . . . . 404 4 Multiplicity-free operators and splitting . . . . . . . . . . . . . . . . . . . . . . . 406 5 Jordan models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 6 The quasi-equivalence of matrices over H ∞ . . . . . . . . . . . . . . . . . . . 416 7 Scalar multiples and Jordan models . . . . . . . . . . . . . . . . . . . . . . . . . . 424 8 Weak contractions of class C0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 Notation Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

Chapter I

Contractions and Their Dilations 1 Unilateral shifts. Wold decomposition In this book we study linear transformations (or “operators”) from a (real or complex) Hilbert space H into a Hilbert space H′ ; if H = H′ we say that the transformation (or operator) is on H. Note that if T is a bounded linear transformation from H into H′ , then its adjoint T ∗ is the bounded linear transformation from H′ into H, defined by the relation (T h, h′ )H′ = (h, T ∗ h′ )H

(h ∈ H, h′ ∈ H′ );

we have kT k = kT ∗ k. A linear transformation V from H into H′ is said to be isometric, or an isometry, if (V h1 ,V h2 )H′ = (h1 , h2 )H for all h1 , h2 ∈ H, or, equivalently, if

V ∗V = IH

(we denote by I the identity transformation on a Hilbert space, indicating this space by a subscript if necessary). Let V be an isometry on H. If a subspace L of H is mapped by V onto itself, then L reduces T . Indeed, L = V L implies V ∗ L = V ∗V L = L; thus L is invariant for V as well as for V ∗ , and hence it reduces V . The transformation V from H into H′ is said to be unitary if V maps H isometrically onto H′ , that is, if V ∗V = IH and V H = H′ . The first of these relations implies (VV ∗ )V = V (V ∗V ) = V , and hence VV ∗ h′ = h′ for every element h′ of the form h′ = V h (h ∈ H). Because V H = H′ , we have VV ∗ = IH′ . Conversely, this relation evidently implies V H = H′ . We conclude that the unitary transformations from H into H′ are characterized by the relations V ∗V = IH

and VV ∗ = IH′ ,

that is, by the relation V ∗ = V −1 .

B.Sz.-Nagy et al., Harmonic Analysis of Operators on Hilbert Space, Universitext, DOI 10.1007/978-1-4419-6094-8_1, © Springer Science + Business Media, LLC 2010

1

2

C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

Let V be an isometry on H. A subspace L of H is called a wandering space for V if V p L ⊥ V q L for every pair of integers p, q ≥ 0, p 6= q; because V is an isometry it suffices to suppose that V n L ⊥ L for n = 1, 2, . . . . One can then form the orthogonal sum in H M+ (L) = Observe that we have V M+ (L) = and hence

∞ L V n L. 0

∞ L V n L = M+ (L) ⊖ L, 1 1

L = M+ (L) ⊖ V M+ (L).

(1.1)

An isometry V on H is called a unilateral shift if there exists in H a subspace L, which is wandering for V and such that M+ (L) = H. This subspace L, called generating for V , is uniquely determined by V : indeed by (1.1) we have L = H⊖V H. The dimension of L is called the multiplicity of the unilateral shift V . A unilateral shift V is determined up to unitary equivalence by its multiplicity. Indeed, let V and V ′ be unilateral shifts on H and on H′ , respectively, such that dim L = dim L′ . Then L′ can be transformed onto L by some unitary map ϕ ; this generates a unitary transformation Φ from H′ onto H:   ∞ ∞ ∞ ′n n ′ 2 Φ ∑V ln = ∑V (ϕ ln ) ln ∈ L , ∑kln k < ∞ , 0

0

0

and we have Φ V ′ = V Φ which implies V ′ = Φ −1V Φ . For a unilateral shift V on H = M+ (L) we have V ∗V n l = V ∗VV n−1 l = V n−1 l (l ∈ L; n ≥ 1) and V ∗ l = 0 (l ∈ L), because (V ∗ l, h) = (l,V h) = 0 (l ∈ L, h ∈ H) owing to the relation L = H ⊖ VH ⊥ V H. Hence for   ∞ ∞ n 2 2 h = ∑V ln ln ∈ L, ∑kln k = khk (1.2) 0

0

we have

∞

∞

0

1

∞

∞

1

0

V h = ∑V n+1 ln = ∑V n ln−1 and

V ∗ h = ∑V n−1 ln = ∑V n ln+1 . 1

(1.3)

(1.3*)

For a subspace B of a Hilbert space A, we denote by A ⊖ B the orthogonal complement of B in A.

1. U NILATERAL SHIFTS . W OLD DECOMPOSITION

3

n Iterating, (1.3*) yields V ∗k h = ∑∞ 0 V ln+k (k = 1, 2, . . .), and hence ∞

∞

∞

n=0

n=0

n=k

kV ∗k hk2 = ∑ kV n ln+k k2 = ∑ kln+k k2 = ∑ kln k2 → 0 as k → ∞. Thus, for a unilateral shift V (k → ∞).

V ∗k → O

(1.4)

The importance of unilateral shifts is shown by the following Theorem 1.1 (Wold decomposition). Let V be an arbitrary isometry on the space H. Then H decomposes into an orthogonal sum H = H0 ⊕ H1 such that H0 and H1 reduce V , the part of V on H0 is unitary and the part of V on H1 is a unilateral shift. This decomposition is uniquely determined; indeed we have H0 =

∞ T

n=0

V nH

and H1 = M+ (H) where

L = H ⊖ V H.

(1.5)

The space H0 or H1 may be absent, that is, equal to {0}. Proof. The space L = H ⊖ V H is wandering for V . Indeed, for n ≥ 1 we have V nL ⊂ V nH ⊂ V H

and V H ⊥ L.

Consider H1 = M+ (L) and H0 = HL ⊖ H1 . Observe that h belongs to H0 if and only if it is orthogonal to all finite sums m−1 V n L (m = 1, 2, . . .). Now we have 0 L ⊕ V L ⊕ · · · ⊕V m−1 L = (H ⊖ V H) ⊕ (VH ⊖ V 2 H) ⊕ · · · ⊕ (V m−1 H ⊖ V m H) = H ⊖ V m H;

thus h ∈ H0 if and only if h ∈ V m H for all m ≥ 0. Hence H0 satisfies the first relation (1.5). Because the subspaces V m H (m = 0, 1, 2, . . .) form a nonincreasing sequence, T∞ n we also have H0 = 1 V H. It follows that V H0 = V

∞ T

n=0

V nH =

∞ T

n=0

V n+1 H =

∞ T

m=1

V m H = H0 ;

thus H0 reduces V and V |H0 is a unitary operator on H0 . Hence H1 also reduces V and the part of V on H1 is evidently a unilateral shift. Thus the subspaces given by (1.5) satisfy our conditions. It remains to prove that if H = H′0 ⊕ H′1 is an arbitrary decomposition satisfying these conditions (i.e. if H′1 = M+ (L′ ), where L′ is wandering with respect to V , and if V H′0 = H′0 ), then H′0 = H0 and H′1 = H1 . This follows readily from the equations L = H ⊖ V H = (H′0 ⊕ H′1 ) ⊖ (V H′0 ⊕ V H′1 ) = (H′0 ⊕ H′1 ) ⊖ (H′0 ⊕ V H′1 ) = H′1 ⊖ V H′1 = L′ .

4

C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

2 Bilateral shifts Let U be a unitary operator on H and let L be a wandering subspace for U. The operator U −1 is also unitary, and hence we have U pL ⊥ U qL for all integers p, q (p 6= q). Thus we can form the two-way orthogonal sum M(L) =

∞ L

−∞

U n L;

it is obvious that M(L) reduces U. Contrary to the case of M+ (L), the orthogonal sum M(L) does not determine L (e.g., we have M(L) = M(UL)). However, the dimension of L is determined uniquely by M(L). This is a corollary of the following proposition: Proposition 2.1. If L′ and L′′ are wandering subspaces for the unitary operator U on H, such that M(L′ ) ⊃ M(L′′ ), (2.1) then If

dim L′

dim L′ ≥ dim L′′ .

(2.2)

is finite, then equality in (2.2) implies equality in (2.1).

Proof. Because dimM(L′ ) = ℵ0 · dim L′ and dim M(L′′ ) = ℵ0 · dim L′′ , (2.1) implies ℵ0 · dim L′ ≥ ℵ0 · dim L′′ . (2.3)

In the case dim L′ ≥ ℵ0 , the left-hand side of (2.3) equals dim L′ , and the right-hand side is ≥ dim L′′ . Thus, in this case, (2.2) holds. It remains to consider the case when dim L′ is a finite number. Choose two orthonormal bases for L′ and L′′ , say {e′n : n ∈ Ω ′ } and observe that

and {e′′m : m ∈ Ω ′′ },

{U k e′n : n ∈ Ω ′ ; k = 0, ±1, . . .}

and

{U k e′′m : m ∈ Ω ′′ ; k = 0, ±1, . . .}

are then orthonormal bases for M(L′ ) and M(L′′ ), respectively. Applying Bessel’s inequality and Parseval’s equality we obtain dim L′ = ∑ke′n k2 ≥ ∑ ∑ |(e′n ,U k e′′m )|2 = ∑ ∑ |(U −k e′n , e′′m )|2 n

=

∑ke′′m k2 m

n m,k

m n,k

′′

= dim L .

2. B ILATERAL SHIFTS

5

Equality holds if and only if e′n is contained in M(L′′ ) for all n ∈ Ω ′ . If this is the case, then L′ ⊂ M(L′′ ), which implies that M(L′ ) ⊂ M(L′′ ) and hence, by (2.1), that M(L′ ) = M(L′′ ). An operator U on the space H is called a bilateral shift if U is unitary and if there exists a subspace L of H, such that L is wandering for U and M(L) = H. Every such subspace L is called a generating subspace, and dim L is called the multiplicity of the bilateral shift U. A bilateral shift is determined by its multiplicity up to unitary equivalence. The proof is analogous to that given for unilateral shifts. Let us note an immediate property of a bilateral shift U, namely that U has no eigenvalue. Indeed, every element of H = M(L) can be written in the form ∞

h = ∑ U n ln , −∞

where ln ∈ L,

and hence

∞

∞

−∞

−∞

and khk2 = ∑ kU n ln k2 = ∑ kln k2 ,

∞

∞

−∞

−∞

Uh = ∑ U n+1 ln = ∑ U n ln−1 . Thus if Uh = λ h, then comparing the components in U n L, we get ln−1 = λ ln for all 2 n. This contradicts the convergence of the series ∑∞ −∞ kln k unless ln = 0 for all n. Hence we have necessarily h = 0. Proposition 2.2. Every unilateral shift V on H can be extended to a bilateral shift U of the same multiplicity, on some space containing H as a subspace. Proof. If we set L = H ⊖V H, then H = are the vectors l = {ln }∞ −∞ ,

where

L∞ n 0 V L. Form the space L whose elements

ln ∈ L

∞

and klk2 = ∑ kln k2 < ∞. −∞

Observe that U{ln } = {ln−1 }

is a bilateral shift on L. One of the generating subspaces for U consists of those vectors {lm } for which ln = 0 if n 6= 0 and l0 is arbitrary (in L); obviously this subspace has the same dimension as L. We embed H in L by identifying the element   ∞ ∞ ∞ h = ∑ V n ln ∈ H ln ∈ L; ∑ kln k2 = ∑ kV n ln k2 = khk2 0

0

0

with the element {ln′ } ∈ L

for which

ln′ = ln (n ≥ 0) and ln′ = 0

(n < 0).

6

C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

This identification is possible because it preserves the linear and metric structure of n+1 l = ∞ V n l H. Moreover, the element V h = ∑∞ ∑1 n n−1 of H will then be identified 0V ′ ′ with the element {ln−1 } = U{ln } of L, which proves that U is an extension of V . This finishes the proof. Observe that in virtue of the above identifications we L n have L = ∞ −∞ U L.

Proposition 2.3. Every isometry V on the space H can be extended to a unitary operator U on some space K containing H as a subspace. Proof. By virtue of the Wold decomposition, we have V = V0 ⊕ V1 , where V0 is unitary and V1 is a unilateral shift. By Proposition 2.2, V1 can be extended to a bilateral shift U1 ; then U = V0 ⊕ U1 is a unitary extension of V .

3 Contractions. Canonical decomposition 1. By a contraction from a Hilbert space H into a Hilbert space H′ we mean a linear transformation from H into H′ such that kT hkH′ ≤ khkH

for all

h ∈ H,

(3.1)

that is, kT k ≤ 1. We always have kT k = kT ∗ k, therefore T ∗ will also be a contraction from H′ into H. Inequality (3.1) implies (T ∗ T h, h) ≤ (h, h) for all h ∈ H, and the analogous inequality for T ∗ implies (T T ∗ h′ , h′ ) ≤ (h′ , h′ ) for all h′ ∈ H′ . Thus, for any contraction T of H into H′ we have T ∗ T ≤ IH and T T ∗ ≤ IH′ , and so one can form the operators DT = (IH − T ∗ T )1/2

and DT ∗ = (IH′ − T T ∗ )1/2 ,

(3.2)

which are self-adjoint (DT on H and DT ∗ on H′ ) and bounded by 0 and 1. We have T D2T = T (IH − T ∗ T ) = T − T T ∗ T = (IH′ − T T ∗ )T = D2T ∗ T and hence it follows by iteration that T (D2T )n = (D2T ∗ )n T Consequently

for n = 0, 1, 2, . . . .

T p(D2T ) = p(D2T ∗ )T

(3.3)

λ n.

Choose a sequence of polynofor every polynomial p(λ ) = a0 + a1 λ + · · · + an mials pm (λ ) that tends to the function λ 1/2 uniformly on the interval 0 ≤ λ ≤ 1. The sequence of operators pm (A) then tends in norm to A1/2 for any self-adjoint operator A bounded by 0 and 1. This is a simple consequence of the spectral representation of A. Applying (3.3) to these polynomials we obtain in the limit (as m → ∞) T DT = DT ∗ T.

(3.4)

3. C ONTRACTIONS . C ANONICAL DECOMPOSITION

7

This relation, and the dual one resulting by taking adjoints, D T T ∗ = T ∗ DT ∗ ,

(3.4*)

is used repeatedly in the sequel. Let us observe that kDT hk2 = (D2T h, h) = (h − T ∗ T h, h) = khk2 − kT hk2 .

(3.5)

Thus the set {h : h ∈ H, kT hk = khk} coincides with the set NDT = {h : h ∈ H, DT h = 0}, NDT is obviously a subspace of H. We call DT and DT ∗ the defect operators, DT = DT H = N⊥ DT

and DT ∗ = DT ∗ H′ = N⊥ DT ∗

the defect spaces, and dT = dim DT

and dT ∗ = dim DT ∗

the defect indices, of the contraction T . Observe that dT = 0 characterizes the isometric operators, and dT = dT ∗ = 0 characterizes the unitary operators. Thus, the defect indices measure, in a sense, the deviation of the contraction T from being unitary. Equations (3.4) and (3.4*) imply T DT ⊂ DT ∗

and T ∗ DT ∗ ⊂ DT .

(3.6)

More precisely, the following relations hold. DT ∗ = T DT ⊕ NT ∗ ,

where

NT ∗ = {h′ : h′ ∈ H′ , T ∗ h′ = 0},

DT = T ∗ DT ∗ ⊕ NT ,

where

NT = {h : h ∈ H, T h = 0}.

(3.7)

and (3.7*)

By reason of symmetry, it suffices to prove (3.7). Let us observe first that for h′ ∈ NT ∗ we have h′ = h′ − T T ∗ h′ = D2T ∗ h′ and hence NT ∗ ⊂ DT ∗ H′ . On the other hand, NT ∗ is orthogonal to T DT , because (T DT h, h′ ) = (DT h, T ∗ h′ ) = 0

for h ∈ H, h′ ∈ NT ∗ .

Thus (3.7) is proved if we show that any element g ∈ DT ∗ , which is orthogonal to T DT , belongs necessarily to NT ∗ . Now, indeed, our hypotheses g ∈ DT ∗ , imply

g ⊥ T DT

T ∗ g ∈ T ∗ DT ∗ ⊂ DT and T ∗ g ⊥ DT .

Thus T ∗ g = 0 and g ∈ NT ∗ .

8

C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

Observe that the restriction T |H ⊖ DT of T to H ⊖ DT is a unitary operator with range H ⊖ DT ∗ and inverse T ∗ |H ⊖ DT ∗ . 2. In the sequel we mainly consider the case H = H′ , that is, contractions on the space H. A simple but useful property of these contractions is expressed by the following result. Proposition 3.1. A contraction T on H and its adjoint T ∗ have the same invariant vectors, that is, T h = h implies T ∗ h = h, and conversely. Proof. If T h = h, then (h, T ∗ h) = (T h, h) = (h, h) = khk2 ; hence it follows kh − T ∗ hk2 = khk2 − 2 Re(h, T ∗ h) + kT ∗ hk2 = khk2 − 2khk2 + kT ∗ hk2 ≤ 0, because kT ∗ hk ≤ khk. This proves that T ∗ h = h. The converse assertion follows by symmetry. 3. Two important types of contractions on a Hilbert space are the unitary operators and the completely nonunitary (c.n.u.) contractions. A contraction T on H is said to be c.n.u. if for no nonzero reducing subspace L for T is T |L a unitary operator. The structure of the unitary operators is well known: for them a spectral theory and an effective functional calculus are available. For these theories, we refer the reader to [Funct. Anal.]. As regards c.n.u. contractions, one of the principal aims of the present book is to develop a theory for them that corresponds in some sense to the spectral theory and to the functional calculus for unitary operators. Our theory is based on a simple theorem concerning “unitary dilations of contractions,” which we formulate and prove in Sec. 4. Let us recall that the bilateral shifts are unitary operators. In contrast, the unilateral shifts are c.n.u. In fact, if the unilateral shift V in H were reduced by some subspace H0 6= {0} to a unitary operator V0 = V |H0 , then we would have kV ∗n hk = khk for all h ∈ H0 . Relation (1.4) implies, however, that V ∗n h → 0 (n → ∞) which is a contradiction for h 6= 0. It is an important fact that every contraction can be decomposed into the orthogonal sum of a unitary operator and a c.n.u. contraction. As a consequence, the study of contractions of general type can be reduced to the study of contractions of these two particular types. Theorem 3.2. To every contraction T on the space H there corresponds a decomposition of H into an orthogonal sum of two subspaces reducing T , say H = H0 ⊕ H1 , such that the part of T on H0 is unitary, and the part of T on H1 is completely nonunitary; H0 or H1 may equal the trivial subspace {0}. This decomposition is uniquely determined. Indeed, H0 consists of those elements h of H for which kT n hk = khk = kT ∗n hk

(n = 1, 2, . . .).

T0 = T |H0 and T1 = T |H1 are called the unitary part and the completely nonunitary part of T , respectively, and T = T0 ⊕ T1 is called the canonical decomposition of T . In particular, for an isometry, the canonical decomposition coincides with the Wold decomposition.

4. I SOMETRIC AND UNITARY DILATIONS

9

Proof. Let us introduce the notation T (n) = T n

(n ≥ 1),

T (n) = T ∗|n|

T (0) = I,

(n ≤ −1).

(3.8)

Because T (n) is a contraction on H for every integer n, the set of vectors h for which kT (n)hk = khk (n fixed) is equal to the subspace NDT (n) formed by the vectors h for which DT (n) h = 0. As a consequence the set H0 = {h : kT (n)hk = khk T

(n = 0, ±1, . . .)}

can be expressed as H0 = ∞ n=−∞ NDT (n) . It follows that H0 is also a subspace of H. ∗ Both T and T transform H0 into itself. Indeed, for h ∈ H0 we have kT n T hk = kT n+1 hk = khk = kT hk (n = 0, 1, . . .),

kT ∗n T hk = kT ∗n−1 T ∗ T hk = kT ∗n−1 hk = khk = kT hk (n = 1, 2, . . .); here we have made use of the fact that, for a contraction T , kT hk = khk implies T ∗ T h = h. Hence T h ∈ H0 . One shows analogously that T ∗ h ∈ H0 . Thus H0 reduces T . If we set T0 = T |H0 , then T0∗ = T ∗ |H0 , and T0∗ T0 = T ∗ T |H0 = IH0 ,

T0 T0∗ = T T ∗ |H0 = IH0 ;

thus T0 is unitary. The subspace H1 = H ⊖ H0 also reduces T , and T1 = T |H1 is c.n.u. Indeed, suppose H2 is a nonzero subspace of H1 , reducing T , and such that T |H2 is unitary. Then for every h ∈ H2 we have kT (n)hk = khk and hence h ∈ H0 . Therefore H2 ⊂ H0 : a contradiction. It remains to prove the uniqueness of the decomposition. Let H = H′0 ⊕ H′1 be an arbitrary decomposition of H with the properties in question. Because T is unitary on H′0 , we have kT (n)hk = khk for all h ∈ H′0 and hence H′0 ⊂ H0 . The spaces H0 and H′0 reduce T , therefore the same is true for H0 ⊖ H′0 , and T |H0 ⊖ H′0 is unitary. Because H0 ⊖ H′0 ⊂ H ⊖ H′0 = H′1 and because T is c.n.u. on H′1 , we have necessarily H0 ⊖ H′0 = {0}; that is, H′0 = H0 . The last assertion of the theorem follows from the uniqueness of the decomposition.

4 Isometric and unitary dilations 1. For two operators, A on the Hilbert space A, and B on the Hilbert space B, we indicate by A = pr B the relationship defined by the following two requirements. (i) A is a subspace of B. (ii) (Aa, a′ ) = (Ba, a′ ) for all a, a′ ∈ A.

10

C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

Condition (ii) is obviously equivalent to the condition (ii′ ) Aa = PA Ba for all a ∈ A, where PA denotes the orthogonal projection from B into A. Here are some immediate properties of this relation. (a) (b) (c) (d)

A ⊂ B (i.e., B is an extension of A) implies A = pr B, A = pr B, B = pr C imply A = pr C, A = pr B implies A∗ = pr B∗ , A = pr B and A′ = pr B′ (A, A′ on A; B, B′ on B) imply cA+c′ A′ = pr(cB+c′ B′ ) for arbitrary scalar coefficients c, c′ , (e) A = pr B, A′ = pr B′ imply A ⊕ A′ = pr(B ⊕ B′), (f) An = pr Bn (An or A, Bn on B, n = 1, 2, . . .) and Bn ⇀ B,

or Bn ⇒ B

Bn → B,

(n → ∞),

imply the convergence of An in the same sense (i.e., weakly, strongly, or in norm) to an operator A, and we have A = pr B. Now we make the following definition. Definition. Let A and B be two operators, A on the space A, and B on the space B. We call B a dilation of A if An = pr Bn

for n = 1, 2, . . . .

Two dilations of A, say B on B and B′ on B′ , are said to be isomorphic if there exists a unitary transformation ϕ from B′ onto B, such that (i) ϕ a = a for all a ∈ A, (ii) B′ = ϕ −1 Bϕ . 2. We can now state our first result on dilations. Theorem 4.1. For every contraction T on the Hilbert space H there exists an isometric dilation V on some Hilbert space K+ (⊃ H), which is moreover minimal in the sense that ∞ W K+ = V n H. (4.1) 0

This minimal isometric dilation of T is determined up to isomorphism; thus one can call it “the minimal isometric dilation” of T . The space H is invariant for V ∗ and we have T P+ = P+V and T ∗ = V ∗ |H, (4.2) where P+ denotes the orthogonal projection from K+ onto H.

4. I SOMETRIC AND UNITARY DILATIONS

11

Proof. Let us form the Hilbert space H+ = whose elements are the vectors h = {h0 , h1 , . . .} with

∞ L 0

H,

∞

khk2 = ∑ khn k2 < ∞.

hn ∈ H,

0

We embed H in H+ as a subspace, by identifying the element h ∈ H with the element {h, 0, . . .} ∈ H+ ; this identification is allowed because it obviously preserves the linear and metric structure of H. Observe that we have then PH {h0 , h1 , . . .} = {h0 , 0, 0, . . .} = h0 . We define on H+ an operator V by V{h0 , h1 , . . .} = {T h0 , DT h0 , h1 , . . .},

where

DT = (I − T ∗ T )1/2 .

From the relation kT hk2 + kDT hk2 = khk2 , which holds for every h ∈ H, we deduce kV{h0 , h1 , . . .}k2 = kT h0 k2 + kDT h0 k2 + kh1k2 + kh2k2 + · · ·

= kh0 k2 + kh1k2 + kh2k2 + · · · = k{h0 , h1 , . . .}k2 ;

because V is obviously linear, it is an isometry on H+ . Moreover, we obtain for n = 1, 2, . . . by induction: Vn {h0 , h1 , . . .} = {T n h0 , DT T n−1 h0 , DT T n−2 h0 , . . . , DT h0 , h1 , h2 , . . .}, and hence it follows for h ∈ H that PH Vn h = PH Vn {h, 0, 0, . . .} = PH {T n h, DT T n−1 h, . . . , DT h, 0, . . .} = {T n h, 0, 0, . . .} = T n h. This proves that V is an isometric dilation of T . In general, our V is not minimal. However, it is easy to show that every isometric dilation V0 of T , say on a space K0 (⊃ H), contains a minimal isometric dilation V in the sense that V is the restriction of V0 to some subspace K+ of K0 , invariant for V0 . In fact, one has to take K+ =

∞ W V0n H. 0

Let V be a minimal isometric dilation of T , on the space K+ . From the dilation property it follows for h ∈ H and n = 0, 1, . . ., T P+ ·V n h = T T n h = T n+1 h = P+V n+1 h = P+V ·V n h.

12

C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

On account of (4.1) this implies the first relation (4.2) which we now show implies the second. For h ∈ H and k ∈ K+ we have (T ∗ h, k) = (T ∗ h, P+ k) = (h, T P+ k) = (h, P+V k) = (h,V k) = (V ∗ h, k), and thus T ∗ h = V ∗ h. Hence in particular V ∗ H ⊂ H. Therefore it remains only to show that all minimal isometric dilations of T are isomorphic. To this end let us start by observing that for any isometric dilation V of T and for h, h′ ∈ H we have ( (V n−m h, h′ ) = (T n−m h, h′ ) if n ≥ m ≥ 0, n m ′ (V h,V h ) = (4.3) (h,V m−n h′ ) = (h, T m−n h′ ) if m ≥ n ≥ 0; thus (V n h,V m h′ ) does not depend upon the particular choice of V . Consequently, the scalar product of two finite sums of the form N

N′

n=0

m=0

∑ V n hn ,

∑ V m h′m

(hn , h′m ∈ H)

depends only upon the vectors hn , h′m , and not upon the particular choice of V . Thus, if V1 and V2 are two isometric dilations of T on the spaces K1 and K2 , respectively, then, setting   N N n ϕ ∑ V2 hn = ∑ V1n hn (N = 0, 1, . . . ; hn ∈ H), (4.4) 0

0

we define an isometric (and consequently, a well-defined and linear) transformation from the linear manifold L2 of the elements of the form ∑N0 V2n hn , onto the linear manifold L1 of the elements of the form ∑N0 V1n hn . If the dilations V1 and V2 are minimal, that is, if Ki =

∞ W Vin H 0

(i = 1, 2),

then Ki = Li (i = 1, 2) and consequently ϕ can be extended by continuity to a unitary transformation from K2 onto K1 . We have ϕ h = ϕ (V20 h) = V10 h = h for h ∈ H. Furthermore, we have ϕ (V2 k) = V1 (ϕ k) first for k ∈ L2 and hence by continuity for all k ∈ K2 . Thus V2 = ϕ −1V1 ϕ , and this proves that the dilations V1 and V2 of T are isomorphic. This finishes the proof of Theorem 4.1. As a supplement to Theorem 4.1, we mention that if V is any isometric operator on some Hilbert space H+ containing H satisfying the condition PH V = T PH

4. I SOMETRIC AND UNITARY DILATIONS

13

then the subspace K+ =

∞ W

n=0

Vn H

is reducing V. To prove this we first notice that the above condition is equivalent to V∗ |H = T ∗ and that the set K0+ of all the finite sums N

k = ∑ Vn h n

(N = 0, 1, . . . ; hn ∈ H)

n=0

is dense in K+ . For the above k, we obviously have Vk ∈ K0+ as well as N

V∗ k = T ∗ h0 + ∑ Vn−1 hn ∈ K0+ , n=1

that is, VK0+ , V∗ K0+ ⊂ K0+ , and subsequently K+ is reducing V.

3. We are now able to prove the following theorem, of fundamental importance for our investigations.

Theorem 4.2. For every contraction T on the Hilbert space H there exists a unitary dilation U on a space K containing H as a subspace, which is minimal, that is, such that ∞ W K = U n H. (4.5) −∞

This minimal unitary dilation is determined up to isomorphism, and thus can be called “the minimal unitary dilation” of T .

Proof. Let us take an arbitrary isometric dilation of T and extend it to a unitary operator: this is possible by virtue of Proposition 2.3. Thus we obtain a unitary dilation of T , which is not necessarily minimal. However, every unitary dilation U0 of T contains a minimal one; we have only to take the restriction of U0 to the subspace K=

∞ W

−∞

U0n H,

which reduces U0 . Let us observe next that if U is a unitary dilation of T , then we have for h, h′ in H, ( (T n−m h, h′ ) if n ≥ m, n m ′ (U h,U h ) = (4.6) (h, T m−n h′ ) if m ≥ n; in this case the integers m, n can be positive, negative, or 0. On the basis of (4.6), one proves that any two minimal unitary dilations of T are isomorphic just as it was

14

C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

done for the isometric dilations, the only difference being that now one has to admit integer exponents m, n of any sign. This finishes the proof of the theorem. Let us state some obvious facts. If we denote by UT the minimal unitary dilation of T then we have UT ∗ = (UT )∗ ,

(4.7) ′

′′

for T = T ⊕ T ,

UT = UT ′ ⊕ UT ′′

(4.8)

and UT = T

for unitary T.

(4.9)

Consequently, if T = T0 ⊕T1 is the canonical decomposition of T into the orthogonal sum of its unitary part T0 and its c.n.u. part T1 , then UT = T0 ⊕ UT1 . 4. If T is a contraction on H then so is Ta = (T − aI)(I − aT ¯ )−1

(|a| < 1).

(4.10)

In fact, for any h ∈ H we have, setting g = (I − aT ¯ )−1 h, khk2 − kTahk2 = k(I − aT ¯ )gk2 − k(T − aI)gk2 = (1 − |a|2)(kgk2 − kT gk2 ) ≥ 0. This shows, moreover, that if T is an isometry then so is Ta . We obviously have (Ta )∗ = (T ∗ )a¯ , therefore we also obtain that if T ∗ is an isometry then so is (Ta )∗ . Consequently, if T is unitary then so is Ta . Proposition 4.3. Let T be a contraction on H, and let |a| < 1. If V is an isometric dilation of T then Va is an isometric dilation of Ta ; moreover, if V is minimal, then Va is also minimal. Similarly, if U is a unitary dilation of T then Ua is a unitary dilation of Ta ; moreover, if U is minimal, then Ua is also minimal. Proof. Consider the Taylor expansions   ∞ λ −a n = ∑ cν (a; n)λ ν 1 − a¯λ ν =0

(n = 0, 1, 2, . . .);

because their radius of convergence is larger than 1, we have ∑ν |cν (a; n)| < ∞. This implies that the operator series ∞

∑ cν (a; n)T ν

ν =0

converges in norm and its sum is equal to Tan (n = 0, 1, 2, . . .). (A functional calculus ∞ ν ν ∑∞ 0 cν λ → ∑0 cν T is studied for the case ∑ν |cν | < ∞, and for analytic functions

5. M ATRIX CONSTRUCTION OF THE UNITARY DILATION

15

of still more general type, in Chaps. III and IV.) Because we also have ∞

∑ cν (a; n)V ν = Van ,

(4.11)

ν =0

we conclude that ∞

∞

∞

ν =0

ν =0

ν =0

Tan = ∑ cν (a; n)T ν = ∑ cν (a; n)pr V ν = pr ∑ cν (a; n)V ν = pr Van for n ≥ 0, and hence Va is an isometric dilation of Ta . Relation (4.11) implies that ∞ W

n=0

Van H ⊂

∞ W

n=0

V n H.

(4.12)

Now, (Ta )−a = T and (Va )−a = V , and substituting V → Va , a → −a in (4.12) it follows that ∞ ∞ ∞ W W W V nH = [(Va )−a ]n H ⊂ Van H, n=0

n=0

and thus we have

∞ W

n=0

n=0

Van H =

∞ W

n=0

V n H.

(4.13)

These relations are valid in particular for a unitary dilation U of T , and for the unitary dilation U ∗ = U −1 of T ∗ . Because Ua∗ = (U ∗ )a¯ , (4.13) yields, when V and a are replaced by U ∗ and a, ¯ the relation ∞ W

n=0

Ua∗n H =

n=0

and consequently

Uan H =

∞ W

n=0

[(U ∗ )a¯ ]n H =

n=0

Hence we have in this case ∞ W

∞ W

U n H,

∞ W

−∞

∞ W

n=0

Uan H =

∞ W

∞ W

n=0

Ua−n H =

−∞

U ∗n H.

∞ W

n=0

U −n H

U n H.

(4.14)

(4.15)

Clearly, (4.13) and (4.15) imply our assertions concerning minimality of the corresponding dilations.

5 Matrix construction of the unitary dilation 1. It is possible to construct a unitary dilation of the contraction T on H by the L following matrix method. Consider the Hilbert space H = ∞ −∞ H the elements of which are the vectors h = {hi }∞ −∞

with

∞

hi ∈ H and khk2 = ∑ khi k2 < ∞. −∞

16

C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

We embed H in H by identifying the element h ∈ H with the vector h = {hi } for which h0 = h and hi = 0 (i 6= 0); H becomes a subspace of H, and the orthogonal projection from H into H is given by (5.1)

PH {hi } = h0 .

Every (bounded, linear) operator S on H can be represented by the matrix (Si j ) (−∞ < i, j < ∞), whose entries Si j are the operators on H satisfying (Sh)i = ∑∞j=−∞ Si j h j ; to the sums, products, and adjoints of operators S there correspond the sums, products, and adjoints of the matrices, where, by definition, we set (Si j )∗ = (S∗ji ). It is important to note that (5.1) implies PH Sh = PH {Si0 h} = S00 h

for h ∈ H.

Consider now the matrix (Ui j ) with entries U00 = T,

U01 = DT ∗ ,

U−1,1 = −T ∗ ,

U−1,0 = DT ,

Ui,i+1 = IH ,

for i 6= 0, 1, and Ui j = O for all other i, j; that is, the matrix              

..



. I I DT −T ∗ T DT ∗

I I ..

.

      ,      

(5.2)

where (in order to indicate the indices of rows and columns) we have drawn a square around the central entry U00 . All the entries not indicated are O, with the exception of the entries just above the diagonal, which are all equal to I = IH . Setting h′i = ∑ Ui j h j j

(i = 0, ±1, ±2, . . .),

that is, h′−1 = DT h0 − T ∗ h1 ,

h′0 = T h0 + DT ∗ h1 ,

h′i = hi+1

(i 6= 0, −1),

(5.3)

one shows by elementary calculations based on the relations (3.4), (3.4*), and (3.5), that kDT h0 − T ∗ h1 k2 + kT h0 + DT ∗ h1 k2 = kh0 k2 + kh1k2

5. M ATRIX CONSTRUCTION OF THE UNITARY DILATION

17

∞ ′ 2 2 and, consequently, ∑∞ −∞ khi k = ∑−∞ khi k . Thus the matrix (Ui j ) defines an isometry U in H. Moreover, U is unitary, because the system of equations (5.3) has for every given vector {h′i } ∈ H the solution {hi } ∈ H with

h0 = DT h′−1 + T ∗ h′0 ,

h1 = −T h′−1 + DT ∗ h′0 ,

hi = h′i−1

(i 6= 0, 1);

(5.4)

this can be proved easily by means of relations (3.4) and (3.4*). The matrix (Ui j ) is triangular; indeed it is superdiagonal (i.e., Ui j = O for i > j). Now, the product (Ci j ) of two superdiagonal matrices, say (Ai j ) and (Bi j ), is also superdiagonal, and we have Cii = Aii Bii . Hence we conclude that the central entry in the matrix of Un (n ≥ 1) is equal to T n , that is, T n = pr Un (n ≥ 1) : U is a dilation of T . 2. This unitary dilation need not, however, be minimal. In order to obtain a minimal unitary dilation, we modify the above construction as follows. Instead of the L space H = ∞ −∞ H we consider its subspace K consisting of the vectors {hn } ∈ H for which hn ∈ DT (n ≤ −1), h0 ∈ H, hn ∈ DT ∗ (n ≥ 1);

obviously H ⊂ K ⊂ H. The subspace K is invariant for U. By virtue of the formulas (5.3) this is established if we prove that h0 ∈ H

and h1 ∈ DT ∗

imply DT h0 − T ∗ h1 ∈ DT .

But this follows from the relations DT H ⊂ DT and T ∗ DT ∗ ⊂ DT (cf. (3.6)). Secondly, U maps K onto K. By virtue of the formulas (5.4) one just has to show that h′0 ∈ H and h′−1 ∈ DT imply − T h′−1 + DT ∗ h′0 ∈ DT ∗ .

But this follows from the relations DT ∗ H ⊂ DT ∗ and T DT ⊂ DT ∗ (cf. (3.6)). It follows that U0 = U|K is a unitary dilation of T . Moreover, it is a minimal one. −n To prove this, first we calculate Un0 h and U∗n 0 h = U0 h for h ∈ H and n = 1, 2, . . . . ∗ From formulas (5.3) (for U0 ) and (5.4) (for U0 ) it follows by iteration that Un0 h = {. . . , 0, DT h, DT T h, . . . , DT T n−1 h, T n h , 0, . . .}

and

∗n ∗n−1 ∗ U−n h, . . . , DT ∗ T ∗ h, DT ∗ h, 0, . . .} 0 h = {. . . , 0, T h , DT T

(n = 1, 2, . . .), where the components are arranged in order of increasing subscripts, the central component (i.e., the one with subscript 0) being indicated by a square. From these formulas we deduce −n

⌣

Un0 h − Un−1 0 T h = {. . . , 0, DT h, 0, . . . , 0 , . . .}

18

C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

and n

⌣

−n+1 ∗ T h = {. . . , 0 , . . . , 0, DT ∗ h, 0, . . .} U−n 0 h − U0

for n ≥ 1. It follows that the (closed linear) span of the subspaces Un0 H (−∞ < n < ∞) contains all the vectors {hn } ∈ K whose components are all 0 except the nth one, which is an arbitrary element of DT H, H, or DT ∗ H, according to the sign of n (n < 0, n = 0, or n > 0). These vectors obviously span the whole space K when n varies over all integers, and therefore we have K=

∞ W

−∞

Un0 H,

and thus the unitary dilation U0 of T is a minimal one. As an elementary example, let us observe that if T = aIH with |a| < 1, then the minimal unitary dilation of T is given by the matrix   .. .     I     I     dI − aI ¯ ,    aI dI     I     I   .. . where d = (1 − |a|2)1/2 acting on the space

··· ⊕ H ⊕ H ⊕ H ⊕ H ⊕··· . 3. The unitary dilation U = UT constructed in Subsec. 1 is in general not minimal, but has the advantage that it is defined on a space depending only on the space H and not upon the particular choice of the contraction T on H. The corresponding matrices (UT ;i, j ) are all superdiagonal. It follows that for an arbitrary sequence Ti (i = 1, . . . , r) of (not necessarily different) contractions on H we have (UT1 · · · UTr )00 = (UT1 )00 · · · (UTr )00 = T1 · · · Tr , that is, T1 · · · Tr = pr UT1 · · · UTr , and also

T1n1 · · · Trnr = pr UnT11 · · · UnTrr

(ni ≥ 0).

(5.5)

However, this property of the operators UT is of very limited practical value, because, in general, the relation UT n = (UT )n is not valid, and UT1 and UT2 do not commute even when T1 and T2 do. This raises the problem of considering commuta-

6. C OMMUTATIVE SYSTEMS OF CONTRACTIONS

19

tive systems of contractions and trying to find a corresponding commutative system of unitary operators so that (5.5) holds. We investigate this problem in Sec. 6 and return to it again in Sec. 9.

6 Commutative systems of contractions 1. Let us start with a generalization of the notion of dilation for systems of operators. Let A = {Aω }ω ∈Ω be a commutative system of bounded operators on the space H. A system B = {Bω }ω ∈Ω of bounded operators on a space K is called a dilation of the system A , if (i) H is a subspace of K, (ii) the system B is commutative, and (iii) Anω11 · · · Anωrr = pr Bnω11 · · · Bnωrr

(ni ≥ 0; i = 1, . . . , r)

for every finite set of subscripts ωi ∈ Ω . The dilation B is said to be isometric, unitary, and so on, when it consists of operators Bω of the type in question. Theorems 4.1 and 4.2 raise the question of whether every commutative system of contractions possesses an isometric or unitary dilation. In this section we show that the answer is positive for every system of two commuting contractions, and negative for some commutative systems of more than two contractions. In Sec. 9 we consider commuting systems of more than two contractions, satisfying certain additional conditions, which do admit isometric and unitary dilations. Theorem 6.1. For every commuting pair T = {T1 , T2 } of contractions on a Hilbert space H there exists an isometric dilation. L

Proof. Let us consider the space H+ = ∞ 0 H as in the proof of Theorem 4.1, H being embedded in H+ as a subspace as indicated there. We define on H+ the operators W1 and W2 by Wi {h0 , h1 , h2 , . . .} = {Ti h0 , DTi h0 , 0, h1 , h2 , . . .}

(i = 1, 2);

(6.1)

these operators are isometric because kTi h0 k2 + kDTi h0 k2 = kh0 k2 , but in general they do not commute. Let us form the space G = H ⊕ H ⊕ H ⊕ H. By the natural identification {h0 , h1 , h2 , . . .} = {h0 , {h1 , h2 , h3 , h4 }, {h5, h6 , h7 , h8 }, . . .} we have H+ = H ⊕ G ⊕ G ⊕ · · · .

Let G be a unitary operator on G, determined later, and define an operator G on H+ by G{h0 , h1 , . . .} = {h0 , G{h1 , . . . , h4 }, G{h5 , . . . , h8 }, . . .}. (6.2) Then G is also unitary and its inverse is given by

G−1 {h0 , h1 , . . .} = {h0 , G−1 {h1 , . . . , h4 }, G−1 {h5 , . . . , h8 }, . . .}.

(6.3)

20

C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

Set V1 = GW1

and V2 = W2 G−1 .

(6.4)

These are isometries on H+ . Let us try to find a G such that V1 and V2 commute. First we calculate V1 V2 and V2 V1 . By virtue of (6.1)–(6.4) we have V1 V2 {h0 , h1 , . . .}

= GW1 W2 G−1 {h0 , h1 , . . .}

= GW1 W2 {h0 , G−1 {h1 , . . . , h4 }, G−1 {h5 , . . . , h8 }, . . .}

= GW1 {T2 h0 , DT2 h0 , 0, G−1 {h1 , . . . , h4 }, G−1 {h5 , . . . , h8 }, . . .}

= G{T1 T2 h0 , DT1 T2 h0 , 0, DT2 h0 , 0, G−1 {h1 , . . . , h4 }, G−1 {h5 , . . . , h8 }, . . .} = {T1 T2 h0 , G{DT1 T2 h0 , 0, DT2 h0 , 0}, {h1 , . . . , h4 }, {h5 , . . . , h8 }, . . .} and V2 V1 {h0 , h1 , . . .} = W2 G−1 GW1 {h0 , h1 , . . .} = W2 W1 {h0 , h1 , . . .} = W2 {T1 h0 , DT1 h0 , 0, h1 , h2 , . . .} = {T2 T1 h0 , DT2 T1 h0 , 0, DT1 h0 , 0, h1 , h2 , . . .}.

Because T1 T2 = T2 T1 , V1 will commute with V2 if, and only if, G satisfies G{DT1 T2 h, 0, DT2 h, 0} = {DT2 T1 h, 0, DT1 h, 0}

(6.5)

for every h ∈ H. Now a simple calculation yields kDT1 T2 hk2 + kDT2 hk2 = khk2 − kT1 T2 hk2

= khk2 − kT2 T1 hk2 = kDT2 T1 hk2 + kDT1 hk2,

and hence k{DT1 T2 h, 0, DT2 h, 0}k = k{DT2 T1 h, 0, DT1 h, 0}k for all h ∈ H. This means that (6.5) determines G as an isometric transformation of the linear manifold L1 of the vectors of the form {DT1 T2 h, 0, DT2 h, 0}, onto the linear manifold L2 of the vectors of the form {DT2 T1 h, 0, DT1 h, 0}; G extends by continuity to an isometry from M1 = L1 onto M2 = L2 . It remains to show that G can be extended to an isometry of the whole space G onto itself. This is equivalent to the assertion that the ⊥ subspaces M⊥ 1 = G ⊖ M1 and M2 = G ⊖ M2 have the same dimension. When H and hence also G have finite dimension, this is obvious. When dimH is infinite, we have dim H = dim G ≥ dim M⊥ (i = 1, 2), i ≥ dim H

⊥ because both M⊥ 1 and M2 contain subspaces of the same dimension as H, for example, the subspace formed by the vectors {0, h, 0, 0} (h ∈ H). This proves that ⊥ dim M⊥ 1 = dim M2 .

6. C OMMUTATIVE SYSTEMS OF CONTRACTIONS

21

If the unitary operator G is determined in this way, the operators V1 and V2 will be two commuting isometries on H+ . They satisfy Vi {h0 , h1 , . . .} = {Ti h0 , . . .} (i = 1, 2), and hence Vn11 Vn22 {h0 , h1 , . . .} = {T1n1 T2n2 h0 , . . .}

for n1 , n2 ≥ 0,

and consequently, PH Vn11 Vn22 h = T1n1 T2n2 h for every

h∈H

and n1 , n2 ≥ 0.

Thus {V1 , V2 } is an isometric dilation of {T1 , T2 }. Remark. For an arbitrary isometric dilation {V1,V2 } of {T1 , T2 } on the space K, the subspace W K′ = V1n1 V2n2 H n1 ,n2 ≥0

is invariant for V1 and V2 , and contains H as a subspace; thus the restrictions of V1 and V2 to K′ also form an isometric dilation {V1′ ,V2′ } of {T1 , T2 }, which is, moreover, minimal, that is, such that K′ =

W

n1 ,n2 ≥0

V1′ n1 V2′ n2 H.

(6.6)

However, contrary to the case of a single contraction, one cannot assert that all the minimal isometric dilations are isomorphic. 2. The existence of a unitary dilation follows from the existence of an isometric dilation by virtue of the following result. Proposition 6.2. For every commutative system {Vω }ω ∈Ω of isometric operators on H there exists a commutative system {Uω }ω ∈Ω of unitary operators on a space K containing H as a subspace, such that Uω ⊃ Vω for every ω ∈ Ω . In brief, every commutative system of isometries can be extended to a commutative system of unitary operators. This proposition holds for finite and infinite systems as well. For finite systems one obtains a proof by applying a finite number of times a process that at every stage diminishes the number of the nonunitary operators. (By the way, iteration of this process a transfinite number of times would also yield a proof for infinite systems; however, we are momentarily interested in the finite case only, and in fact, in the case of two isometries. We return to the infinite case in Sec. 9, using another method.) The process in question is founded on the following result. Proposition 6.3. Let {V,Wν (ν ∈ N)} be a commutative system of isometries on a space H. Then there exists a commutative system {V, Wν (ν ∈ N)} of isometries on a space H containing H as a subspace, such that (i) V ⊂ V and Wν ⊂ Wν (ν ∈ N),

22

C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

(ii) V is unitary, and (iii) Wν is unitary (on H) for every ν such that Wν is unitary (on H). Proof. First we extend the isometry V to a unitary operator V on some space H ⊃ H, this being possible by virtue of Proposition 2.3; this extension can be chosen to be minimal in the sense that ∞ W H = Vn H (6.7) −∞

(this condition is obviously fulfilled by the unitary extension constructed in the proof of Proposition 2.3). Let us observe that for every finite sum ∑ Vn hn

(6.8)

(hn ∈ H; n runs over a finite set of integers, of arbitrary sign) and for every ν ∈ N we have

2

∑ VnWν hn = ∑ ∑(VnWν hn , VmWν hm )

n

n m

= ∑ ∑(Vn−mWν hn ,Wν hm ) + ∑∑(Wν hn , Vm−nWν hm ). n<m

n≥m

Because V extends V , because V and Wν commute, and because Wν is an isometry the preceding expression is equal to ∑ ∑(V n−m hn , hm ) + ∑∑(hn ,V m−n hm ) = ∑ ∑(Vn−m hn , hm ) + ∑∑(hn , Vm−n hm ) n≥m

n<m

n≥m

n<m

n m

n

2

n 2 = ∑ ∑(Vn hn , Vm hm ) = V h n = khk .

∑

Consequently, by setting

Wν ∑ Vn hn = ∑ VnWν hn n

n

(6.9)

we obtain an isometric transformation Wν of the linear manifold M of vectors of the form (6.8) into itself. Because M = H (6.7), Wν extends by continuity to an isometry on H. If Wν is unitary on H, that is, Wν H = H, (6.9) implies that Wν M = M and hence Wν H = H. Thus Wν is unitary on H. Let us show finally that the system {V, Wν (ν ∈ N)} is commutative. In fact, one has VWν (Vn h) = V(VnWν h) = Vn+1Wν h = Wν Vn+1 h = Wν V(Vn h), Wν1 Wν2 (Vn h) = Wν1 VnWν2 h = VnWν1 Wν2 h and, by the same reasoning, Wν2 Wν1 (Vn h) = Wν2 VnWν1 h = VnWν2 Wν1 h

6. C OMMUTATIVE SYSTEMS OF CONTRACTIONS

23

for ν , ν1 , ν2 ∈ N, for an arbitrary integer n, and for h ∈ H. By virtue of (6.7) these relations imply VWν = Wν V and Wν1 Wν2 = Wν2 Wν1 ; thus the system is indeed commutative. Starting with Theorem 6.1, and applying Proposition 6.2 for two commuting isometries, we arrive at the following result. Theorem 6.4. For every commuting pair of contractions there exists a unitary dilation. 3. It is a rather striking fact that the above theorem does not hold for more than two commuting contractions. In fact, we construct a system {T1 , T2 , T3 } of commuting contractions for which no system {U1 ,U2 ,U3 } of commuting unitary operators can be found such that Ti = pr Ui (i = 1, 2, 3). (6.10) To this end we choose unitary operators A1 , A2 , A3 on a Hilbert space A, such that −1 A1 A−1 2 A3 6= A3 A2 A1 .

(6.11)

(We choose, e.g., A2 = I, and for A1 and A3 any two noncommuting unitary operators on A.) We consider the space H = A ⊕ A of elements {a1 , a2 } (a1 , a2 ∈ A), and define the operators Ti (i = 1, 2, 3) on H by Ti {a1 , a2 } = {0, Ai a1 }. Clearly kTi k = 1 and Ti T j = O = T j Ti for i, j = 1, 2, 3. Suppose there exist commuting unitary operators U1 ,U2 ,U3 on some Hilbert space K(⊃ H), for which the relations (6.10) hold. Then we have PHUi {a, 0} = Ti {a, 0} = {0, Ai a}

(a ∈ A).

(6.12)

Because kUi {a, 0}k = k{a, 0}k = kak = kAi ak = k{0, Ai a}k by virtue of the isometry property of Ui and Ai , we infer from (6.12) that Ui {a, 0} = {0, Ai a}

(a ∈ A).

Hence we deduce −1 U j−1Ui {a, 0} = U j−1 {0, Ai a} = U j−1 {0, A j (A−1 j Ai )a} = {A j Ai a, 0}

and

−1 UkU j−1Ui {a, 0} = Uk {A−1 j Ai a, 0} = {0, Ak A j Ai a}.

−1 Because the Us commute, we conclude that Ak A−1 j Ai = Ai A j Ak (i, j, k = 1, 2, 3). This contradicts the assumption (6.11) and thus proves that no commuting Us exist that satisfy (6.10).

24

C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

7 Positive definite functions on a group The constructions of the unitary dilation of a contraction, given in Secs. 4 and 5, although rather simple, have the disadvantage of being overly tied to the particular problem in question. By contrast, the method which we follow in Sec. 8 is based on a general theorem on positive definite operator-valued functions on a group. Definitions. Let G be a group. (i) A function T (s) on G, whose values are bounded operators on a Hilbert space H, is said to be positive definite if T (s−1 ) = T (s)∗ for every s ∈ G, and ∑ ∑ (T (t −1 s)h(s), h(t)) ≥ 0

s∈G t∈G

(7.1)

for every finitely nonzero function h(s) from G to H, that is, which has values different from 0 on a finite subset of G only. (If the space H is complex, the condition T (s−1 ) = T (s)∗ (s ∈ G) is a consequence of (7.1). The proof is elementary and we omit it.) (ii) By a unitary representation of the group G we mean a function U(s) on G, whose values are unitary operators on a Hilbert space K and that satisfies the conditions U(e) = I (e being the identity element of G) and U(s)U(t) = U(st) for s,t ∈ G. There is a connection between the two notions just defined, which we now establish.

Theorem 7.1. (a) If U (s) is a unitary representation of the group G in the space K, and if H is a subspace of K, then T (s) = PHU(s)|H is a positive definite function on G such that T (e) = IH . If, moreover, G has a topology and U (s) is a continuous function of s (weakly or strongly, which amounts to the same thing because U(s) is unitary), then T (s) is also a continuous function of s. (b) Conversely, for every positive definite function T (s) on G, whose values are operators on H, with T (e) = IH , there exists a unitary representation of G on a space K containing H as a subspace, such that T (s) = pr U(s) and K=

W

s∈G

U(s)H

(s ∈ G)

(minimality condition).

(7.2) (7.3)

This unitary representation of G is determined by the function T (s) up to isomorphism so that one can call it “the minimal unitary dilation” of the function T (s). If, moreover, the group G has a topology and T (s) is a (weakly) continuous function of s, then U(s) is also a (weakly, hence also strongly) continuous function of s. Proof. Part (a) of the theorem is easy. In fact, we have T (e) = PHU(e)|H = PH |H = IH , T (s−1 ) = PHU(s−1 )|H = PHU(s)∗ |H = (PHU(s)|H)∗ = T (s)∗ ,

7. P OSITIVE DEFINITE FUNCTIONS ON A GROUP

25

and ∑ ∑ (PHU(t −1 s)h(s), h(t)) = ∑ ∑ (U(t)∗U(s)h(s), h(t))

s∈G t∈G

s∈G t∈G

2

= ∑ U(s)h(s)

≥0 s∈G

for every finitely nonzero function h(s) from G to H. The assertion concerning continuity is obvious. Part (b). Let us consider the set H, obviously linear, of the finitely nonzero functions h(s) from G to H, and let us define on H a bilinear form2 by hh, h′ i = ∑ ∑(T (t −1 s)h(s), h′ (t)) s t

[h = h(s), h′ = h′ (s)].

By virtue of (7.1) we have hh, hi ≥ 0 and hence it follows, using Schwarz’s inequality |hh, h′ i|2 ≤ hh, hi · hh′ , h′ i, that the vectors h for which hh, hi = 0 form a linear manifold N in H. It also follows that the value of hh, h′ i does not change if we replace the functions h, h′ by equivalent ones modulo N. In other words, the form hh, hi defines in the natural way a bilinear form (k, k′ ) on the quotient space K0 = H/N. The corresponding quadratic form (k, k) is positive definite on K0 , therefore kkk = (k, k)1/2 is a norm on K0 ; by completing K0 with respect to this norm we obtain a Hilbert space K. Now we embed H in K (and even in K0 ) by identifying the element h of H with the function h = δe (s)h (where δe (e) = 1 and δe (s) = 0 for s 6= e), or, more precisely, with the equivalence class modulo N determined by this function. This identification is allowed because it preserves the linear and metric structure of H. Indeed, we have hδe h, δe h′ i = ∑ ∑(T (t −1 s)δe (s)h, δe (t)h′ )H = (T (e)h, h′ )H = (h, h′ )H . s t

Now we set, for h = h(s) ∈ H and a ∈ G, ha = h(a−1 s). We obviously have (h + h′ )a = ha + h′a , (ch)a = cha , he = h, (hb )a = hab , and furthermore, hha , h′a i = ∑ ∑(T (t −1 s)h(a−1 s), h′ (a−1t)) s t

= ∑ ∑(T (τ −1 σ )h(σ ), h′ (τ )) = hh, h′ i. σ τ

Therefore h ∈ N implies ha ∈ N and consequently the transformation h → ha in H generates a transformation k → ka of the equivalence classes modulo N. Setting U(a)k = ka , thus we define for every a ∈ G a linear transformation of K0 onto K0 , 2

In the complex case, the bilinearity means linearity in the first variable and conjugate linearity in the second variable.

26

C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

such that U(e) = I, U(a)U(b) = U(ab), and (U(a)k,U(a)k′ ) = (k, k′ ). These transformations on K0 extend by continuity to unitary transformations on K, forming a representation of the group G. For h, h′ ∈ H we obtain (setting δa (s) = δe (a−1 s)) (U(a)h, h′ )K = hδa h, δe h′ i = ∑ ∑(T (t −1 s)δa (s)h, δe (t)h′ )H = (T (a)h, h′ )H , s t

and hence T (a) = pr U(a) for every

a ∈ G.

Let us observe next that every function h = h(s) ∈ H can be considered as a finite sum of terms of the type δσ (s)h (i.e., the type (δe (s)h)σ (σ ∈ G)), and hence every element k of K0 can be decomposed into a finite sum of terms of the type U(σ )h (σ ∈ G, h ∈ H). This implies (7.3). The isomorphism of the unitary representations of G satisfying (7.2) and (7.3) is a consequence of the relation (U(s)h,U(t)h′ ) = (U(t)∗U(s)h, h′ ) = (U(t −1 )U(s)h, h′ ) = (U(t −1 s)h, h′ ) = (T (t −1 s)h, h′ ), which shows that the scalar products of the elements of K of the form U(s)h, U(t)h′ (s,t ∈ G, h, h′ ∈ H) do not depend upon the particular choice of the unitary representation U(s) satisfying our conditions. It remains to consider the case when G has a topology and T (s) is a weakly continuous function of s. Let us show that U(s) is then also a weakly continuous function of s, that is, the scalar-valued function (U(s)k, k′ ) is a continuous function of s, for any fixed k, k′ ∈ K. Because U(s) has a bound independent of s (in fact, kU(s)k = 1), and because, moreover, the linear combinations of the functions of the form δσ h (σ ∈ G, h ∈ H) (or, to be more exact, the corresponding equivalence classes modulo N) are dense in K, one concludes that it suffices to prove that (U(s)δσ h, δτ h′ ) is a continuous function of s for any fixed h, h′ ∈ H and σ , τ ∈ G. Now, this scalar product is equal to (U(s)U(σ )h,U(τ )h′ ) = (U(τ −1 sσ )h, h′ ) = (T (τ −1 sσ )h, h′ ), and this is a continuous function of s because T (s) was assumed to be a weakly continuous function of s. This finishes the proof of the theorem.

8. S OME APPLICATIONS

27

8 Some applications 1. Let us consider a function T (n) on the additive group Z of the integers n, whose values are bounded operators on a Hilbert space H and for which T (0) = I and T (−n) = T (n)∗ . According to the general definition, T (n) is positive definite on Z if ∞ ∞ (8.1) ∑ ∑ (T (n − m)hn, hm ) ≥ 0 n=−∞ m=−∞

for every two-way sequence {hn }∞ −∞ of elements of H, which is finitely nonzero, that is, such that hn 6= 0 for a finite set of subscripts only. For such a sequence we can choose an integer a so that the sequence {h′ν }∞ −∞ defined by h′ν = hν +a satisfies h′ν = 0 for ν < 0. When ν = n − a and µ = m − a we have ν − µ = n − m, and therefore (8.1) holds for the finitely nonzero two-way sequences if and only if it holds for the finitely nonzero one-way sequences {hn }∞ 0 , that is, if we have ∞

∞

∑ ∑ (T (n − m)hn, hm ) ≥ 0.

(8.1′)

n=0 m=0

As a first application let us consider the function T (n) which derives from an operator T in the Hilbert space H as follows: T (n) = T ∗|n|

(n ≤ −1),

T (0) = I,

T (n) = T n

(n ≥ 1).

(8.2)

Let us observe that the reciprocal formulas gn =

∑

n≤m<∞

T m−n hm ,

hn = gn − T gn+1

(n ≥ 0)

(8.3)

∞ define a one-to-one transformation {hn }∞ 0 → {gn }0 of the set of finitely nonzero sequences onto itself. Thus our function T (n) satisfies (8.1′ ) if and only if it satisfies ∞

∞

∑ ∑ (T (n − m)(gn − T gn+1), gm − T gm+1 ) ≥ 0

n=0 m=0

(8.4)

for every finitely nonzero sequence {gn }∞ 0 . Now the sum (8.4) can be rearranged into the sum ∞ ∞ (8.5) ∑ ∑ (D(n, m)gn , gm ), n=0 m=0

28

C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

where D(0, 0) = T (0) = I, D(1, 0) = T (1) − T (0)T = T − T = O,

D(0, 1) = T (−1) − T ∗ T (0) = T ∗ − T ∗ = O,

D(n, n) = T (0) − T (−1)T − T ∗ T (1) + T ∗ T (0)T = I − T ∗ T − T ∗ T + T ∗ T = I − T ∗ T for n ≥ 1,

D(n, m) = T (k) − T (k − 1)T − T ∗ T (k + 1) + T ∗ T (k)T = T k − T k−1 T − T ∗ T k+1 + T ∗ T k T = O ∗

for n − m = k ≥ 1,

D(n, m) = T (−k) − T (−k − 1)T − T T (−k + 1) + T ∗ T (−k)T = T ∗k − T ∗k+1 T − T ∗ T ∗k−1 + T ∗ T ∗k T = O

for n − m = −k ≤ −1.

Hence the sum (8.5) equals ∞

(g0 , g0 ) + ∑ ((I − T ∗ T )gn , gn ). n=1

In order that this sum be nonnegative for every finitely nonzero sequence {gn }∞ 0, it is necessary and sufficient that we have I − T ∗ T ≥ O; that is, kT k ≤ 1. So we have obtained: The contractions T are characterized by the property that T (n) is a positive definite function on the group Z. It follows by applying Theorem 7.1 that for every contraction T on H there exists a unitary representation U(n) of the group Z on some space K ⊃ H, such that T (n) = pr U(n) holds for every integer n and that K be spanned by the subspaces U(n)H (−∞ < n < ∞). Setting U(1) = U we have U(n) = U n for each n, and hence we have in particular T n = pr U n for n ≥ 0; that is, U is a minimal unitary dilation of T. So we have obtained a new proof of the existence of a minimal unitary dilation of a contraction. 2. Let us consider now a continuous one-parameter semigroup {T (s)}s≥0 of contractions on H. That is, T (s) is, for every value of the real parameter s ≥ 0, an operator on H such that  T (0) = I;    T (s1 + s2 ) = T (s1 )T (s2 ) for s1 , s2 ≥ 0; (8.6) kT (s)k ≤ 1 for s ≥ 0;    T (s) ⇀ I as s → +0.

8. S OME APPLICATIONS

29

These conditions imply the strong continuity of T (s). In fact, if 0 ≤ s1 < s2 and h ∈ H, we have for σ = s2 − s1 , kT (s2 )h − T (s1 )hk2 = kT (s1 )[T (σ )h − h]k2

≤ kT (σ )h − hk2 = kT (σ )hk2 − 2 Re(T (σ )h, h) + khk2

≤ 2khk2 − 2 Re(T (σ )h, h) = 2 Re(h − T (σ )h, h);

the last term tends to 0 as σ → +0, and hence the assertion follows. Relations (8.6) are obviously invariant with respect to taking adjoints: if the oneparameter semigroup {T (s)}s≥0 is continuous, then so is {T (s)∗ }s≥0 . In particular T (s)∗ is also a strongly continuous function of s (s ≥ 0). Let {T (s)}s≥0 be a continuous one-parameter semigroup of contractions. We extend it to a function T (s) on the whole real line R by setting T (−s) = T (s)∗ . We show that the function thus obtained is positive definite on the additive group R, that is, (8.7) ∑ ∑(T (s − t)h(s), h(t)) ≥ 0 s t

for every finitely nonzero function h(s) from R to H. Suppose h(sn ) = hn 6= 0 for the finite subset S = {sn } of points of R and h(s) = 0 for the other points. Then we have to show that the sum (8.7′) ∑ ∑(T (sn − sm )hn , hm ) n m

is nonnegative. In the particular case that the values sn are commensurable (i.e., of the form sn = vn · d for some positive real number d and integers vn of any sign), the sum (8.7′ ) can be written in the form ∑ ∑(Td (vn − vm )hn , hm ), n m

where Td (n) is the function associated with the contraction Td = T (d) in the sense of Subsec. 1. Hence, positivity of the sum (8.7′) follows in this case from the results of Subsec. 1. When the points of the set S are not commensurable, we replace S by a (k) set S(k) = {sv } of commensurable (e.g., rational) points converging to S as k → ∞ (k) (i.e., sv → sv for each v). For S(k) , the sum (8.7′) is nonnegative, and this property is preserved when we pass to the limit S, owing to the weak continuity of T (s). One can therefore apply Theorem 7.1 to arrive at the following result. Theorem 8.1. For every continuous one-parameter semigroup {T (s)}s≥0 of contractions on H there exists a continuous one-parameter group {U(s)}∞ −∞ of unitary operators on a space K ⊃ H, such that T (s) = pr U(s) and K=

W

s∈R

U(s)H

(0 ≤ s < ∞)

(minimality condition).

30

C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

These conditions determine U(s) up to an isomorphism and we call it “the minimal unitary dilation” of the given semigroup of contractions. 3. Let Bλ be an operator-valued distribution function on the interval 0 < λ ≤ 2π ; thus Bλ is, for every value of λ , a bounded self-adjoint operator on the complex Hilbert space H, such that Bλ ≤ Bµ for λ < µ , Bλ = Bλ +0 , B+0 = O, and B2π = I. The integrals T (n) =

Z 2π 0

einλ dBλ

(n = 0, ±1, . . .)

exist in an obvious sense (as limits in operator norm of Riemann-type sums) and define an operator-valued function T (n) on the group Z, such that T (0) = I and T (−n) = T (n)∗ . Moreover, ∑ ∑(T (n − m)hn, hm ) = n m

Z 2π 0

∑ ∑ ei(n−m)λ d(Bλ hn , hm )

Z 2π

n m

∑ ∑ ei(n−m)λ (B(d λ )hn , hm ) n m  Z 2π  B(d λ ) ∑ einλ hn , ∑ eimλ hm ≥ 0, = =

0

n

0

m

the last integral denoting the limit of the sums   λ λ in in k k ∑ (B(λk+1 ) − B(λk )) ∑ e hn , ∑ e hn , n

k

n

where λ0 = 0 < λ1 < · · · < λk < · · · < λl = 2π and max(λk+1 − λk ) → 0. Thus we can R apply Theorem 7.1; it follows that there exists a unitary operator U = 02π eiλ dEλ on a complex Hilbert space K ⊃ H, such that T (n) = pr U(n) (n = 0, ±1, . . .); that is, Z Z 2π

0

einλ d(Bλ h, h′ ) =

2π

0

einλ d(Eλ h, h′ )

(h, h′ ∈ H)

(8.8)

for all integers n. If we choose {Eλ } so that it satisfies the same condition of normalization as {Bλ } (i.e., Eλ = Eλ +0 , E+0 = O, E2π = IK ), then (8.8) implies Bλ = pr Eλ

for 0 < λ ≤ 2π .

So we have proved the following result. Theorem 8.2. For every operator-valued distribution function Bλ there exists an orthogonal projection-valued distribution function Eλ (i.e. a spectral family) in some larger space such that Bλ = pr Eλ . Let us note that, in the above proof, the interval of variation of the parameter λ was (0, 2π ], but the result extends to the case of any finite or infinite interval, by using a continuous monotonic change of variable. Let us also note that the case of real spaces can be reduced to the case of complex spaces by an obvious “complexification.”

9. R EGULAR UNITARY DILATIONS OF COMMUTATIVE SYSTEMS

31

4. Finally we give an important inequality that can be easily derived from Theorem 4.2. Let T be a contraction on the complex space H, and let U be a unitary dilation of T on the (complex) space K ⊃ H. The relations T n = pr U n (n = 0, 1, . . .) imply p(T ) = pr p(U) for every polynomial p(λ ) = c0 + c1 λ + · · · + cn λ n with real or complex coefficients, and hence kp(T )k ≤ kp(U)k. Now it follows from the spectral representation of unitary operators, that kp(U)k is equal to the maximum of |p(λ )| on the spectrum of U, and thus kp(U)k ≤ max|λ |=1 |p(λ )|. Consequently, kp(T )k ≤ max|λ |=1 |p(λ )|. So we have proved the following proposition. Proposition 8.3 (von Neumann inequality). For every contraction T on the complex space H and for every polynomial p(λ ) we have kp(T )k ≤ max |p(λ )|. |λ |≤1

(8.9)

9 Regular unitary dilations of commutative systems Let T = {Tω }ω ∈Ω be a commutative system of contractions on the space H. Recall that a corresponding system U = (Uω )ω ∈Ω of operators on a space K ⊃ H is a unitary dilation of the system T if U is also commutative, consists of unitary operators Uω , and satisfies r

r

1

1

∏ Tωnii = pr ∏ Uωnii

(9.1)

for every finite set of subscripts ωi ∈ Ω and of corresponding integers ni ≥ 0. This definition can be expressed in a more convenient form if we introduce the class Z Ω of the “vectors” n with the components nω (ω ∈ Ω ), where ω 7→ nω is a finitely nonzero function from Ω to the set of integers (of any sign). Z Ω is an Abelian group with respect to the addition by components, the identity element being the vector o all of whose components are 0. When we speak in this section of “vectors” n, m, and so on, we always mean vectors in Z Ω . If nω ≥ 0 for all ω , we write n ≥ o; n ≥ m means that n − m ≥ o. For arbitrary n, m, we set n ∪ m = {max{nω , mω }} and n ∩ m = {min{nω , mω }}; finally, we define n+ = n ∪ o, n− = −(n ∩ o). Let us set, for a vector n ≥ o, Tn =

∏ Tωnω ;

(9.2)

ω ∈Ω

this product is well defined because T is commutative, and because there are only a finite number of factors different from I. Let us also set Un =

∏ Uωnω ,

(9.3)

ω ∈Ω

where the restriction n ≥ o is no longer necessary, because the unitary operators Uω have (unitary) inverses Uω−1 . Obviously, the operators U n yield a unitary representation of the group Z Ω . Conversely, every unitary representation U(n) of the group

32

C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

Z Ω can be obtained this way. This follows from the fact that n can be written in the form n = ∑ n ρ eρ , ρ ∈Ω

where eρ is the vector whose only nonzero component is the ρ th one, this being equal to 1; indeed, one has now only to set Uρ = U(eρ ). Thus the problem of finding the unitary dilations U of T is equivalent to finding the unitary representations U(n) of the group Z Ω for which T n = pr U(n) for n ≥ o;

(9.1∗)

that is, T n = PHU(n)|H. Now if U(n) is a unitary representation of Z Ω having this property it is possible to extend the function T n (n ≥ o) to a function T (n) that is defined and positive definite on the whole group Z Ω : one has only to set T (n) = PHU(n)|H,

n ∈ ZΩ .

(9.4)

Conversely, for every extension of T n to a positive definite function T (n) on Z Ω there exists, by Theorem 7.1, a unitary representation U(n) of Z Ω (even a minimal one) so that (9.4) holds. Thus the problem of finding the (minimal) unitary dilations of the given system T of contractions is equivalent to the problem of finding the extensions of the function T n (n ≥ o) to a positive definite function T (n) on Z Ω . Clearly, to different extensions there correspond nonisomorphic unitary dilations. When T consists of a single contraction T , the function T n (n ≥ 0) has the unique positive definite extension to Z 1 = Z defined by T (−n) = T (n)∗ = T ∗n for n > 0; see Subsec. 1 of the preceding section. For T consisting of two (commuting) contractions the existence of at least one positive definite extension follows from Theorem 6.4. In contrast, it follows from Sec. 6.3 that for commutative systems of more than two contractions there exist in general no positive definite extensions of T n . However, we are able to exhibit certain particular cases in which such extensions do exist. We proceed as follows. We extend T n to a function T (n) on Z Ω by some simple rule which is amenable to calculations, and determine under which additional conditions this extension is positive definite. One simple rule is to set −

+

T (n) = (T n )∗ T n ;

(9.5)

let us call it the regular extension. It has a dual one which results from changing the order of the two factors on the right; however, because this means essentially replacing the system {Tω } by the system {Tω∗ }, the study of this dual extension reduces to the study of the regular one. One sees immediately that the regular extension satisfies the condition T (−n) = T (n)∗ . By a reasoning analogous to that given at the beginning of the preceding section (for the case of a single contraction), T (n) is

9. R EGULAR UNITARY DILATIONS OF COMMUTATIVE SYSTEMS

33

positive definite on Z Ω if (and only if) ∑ ∑ (T (n − m)h(n), h(m)) ≥ 0

n≥o m≥o

(9.6)

for every finitely nonzero function h(n) defined for n ≥ o. Now we make use of a generalization of the reciprocal formulas (8.3). Let us observe first that for every function h(n) of the above type the function g(n) = ∑ T m−n h(m)

(n ≥ o)

m≥n

(9.7)

is also finitely nonzero. We can retrieve the function h(n) from the function g(n) in the following way. For each finite subset v of Ω let us set ( 1 if ω ∈ v, e(v) = {eω (v)}, eω (v) = 0 if ω ∈ Ω \v, and let |v| denote the number of the elements of v. With these notations, the reciprocal of the formula (9.7) is h(n) = ∑ (−1)|v| T e(v) g(n + e(v)) v⊂Ω

(n ≥ o),

(9.8)

where, as indicated by the notation, v runs over the set of all the finite subsets of Ω .3 In fact, for any fixed n (≥ o) we have ∑ (−1)|v| T e(v)

v⊂Ω

∑

m≥n+e(v)

= ∑ (−1)|v| v⊂Ω

= ∑

"

∑

T m−n−e(v) h(m) ∑

m≥n+e(v)

|v|

(−1)

m≥n e(v)≤m−n

T m−n h(m) #

T m−n h(m) = h(n);

here we have used the elementary theorem of combinatorics asserting that if v runs through all subsets of a finite set v0 (the whole set and the empty set also admitted) then ( 1 if v0 is empty, |v| (9.9) ∑ (−1) = 0 if v0 is not empty. v⊂v0 Conversely, if one starts with an arbitrary, finitely nonzero function g(n) (n ≥ o) then the function h(n) which it generates by formula (9.8) is also finitely nonzero, 3

The function g(n) is finitely nonzero, and hence there are only a finite number of nonzero terms of the sum (9.8).

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C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

and for every fixed n (≥ o) we have ∑ T m−n h(m) = ∑ (−1)|v| ∑ T m−n+e(v) g(m + e(v)) v⊂Ω

m≥n

m≥n

= ∑

(−1)|v|

= ∑

∑

v⊂Ω

p≥n

∑

p≥n+e(v)

T p−n g(p)

(−1)|v|

e(v)≤p−n

!

g(p) = g(n),

again by virtue of (9.9). We conclude that formulas (9.7)–(9.8) give a transformation of the set of finitely nonzero functions (defined for n ≥ o) onto itself, and the inverse of this transformation. Consequently, in order that (9.6) hold for every finitely nonzero function h(n) (n ≥ o) it is necessary and sufficient that the sum   ∑ ∑ T (n − m) ∑ (−1)|v| T e(v) g(n + e(v)), ∑ (−1)|w| T e(w) g(m + e(w)) v⊂Ω

n≥o m≥o

w⊂Ω

be ≥ 0 for every finitely nonzero function g(n) (n ≥ o). Now this sum can be written in the form (9.10) ∑ ∑ (D(p, q)g(p), g(q)), p≥o q≥o

where D(p, q) =

∑

∑ (−1)|v|+|w| (T e(w) )∗ T (p − e(v) − q + e(w))T e(v)

v⊂π (p) w⊂π (q)

(9.11)

and π (n) is the set defined for every vector n by

π (n) = {ω : nω > 0}. We now prove that D(p, q) = O if p 6= q. Observe first that if p 6= q then the sets π (p − q) and π (q − p) cannot both be empty. By reason of symmetry, it suffices therefore to consider the case when π (p − q) is not empty. The set

δ (w) = π (p) ∩ π (p − q + e(w)) is then nonempty for every finite subset w of Ω and, in fact, contains the set π (p − q). Next we observe that for every fixed w one obtains all subsets v of π (p) by taking v = v′ ∪ v′′ , where v′ and v′′ satisfy the conditions v′ ⊂ π (p)\δ (w) and v′′ ⊂ δ (w).

9. R EGULAR UNITARY DILATIONS OF COMMUTATIVE SYSTEMS

35

We have |v| = |v′ | + |v′′ | and e(v) = e(v′ ) + e(v′′ ); then D(p, q) is equal to ∑ (−1)|w| (T e(w) )∗

w⊂π (q)

∑

′

(−1)|v |

v′ ⊂π (p)\δ (w)



′′

∑

(−1)|v | T (p − q + e(w)

v′′ ⊂δ (w)

′

′′

− e(v ) − e(v ))T

e(v′′ )



′

T e(v ) .

Now we show that the sum between the brackets [ ] equals O for every fixed w and v′ . In fact, it follows from definition (9.5) of the regular extension that ′′

T (p − q + e(w) − e(v′) − e(v′′ ))T e(v ) = (T a )∗ T b , where

a = [p − q + e(w) − e(v′) − e(v′′)]−

and

b = [p − q + e(w) − e(v′) − e(v′′)]+ + e(v′′ ).

Now a and b do not depend on v′′ ; namely we have a = [p − q + e(w) − e(v′)]−

and b = [p − q + e(w) − e(v′)]+ ,

which is a consequence of the fact that for ω ∈ v′′ we have pω − qω + eω (w) ≥ 1 and eω (v′ ) = 0. Thus the above sum between the brackets [ ] is equal to ′′ (T a )∗ T b ∑v′′ ⊂δ (w) (−1)|v | , and this is equal to O because the set δ (w) is not empty. This proves that D(p, q) = O. The sum (9.10) reduces thus to the form ∑ (D(p, p)g(p), g(p)).

p≥o

In order that this sum be nonnegative for every finitely nonzero function g(p) (p ≥ o) it is necessary and sufficient that we have D(p, p) ≥ O for every p ≥ o. Now, (9.11) gives D(p, p) =

∑

−

+ +e(v)

∑ (−1)|v|+|w| (T e(w)+[e(w)−e(v)] )∗ T [e(w)−e(v)]

v⊂π (p) w⊂π (p)

.

It is easily seen that both exponents of T are equal to e(u) where u = v ∪ w. Using (9.9) again we obtain ∑

(v,w) (−1)|v|+|w|

v∪w=u

= ∑

(v) (−1)|v|

= ∑

(v) (−1)|v|+|u\v|

v⊂u

v⊂u

∑

(w) (−1)|w|

w=(u\v)∪v′ v′ ⊂v

∑

v′ ⊂v

(v′ ) (−1)|v′ |

= (−1)|u| ,

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C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

and finally D(p, p) =

∑ (−1)|u| (T e(u) )∗ T e(u) .

u⊂π (p)

Thus we have obtained the following result. Theorem 9.1. Let T = {Tω }ω ∈Ω be a commutative system of contractions on the space H. In order that T have a unitary dilation U = {Uω }ω ∈Ω on a space K ⊃ H, which is regular, that is, such that −

+

(T n )∗ T n = pr U n

for all n = {nω }ω ∈Ω ∈ Z Ω ,

where Un =

∏ Uωnω

for all n,

ω ∈Ω

Tn =

∏ Tωnω

ω ∈Ω

for n ≥ o,

it is necessary and sufficient that we have S(u) = ∑ (−1)|v| (T e(v) )∗ T e(v) ≥ O v⊂u

(9.12)

for every finite subset u of Ω . Moreover, one can require that U be minimal, that is, the subspaces U n H (n ∈ Z Ω ) span the space K. In this case the regular unitary dilation U is determined up to isomorphism. Remark 1. Suppose the system T contains an isometry, say Tω0 . Then S(u) = O for every finite subset u of Ω containing ω0 . In fact, for any such u we obtain all subsets v of u by taking v = v0 ∪ v1 , where v0 and v1 satisfy v0 ⊂ u0 = {ω0 } and v1 ⊂ u1 = u\u0 . We have T e(v)∗ T e(v) = T e(v1 )∗ T e(v0 )∗ T e(v0 ) T e(v1 ) = T e(v1 )∗ T e(v1 ) because T e(v0 ) equals Tω0 or I accordingly as v0 is the set u0 or the empty set (the one-point set u0 has just these two subsets). Because |v| = |v0 | + |v1 |, we have   S(u) = ∑ (−1)|v1 | ∑ (−1)|v0 | T e(v1 )∗ T e(v1 ) = O, v1 ⊂u1

v0 ⊂u0

where we have used (9.9), in this case for the one point set u0 . Let us introduce the following notion. We say that the operators A and B doubly commute if A commutes with B and B∗ (and, therefore, B commutes with A and A∗ ). Remark 2. Let ud be a nonempty subset of a finite subset u of Ω such that Tω and Tω ′ doubly commute whenever ω ∈ ud and ω ′ ∈ u, ω ′ 6= ω . Let uc = u\ud . If S(uc ) ≥ O then S(u) ≥ O also.

9. R EGULAR UNITARY DILATIONS OF COMMUTATIVE SYSTEMS

37

In fact, one has ∑ (−1)|vd |+|vc | T e(vd )∗ T e(vc )∗ T e(vc ) T e(vd )

S(u) = ∑

vd ⊂ud vc ⊂uc

= ∑ (−1)|vc | T e(vc )∗ T e(vc ) ∑ (−1)|vd | ∏ω ∈vd Tω∗ Tω vc ⊂uc

= ∑

vc ⊂uc

(−1)|vc | T e(vc )∗ T e(vc )

= S(uc ) ·

vd ⊂ud

∏ω ∈ud (I − Tω∗ Tω )

∏ (I − Tω∗ Tω ).

ω ∈ud

Because the factors I − Tω∗ Tω (ω ∈ ud ) are nonnegative and they commute with each other as well as with S(uc ), we see that S(uc ) ≥ O implies S(u) ≥ O. Remark 3. If ∑ω ∈u kTω k2 ≤ 1, then S(u) ≥ 0. In fact, let u = {ω1 , . . . , ωr } and write, for the sake of brevity, Ti in place of Tωi . For 0 ≤ p ≤ r and for h ∈ H, set a p (h) = ∑ kT e(v) hk2 . v⊂u |v|=p

Then for 1 ≤ p ≤ r we have a p (h) = = ≤

∑

1≤i1 <...
kTi p . . . Ti1 hk2 ≤

∑

1≤i1 <···
kTi p k2 kTi p−1 . . . Ti1 hk2

∑

kTi p−1 . . . Ti1 hk2

∑

kTi p−1 . . . Ti1 hk2 = a p−1 (h),

1≤i1 <...
∑

i p−1
kTi p k2

and hence r

(S(u)h, h) = ∑ (−1)|v| kT e(v) hk2 = ∑ (−1) p a p (h) v⊂u p=0   r r ≥ a0 (h) − a1(h) = khk2 − ∑ kTi hk2 ≥ 1 − ∑ kTi k2 khk2 ≥ 0. i=1

i=1

By virtue of the above remarks, Theorem 9.1 implies the following result. Proposition 9.2. Let T be a commutative system of contractions. Delete from T the isometries, and from the rest, denoted by T1 , delete those operators that doubly commute with every other operator in T1 ; let the rest be denoted by T2 . If T2 has a regular unitary dilation (in particular, if T2 is empty) then so does T . In particular T = {Tω } has a regular unitary dilation in each of the cases below: (i) T consists of isometries, (ii) T consists of doubly commuting contractions, (iii) T is countable and ∑ω kTω k2 ≤ 1.

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C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

When V and U are isometries, the relation V = pr U implies V ⊂ U and hence the part (i) of the above proposition implies that for every commutative system {Vω } of isometries there exists a commutative system {Uω } of unitary operators on some larger space such that Uω is an extension of Vω for every ω . Thus we have obtained a new proof of Proposition 6.2, valid for finite as well as for infinite systems.

10 Another method to construct isometric dilations 1. We sketch one more method of constructing an isometric dilation of a contraction. Its interest is due mainly to the fact that it carries over, with obvious modifications, to one-parameter semigroups of contractions too. Let T be a contraction on the Hilbert space H. For each h ∈ H we have khk2 = kDT hk2 + kT hk2 = kDT hk2 + kDT T hk2 + kT 2 hk2 = · · · = kDT hk2 + kDT T hk2 + · · · + kDT T n−1 hk2 + kT n hk2 .

The inequalities imply therefore

khk ≥ kT hk ≥ · · · ≥ kT n hk ≥ · · · ≥ 0 ∞

khk2 = ∑ kDT T j hk2 + limn→∞ kT n hk2 . j=0

(10.1) (10.2)

From (10.1) it also follows that I ≥ T ∗ T ≥ · · · ≥ T ∗n T n ≥ · · · ≥ O, and hence S = limn→∞ T ∗n T n exists in the sense of strong operator convergence. Set Q = S1/2 . Then T ∗ Q2 T = lim T ∗n+1 T n+1 = Q2 , n

kQT hk2

kQhk2

so we have = (h ∈ H), and the transformation Qh → QT h is isometric. It extends by continuity to an isometry W from Q = QH into Q. Thus we have QT = W Q. (10.3) Let us now consider the space H of vectors h = {hn }∞ −∞ with hn ∈ DT and ∞

khk2 = ∑ khn k2 < ∞. −∞

We embed H in K = H ⊕ Q by identifying the element h of H with the element {. . . , 0, 0, DT h , DT T h, DT T 2 h, . . .} ⊕ Qh of K, this being justified by the isometry expressed by (10.2). (We have put a square around the 0th component.)

10. A NOTHER METHOD TO CONSTRUCT ISOMETRIC DILATIONS

39

Let V be the bilateral shift on H defined by V {hn} = {h′n },

where

h′n = hn+1

(n = 0, ±1, . . .);

then U = V ⊕ W is an isometry on K, and by (10.3) it follows for h ∈ H and m = 1, 2, . . . that (m) U m h − T m h = {hn }∞ (10.4) n=−∞ ⊕ 0,

where

(m) hn

( DT T m+n h = 0

if −m ≤ n ≤ −1, in the other cases.

Clearly, the vector on the right-hand side of (10.4) is orthogonal to H and hence T m = pr U m (m = 1, 2, . . .). Thus U is an isometric dilation of T . 2. Consider the case of a continuous one-parameter semigroup {T (s)}s≥0 of contractions on H; see Sec. 8.2. This semigroup has an infinitesimal generator A, defined by 1 Ah = lim [T (s) − I]h (10.5) s→+0 s whenever this limit exists (in the strong sense); A is a closed linear operator with domain D(A) dense in H and we have (d/ds)T (s)h = AT (s)h = T (s)Ah for h ∈ D(A), s ≥ 0; see [Func. Anal.] Sec. 142. Let us note that kT (s)k ≤ 1 implies Re((T (s) − I)h, h) = Re(T (s)h, h) − (h, h) ≤ kT (s)kkhk2 − khk2 ≤ 0 for h ∈ H; hence it follows Re(Ah, h) ≤ 0

for h ∈ D(A).

(10.6)

We define by [h, k] = −(Ah, k) − (h, Ak)

(10.7)

a bilinear form on D(A); by virtue of (10.6) it is semidefinite, that is, [h, h] = −2 Re(Ah, h) ≥ 0

(h ∈ D(A)).

It follows from the definition of A that for h ∈ D(A) and s ≥ 0, d kT (s)hk2 = 2 Re(T (s)Ah, T (s)h) ds = 2 Re(AT (s)h, T (s)h) = −[T (s)h, T (s)h] and consequently khk2 =

Z t 0

[T (s)h, T (s)h]ds + kT (t)hk2 .

(10.8)

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C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

Thus khk2 =

Z ∞ 0

[T (s)h, T (s)h]ds + lim kT (t)hk2 , t→∞

(10.9)

where the limit exists because kT (t)hk2 is a nonincreasing function of t. It also follows that limt→∞ T (t)∗ T (t) exists in the sense of strong operator convergence; let Q(≥ O) be the square root of this limit. Then we construct, in analogy with the discrete case, a continuous semigroup {W (s)}s≥0 of isometries on Q = QH such that QT (s) = W (s)Q (s ≥ 0). (10.10)

Let D be the completion, with respect to the metric (10.7)–(10.8), of the preHilbert space D(A) modulo the subspace formed by the vectors h for which [h, h] = 0. Then we consider the functions h = h(s) (−∞ < s < ∞) with values in D, strongly measurable and such that khk2 =

Z ∞

−∞

kh(s)k2 ds < ∞.

These functions form a Hilbert space H, where as usual we do not distinguish two functions as elements of this space if they coincide almost everywhere. We embed the space H in the space K = H⊕Q

by identifying the element h of H with the element h ⊕ Qh of K, where ( 0 for s < 0, h(s) = T (s)h for s ≥ 0. Let {V (s)}s≥0 be the continuous semigroup defined by (V (t)h)(s) = h(s + t)

(−∞ < s < ∞;t ≥ 0; h ∈ H).

Then the operators U(s) = V (s) ⊕ W (s) (s ≥ 0) form a continuous semigroup of isometries on K. For h ∈ H and t ≥ 0 we have U(t)h − T(t)h = h(t) ⊕ 0, where (t)

h (s) =

(10.11)

(

T (s + t)h if −t ≤ s < 0, 0 in the other cases.

The right-hand side of (10.11) is obviously orthogonal to H, and hence we have T (t) = pr U(t) (t ≥ 0). Thus the semigroup {U(s)}s≥0 is a dilation of the semigroup {T (s)}s≥0 .

3. For an example we construct an isometric dilation of a concrete continuous semigroup of contractions, generated by a system of differential equations. In the particular case we consider, the use of the infinitesimal generator can be avoided.

10. A NOTHER METHOD TO CONSTRUCT ISOMETRIC DILATIONS

41

The system we consider is of the form dxi = Xi (x) dt

(i = 1, . . . , n),

(10.12)

where x = (x1 , . . . , xn ) lies in the n-dimensional real Euclidean space Rn . We suppose that the functions Xi are continuously differentiable and that the divergence of (X1 , . . . , Xn ) is nonpositive, that is,

∂ Xi (x) ≥ 0. ∂ i=1 xi n

ρ (x) = − ∑

We also suppose that for every x ∈ Rn the solution x(t) of (10.12) with x(0) = x exists not only on a small neighborhood of t = 0 but on the whole t-axis. In this case

τt : x → x(t) is a differentiable transformation of Rn onto itself. The functional determinant

δt (x) =

D(τt x) D((τt x)1 , . . . , (τt x)n ) = D(x) D(x1 , . . . , xn )

can be calculated easily: in fact, δt (x) is the Wronskian of the solutions   ∂ (τt x)1 ∂ (τt x)n ( j = 1, . . . , n) ,..., ∂xj ∂xj of the system of linear differential equations n ∂ Xi (τt x) dui = ∑ uk dt k=1 ∂ xk

(i = 1, . . . , n),

associated with the system (10.12), and hence one derives, using Liouville’s theorem and the fact that δ0 (x) = 1, the formula  Zt  (10.13) δt (x) = exp − ρ (τs x)ds . 0

Thus for any Borel-measurable function ϕ (x) integrable on Rn we have Z

Rn

ϕ (τ−t x) dx =

Z

Rn

ϕ (x)δt (x) dx

(dx = dx1 . . . dxn ).

(10.14)

Set, for f (x) ∈ L2 (Rn ) and for t ≥ 0, (T (t) f )(x) = f (τ−t x). Choosing ϕ (x) = | f (x)|2 formulas (10.13) and (10.14) imply that T (t) is a contraction of L2 (Rn ) for t ≥ 0. Moreover, the obvious relations τt+s = τt ◦ τs and τ0 = the

42

C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

identity transformation on Rn , imply that {T (t)}t≥0 is a semigroup of contractions on L2 (Rn ). For continuous f (x) with compact support it is obvious that T (t) f → f strongly, as t → +0. These functions being dense in L2 (Rn ) we conclude easily that the semigroup {T (t)} is continuous. Let us introduce the measures d ν (x, s) = ρ (x)dxds (in Rn+1 ), where

 Z δ∞ (x) = exp −

0

∞

d µ (x) = δ∞ (x)dx

(in Rn ),

 ρ (τs x)ds = lim δt (x) (≥ 0). t→∞

Proposition 10.1. The continuous one-parameter semigroup {T (t)}t≥0 of contractions attached to the system of differential equations (10.12) with divergence −ρ (x) ≤ 0, has an isometric dilation that is unitarily equivalent to the semigroup {U(t)}t≥0 defined on the space K = L2 (Rn+1 ; ν ) ⊕ L2 (Rn ; µ ) by

U(t)[f(x, s) ⊕ f (x)] = f(x, s + t) ⊕ f (τ−t x).

Proof. Let g(x) be continuously differentiable and have compact support in Rn . Then Z Z n ∂ |g(x)|2 d d 2 2 kT (t)gk = |g(τ−t x)| dx = − X j (x) dx, ∑ dt dt Rn Rn j=1 ∂ x j t=0 t=0

and hence by partial integration Z Z n ∂ Xj d 2 kT (t)gk = dx = − |g|2 ρ dx. |g|2 ∑ n n dt ∂ x R R j j=1 t=0

The function T (t)g is also continuously differentiable and has compact support, thus we obtain from the preceding formula that d d kT (t)gk2 = kT (s)T (t)gk2 dt ds s=0 =−

Z

Rn

|T (t)g|2 ρ dx = −

Z

Rn

|g(τ−t x)|2 ρ (x) dx,

and consequently, by the obvious relation  Z t d kT (s)gk2 ds + kT (t)gk2 , kgk2 = − ds 0

11. U NITARY ρ - DILATIONS

43

we conclude that kgk2 =

Z tZ 0

Rn

|g(τ−s x)|2 ρ (x) dx ds +

Z

Rn

|g(τ−t x)|2 dx.

Thus using first (10.14) (with ϕ = |g|2 ) and then letting t → ∞ we obtain finally kgk2 =

ZZ

Rn ×(0,∞)

|g(τ−s x)|2 d ν (x, s) +

Z

Rn

|g(x)|2 d µ (x).

(10.15)

This formula, valid for every function of the type considered (thus for a set of functions dense in L2 (Rn )), is the concrete form of the formula (10.9) for the semigroup under consideration. The rest of the proof proceeds on the basis of the formula (10.15) in the same way as the construction in the preceding section was derived from formula (10.9). Remark. Finer analysis (which we omit) shows that the semigroup {U(t)}t≥0 obtained consists in fact of unitary operators, and that its natural extension to a group {U(t)}∞ −∞ yields the minimal unitary dilation of {T (t)}.

11 Unitary ρ -dilations 1. As a generalization of the notion of unitary dilation of an operator we introduce the following concept. Definition. We call class Cρ (ρ > 0) the set of operators T on the Hilbert space H for which there exists a unitary operator U on some Hilbert space K(⊃ H) such that T n = ρ · pr U n

(n = 1, 2, . . .);

(11.1)

U is then called a unitary ρ -dilation of T . For T ∈ Cρ we obviously have kT n k ≤ ρ

(n = 1, 2, . . .)

(11.2)

and hence lim kT n k1/n ≤ 1; therefore the spectrum of T is contained in the closed unit disc. The class C1 consists precisely of the contractions. The following theorem characterizes each of the classes Cρ , at least for complex spaces. Theorem 11.1. Let T be a bounded operator on the complex Hilbert space H, and let ρ > 0. In order that T belong to the class Cρ it is necessary and sufficient that the condition     2 2 2 Re(zT h, h) ≤ khk2 , (Iρ ) − 1 kzT hk + 2 − ρ ρ

44

C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

or equivalently, that the condition (ρ − 2)k(I − zT )hk2 + 2 Re((I − zT )h, h) ≥ 0

(I∗ρ )

be satisfied for all h ∈ H, |z| ≤ 1. Proof. The equivalence of the two conditions is obvious. Suppose T has a unitary ρ -dilation U on the space K. Because U is unitary, the series I + 2zU + · · · + 2znU n + · · · converges in the operator norm for |z| < 1, its sum being equal to (I + zU)(I − zU)−1 . From (11.1) it follows that the series IH +

2 2 zT + · · · + zn T n + · · · ρ ρ

also converges in the operator norm. The sum of this series is then necessarily equal to   2 2 IH + (IH − zT )−1 , 1− ρ ρ therefore we have   2 2 IH + (IH − zT )−1 = pr(IK + zU)(IK − zU)−1 1− ρ ρ

(|z| < 1).

(11.3)

On the other hand, we have Re((I + zU)k, (I − zU)k) = kkk2 − |z|2 kUkk2 = (1 − |z|2 )kkk2 ≥ 0 for k ∈ K and |z| < 1, and hence we deduce Re((I + zU)(I − zU)−1k′ , k′ ) ≥ 0

(k′ ∈ K, |z| < 1).

Thus, using (11.3), we obtain    2 2 −1 Re 1 − (l, l) + ((I − zT ) l, l) ≥ 0 (l ∈ H, |z| < 1). ρ ρ

(11.4)

Setting l = lz = (I − zT )h, h ∈ H, and multiplying by ρ we obtain (I∗ρ ) first for |z| < 1 and then by continuity for |z| ≤ 1 also. We now prove that, conversely, the (equivalent) conditions (Iρ ), (Iρ∗ ) imply that T is of class Cρ . To this end we first show that (Iρ ) implies that the spectrum of T is contained in the closed unit disc. Suppose the contrary. Then there exists a point 1/z0 outside the unit circle, belonging to the boundary of the spectrum of T and, consequently, belonging to the approximate point spectrum of T (see H ALMOS [4], Problem 63). That is, there exists a sequence {hn } of elements of H such that khn k = 1 (n = 1, 2, . . .), (I − z0 T )hn → 0 (n → ∞), and hence z0 (T hn , hn ) → 1 and kz0 T hn k → 1.

11. U NITARY ρ - DILATIONS

45

Let 0 < r < 1 − |z0 |. Then z = z0 + rz0 is also inside the unit circle. So we have, by virtue of (I∗ρ ), (ρ − 2)k(I − z0 T )hn − rz0 T hn k2 + 2 Re[((I − z0 T )hn , hn ) − rz0 (T hn , hn )] ≥ 0. Let n → ∞; we obtain in the limit: (ρ − 2)r2 − 2r ≥ 0. Dividing by r and letting r → 0 we obtain the contradiction −2 ≥ 0. This proves the assertion that no point of the spectrum lies outside the unit circle, that is, (I − zT )−1 exists as a bounded operator on H for every z inside the unit circle, and equals the n n sum of the series ∑∞ 0 z T , which converges in the operator norm. Let l be an arbitrary element in H and apply (I∗ρ ) to h = hz = (I − zT )−1 l. Then (ρ − 2)klk2 + 2 Re(l, (I − zT )−1 l) ≥ 0, and hence

(Q(r, ϕ )l, l) ≥ 0

(0 ≤ r < 1; 0 ≤ ϕ ≤ 2π ; l ∈ H),

(11.5)

where Q(r, ϕ ) is the operator-valued function defined by the series Q(r, ϕ ) = I +

1 iϕ 1 r(e T + e−iϕ T ∗ ) + · · · + rn (einϕ T n + e−inϕ T ∗n ) + · · · , ρ ρ

which converges in the operator norm. Let {hn }∞ −∞ be a finitely nonzero sequence of elements of H and let ∞ h(ϕ ) = ∑ hn e−inϕ . −∞

We deduce from (11.5) that 0≤

1 2π ∞

Z 2π 0

(Q(r, ϕ )h(ϕ ), h(ϕ )) d ϕ

= ∑ (hn , hn ) + −∞

1 1 ∑ ∑ rn−m (T n−m hn , hm ) + ∑ ∑ rm−n (T ∗m−n hn , hm ) ρ n>m ρ m>n

for every r, 0 ≤ r < 1. If we let r → 1 − 0, we obtain ∞

∑

∞

∑ (Tρ (n − m)hn, hm ) ≥ 0,

n=−∞ m=−∞

where Tρ (n) is derived from T by the formulas Tρ (0) = I,

Tρ (n) =

1 n 1 T and Tρ (−n) = T ∗n ρ ρ

(n ≥ 1).

(11.6)

This function Tρ (n) of n is thus positive definite on the additive group Z of the integers. By virtue of Theorem 7.1 there exists a unitary operator Uρ on a space Kρ

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C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

containing H as a subspace, such that Tρ (n) = pr Uρn for all n ∈ Z. By definition (11.6) of Tρ (n) this means that Uρ is a unitary ρ -dilation of T . This concludes the proof of the theorem. Observe that for ρ = 1 condition (Iρ ) reduces to the condition kT hk ≤ khk, that is, kT k ≤ 1. Thus Theorem 11.1 constitutes a generalization of Theorem 4.2. For ρ = 2, (Iρ ) reduces to the condition Re z(T h, h) ≤ khk2

(h ∈ H, |z| ≤ 1)

(11.7)

which is obviously equivalent to khk2 ≥ |(T h, h)|

(h ∈ H).

(11.8)

So in this particular case our theorem can be formulated as follows. Proposition 11.2. In order that the operator T belong to the class C2 it is necessary and sufficient that the “numerical radius” of T defined by w(T ) = sup{|(T h, h)| : h ∈ H, khk ≤ 1} satisfy w(T ) ≤ 1.

Let us add some further remarks. Remark 1. When 0 < ρ < 2, ρ 6= 1 condition (Iρ ) reduces to   ρ − 1 |µ | khk h ∈ H, ≤ |µ | < ∞ , k(µ I − T )hk ≤ |ρ − 1| ρ − 2

(Iρ′ )

and when 2 < ρ < ∞ it reduces to k(µ I − T )hk ≥

|µ | khk ρ −1

  ρ −1 h ∈ H, ≤ |µ | < ∞ . ρ −2

(Iρ′′ )

In fact, for 0 < |z| ≤ 1 multiply (Iρ ) by the factor

ρ 1 , 2 − ρ |z|2 and set

µ=

ρ −1 1 . ρ −2 z

By rearranging we arrive at the above alternative forms of (I ρ ), depending on the sign of the factor, that is the form (I′ρ ) if this factor is positive (ρ < 2), and the form (I′′ρ ) if it is negative (ρ > 2). Remark 2. If 1 < ρ < 2, condition (I′ρ ) is equivalent to k µ I − T k ≤ |µ | + 1



 ρ −1 ≤ |µ | < ∞ . 2−ρ

(II′ρ )

11. U NITARY ρ - DILATIONS

47

In fact, (II′ρ ) implies (Iρ′ ) because |µ | + 1 ≤

|µ | ρ −1

for |µ | ≥

ρ −1 . 2−ρ

Conversely, for such a µ we deduce from (I′ρ ), setting ε = µ /|µ |,





ρ − 1

ε ρ − 1 I − T ≤ |µ | − ρ − 1 + 1 = |µ | + 1; + k µ I − T k ≤ µ − ε

2−ρ 2−ρ 2−ρ 2−ρ

thus condition (II′ρ ) is valid. Remark 3. If 2 < ρ < ∞, condition (I′′ρ ) is equivalent to the condition that T has its spectrum in the closed unit disc and   1 ρ −1 −1 k(µ I − T ) k ≤ 1 < |µ | < . (IIρ′′ ) |µ | − 1 ρ −2 In fact, (I′′ρ ) implies for 1 < |µ | ≤ rρ = (ρ − 1)/(ρ − 2) and µ = ε |µ | that k(µ I − T )hk ≥ k(ε rρ I − T )hk − k(ε rρ − µ )hk 1 khk − (rρ − |µ |)khk = (|µ | − 1)khk ≥ |ε rρ | ρ −1 for all h ∈ H. Moreover, because (I′′ρ ) implies that T ∈ Cρ , the spectrum of T is in the closed unit disc, and we obtain that (II′′ρ ) holds. Conversely, if T satisfies (IIρ′′ ) and if, moreover, its spectrum is in the closed unit disc, then (I − zT )−1 exists and is an analytic function of z inside the unit circle, and we obtain by (II′′ρ ) and the maximum principle that max k(I − zT )−1 k = max k(I − zT )−1 k = rρ · max k(ζ I − T )−1 k

|z|≤1/rρ

|z|=1/rρ

rρ = ρ − 1. ≤ rρ − 1

|ζ |=rρ

Thus for rρ ≤ |µ | < ∞ we have

 −1

ρ −1

1 1

, k(µ I − T )−1 k =

≤

I− T

|µ | µ |µ |

and consequently (I′′ρ ). Remark 4. If the spectrum of T lies in the open unit disc, then T ∈ Cρ for ρ large enough. Indeed, then inequality (II′′ρ ) holds for ρ large enough. 2. From condition (Iρ∗ ) it is obvious that the class Cρ is a nondecreasing function of ρ . We show that it is in fact an increasing function.

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C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

Proposition 11.3. If dim H ≥ 2, the class Cρ (0 < ρ < ∞) increases with ρ ; that is, Cρ ⊂ Cρ ′

and Cρ 6= Cρ ′

for 0 < ρ < ρ ′ < ∞.

Proof. We construct for every ρ > 0 an operator Tρ in H such that Tρ ∈ Cρ and kTρ k = ρ ; this operator can belong to none of the classes Cσ with 0 < σ < ρ . To this end choose an orthonormal basis {ϕ1 , ϕ2 , ψν (ν ∈ Ω )} in H (the set Ω may be empty), and we consider the operator Tρ defined by Tρ ϕ1 = ρϕ2 ,

Tρ ϕ2 = 0,

Tρ ψν = 0

(ν ∈ Ω ).

Clearly we have kTρ k = ρ and Tρn = O (n ≥ 2). Let K be a Hilbert space of dimension ℵ0 · dim H and choose any orthonormal basis in K; its elements can be arranged in the following way. {ϕm′ (m = 0, ±1, ±2, . . .);

ψν′ m (ν ∈ Ω ; m = 0, ±1, ±2, . . .)}.

We identify ϕ1 with ϕ1′ , ϕ2 with ϕ2′ , and ψν with ψν′ 0 (ν ∈ Ω ); this defines an isometric embedding of H in K as a subspace. Next we define a unitary operator U on K by setting ′ U ϕm′ = ϕm+1 ,

U ψν′ m = ψν′ ,m+1

(ν ∈ Ω )

for m = 0, ±1, ±2, . . . . If we denote by P the orthonormal projection of K into H, we obtain

ρ PU ϕ1 = ρ Pϕ2 = ρϕ2 ,

ρ PU ϕ2 = ρ Pϕ3′ = 0,

ρ PU ψν = ρ Pψν′ ,1 = 0,

and for n ≥ 2: ′ ρ PU n ϕi = ρ Pϕi+n = 0 (i = 1, 2),

ρ PU n ψν = ρ Pψν′ ,n = 0.

Thus ρ · PU n h = Tρn h for n ≥ 1 and h = ϕ1 , ϕ2 , ψν . This implies the same relation for arbitrary h ∈ H. Thus U is a unitary ρ -dilation of Tρ . This concludes the proof. 3. The von Neumann inequality (8.9) can be extended, in an appropriate form, to the classes Cρ . In fact, (11.1) implies for every polynomial p(λ ) with complex coefficients: p(T ) = pr[ρ · p(U) + (1 − ρ ) · p(0)IK]. (11.9) The fact that U is unitary yields the following result.

Proposition 11.4. For T ∈ Cρ and for any polynomial p(λ ) of the complex variable λ we have kp(T )k ≤ max |ρ · p(z) + (1 − ρ ) · p(0)|. (11.10) |z|≤1

It is possible to complete this proposition as follows.

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49

Proposition 11.5. Let q(λ ) be a polynomial such that q(0) = 0 and |q(λ )| ≤ 1 for |λ | ≤ 1. Then for T ∈ Cρ we also have q(T ) ∈ Cρ (0 < ρ < ∞). Proof. Let U be a unitary ρ -dilation of T . Applying (11.9) to the polynomials p(λ ) = q(λ )n (n = 1, 2, . . .) yields q(T )n = ρ · pr q(U)n

(n = 1, 2, . . .).

(11.11)

Because |q(λ )| ≤ 1 for |λ | ≤ 1, it follows from the spectral theory of unitary operators that kq(U)k ≤ 1. Consequently, there exists a unitary operator V in some larger space such that q(U)n = pr V n (n = 0, 1, . . .). (11.12) Now (11.11) and (11.12) imply q(T )n = ρ · pr V n

(n = 1, 2, . . .),

and hence q(T ) ∈ Cρ . For ρ = 2 this result can also be restated, by virtue of Proposition 11.2, in the following form. Proposition 11.6. If w(T ) ≤ 1 then w(q(T )) ≤ 1 for every polynomial q(λ ) such that q(0) = 0 and |q(λ )| ≤ 1 (|λ | ≤ 1). In particular, w(T ) ≤ 1 implies w(T n ) ≤ 1 (n = 1, 2, . . .). Let us state again that the results of this section relate to operators on complex Hilbert spaces.

12 Notes Theorem 1.1 on the decomposition of a space H induced by an isometry V on H has been formulated in a probabilistic setting by W OLD [1], p. 89. Except for the expression of the subspace H0 of the unitary part as the intersection of the ranges of the iterates of V , the theorem already appears in the fundamental paper on abstract Hilbert space of VON N EUMANN [1], p. 96; in its present form the theorem was stated and proved by H ALMOS [2], Lemma 1. Proposition 2.1 on bilateral shifts can also be derived—at least for complex Hilbert space—from the general theory of spectral multiplicity; the direct proof given here, which is valid without restriction on the field of scalars, is due to H ALPERIN; see Sz.-N.–F. [V]. Proposition 3.1 on the invariant vectors of a contraction was found by S Z .-NAGY in connection with some ergodic theorems (cf. R IESZ–S Z.-NAGY [1] and [Func. Anal.] Sec. 144). Generalizations of this proposition were given in S Z.-N.–F. [1]. Theorem 3.2 on the canonical decomposition of a contraction was proved by L ANGER [1] and S Z.-N.–F. [IV]. The notation A = pr B was introduced by S Z .-NAGY in [P]. For two operators so related, H ALMOS [1] says that A is a “compression” of B, and B is a “dilation” of

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C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

A, whereas S Z.-NAGY [P] says “projection” instead of “compression”. In this book we have preferred to abandon this terminology and preserve for the term “dilation” the meaning given in Sec. 4 (i.e., “power dilation” in the sense of H ALMOS [4]). By the way, let us observe that, because A = pr B if and only if the bilinear form (Bb, b′ ) is an extension of the bilinear form (Aa, a′ ), we would be justified in calling B a “numerical extension” of A, and A a “numerical restriction” of B (in analogy with “numerical range”, “numerical radius”, etc.). The fact that for every contraction T there exists an isometry V such that T = pr V , was observed already by J ULIA [1]–[3]. H ALMOS [1] has shown that V can be chosen to be unitary, V = U. Theorem 4.2, on the existence of a unitary U such that the relations T n = pr U n hold simultaneously for n = 1, 2, . . . (i.e., of a unitary dilation of T ), was found by S Z .-NAGY [I]. The original proof used the theorem of F. R IESZ on the trigonometric moment problem and the theorem of N A˘I MARK [1] (Theorem 8.2) on the existence, for every operator distribution function Bλ , of an orthogonal projectionvalued distribution function Eλ (i.e., of a spectral family) such that Bλ = pr Eλ . The next proof (S Z .-NAGY [I bis], [P], [1]) was based upon the fact that the function T (n), derived from the contraction T by formulas (8.2), is positive definite on the additive group of the integers, so that one can apply the theorem of NA˘I MARK [1] on operator-valued positive definite functions on groups (Theorem 7.1). We have reproduced this method in Sec. 8.1, with the only difference that the positive definiteness of the function T (n) is proved here in a simpler way than in the places indicated. It should be remarked that the NA˘I MARK theorem (Theorem 7.1) was extended to ∗-semigroups by S Z .-NAGY [P]; from among the various applications of this generalized theorem we should mention the proof of a theorem of H ALMOS [1] on subnormal operators. These first two proofs of Theorem 4.2 were followed by the matrix proof due ¨ to S CH AFFER [1], reproduced in Sec. 5.1. The modification of this construction yielding the minimal unitary dilation, given in Sec. 5.2, is due to S Z .-NAGY [2] (cf. also H ALPERIN [1]). The proof in Sec. 4 of this book follows a fourth method: first one finds an isometric dilation, then this is extended to a unitary dilation. Finally, the construction in Sec. 10.1 also yields an isometric dilation, which, moreover, can be shown to be minimal; see D OUGLAS [3]. The problem of finding a unitary (or isometric) dilation of a commutative system of contractions was proposed by S Z .-NAGY; he proved the existence of a unitary dilation under the additional condition that the contractions considered are doubly commuting; see S Z .-NAGY [I bis], [7]. The concept of the regular unitary dilation and the systematic study of the existence problem for such dilations, is due to B REHMER [1]; see also S Z .-NAGY [4] (the terminology used in this book is new). This study was later completed and simplified by H ALPERIN [2], [4]; the proof in Sec. 9 is close to that of H ALPERIN [2]. Theorems 6.1 and 6.4 establishing the existence, for every commutative pair of contractions, of an isometric dilation and of a unitary dilation, is due to A NDO [1]. The example of a system of three commuting contractions for which no unitary dilation exists (Sec. 6.3), was derived by PARROTT [1].

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51

Theorem 8.1, on the unitary dilation of a continuous one-parameter semigroup of contractions, is due to S Z .-NAGY [I], [I bis], [P]. If {Vs }s≥0 is a semigroup of isometric operators and {Us }s≥0 is its unitary dilation, then we have necessarily Vs ⊂ Us (s ≥ 0). So one obtains a new proof of a theorem of C OOPER [1] asserting that every continuous one-parameter semigroup of isometries on the space H can be extended to a continuous one-parameter semigroup of unitary operators on a space K ⊃ H. The fact that every commutative system of isometries on the space H can be extended to a commutative system of unitary operators on a space K ⊃ H (Proposition 6.2), was proved by I T Oˆ [1] and B REHMER [1]. In Sec. 6 we have followed (at least partially) the method of I T Oˆ , and in Sec. 9, Proposition 9.2 (i), the method of B REHMER. See also D OUGLAS [4]. Inequality (8.9) for contractions can be restated by saying that the closed unit disc |λ | ≤ 1 is a “spectral set” for every contraction on a complex Hilbert space. This theorem was obtained first by VON N EUMANN [4]; his proof used some methods of the theory of analytic functions. The proof was later simplified by H EINZ [1] (cf. [Func. Anal.]); in this form the proof is based upon the classical Cauchy–Poisson formula. The proof in Sec. 8, which reduces the problem through the use of unitary dilations to the simpler particular case of unitary operators, was given in S Z .-NAGY [I], [P]. As a natural generalization of von Neumann’s inequality one can conjecture the inequality kp(T1 , . . . , Tn )k ≤ (*) max |p(λ1 , . . . , λn )| |λ1 |≤1,...,|λn |≤1

{Ti }n1

for any commuting system of contractions and any polynomial p of the complex variables λ1 , . . . , λn . For n = 2, inequality (∗ ) follows from A NDO’s theorem (Theorem 6.4) in the same way as inequality (8.9) followed from Theorem 4.2. For n ≥ 3 this method breaks down, because then the system {Ti }n1 has in general no unitary dilation (cf. Sec. 6.3). Although this does not imply that (∗ ) should fail for n ≥ 3, it turned out that (∗ ) also fails for n ≥ 3 (see VAROPOULOS [1]). Let us also mention that inequality (8.9) characterizes complex Hilbert spaces among complex Banach spaces; indeed, if (8.9) is valid for every contraction T on a complex Banach space X and for every polynomial p(λ ), then X is necessarily a Hilbert space ( cf. F OIAS¸ [1]). For p(λ ) = λ , inequality (8.9) reduces to kT k ≤ 1. On the other hand, the contractions on a Hilbert space are characterized by the property of admitting a unitary dilation. Now, in a complex Hilbert space, the unitary operators are those normal operators whose spectrum is situated on the unit circle. Hence, for an operator T on a complex Hilbert space, the validity of von Neumann’s inequality is equivalent to the existence of a normal dilation whose spectrum is situated on the unit circle. This raises the following problem. Let T be a bounded operator on the complex Hilbert space H and let S be a compact subset of the plane of complex numbers. Is the validity of the inequality kp(T )k ≤ max |p(z)|, z∈S

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C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

for every polynomial p(z), equivalent to the existence of a normal dilation N of T whose spectrum lies on the boundary of S? For bounded Jordan sets S (i.e., whose boundary is a simple closed curve) this equivalence was established by S Z .-N.–F. [III], and for general compact sets S with connected complement by F OIAS¸ [4]. (See also L EBOW [1] and B ERGER [1].) Later, S ARASON [1] showed that the proof of this equivalence (for compact S with connected complement) can be reduced to the case of the disc. The method used in Sec. 10 to obtain isometric dilations of a contraction or of a continuous one-parameter semigroup of contractions, is not the only one that allows dealing with these two cases in an analogous way. Indeed, the original method of S Z .-NAGY [I] also permitted the two cases to be treated analogously. The method of Sec. 10.2 has been obtained in connection with the problem of finding the unitary dilation of a system of differential equations of negative divergence (cf. Sec. 10.3); this problem was proposed to one of the authors by A. G. K OSTJU Cˇ ENKO and its solution is contained in Proposition 10.1 and in the remark following it. The first result on the existence of a unitary ρ -dilation (for ρ 6= 1) is due to B ERGER [2] ( cf. also H ALMOS [3]) and concerns the case ρ = 2: this is our Proposition 11.2. The concept of unitary ρ -dilations and the first general results in this direction are in S Z .-N.–F. [6]; however, the proof of Theorem 11.1 as given in Sec. 11 is in part different from the original one, and follows the line of the proof given in S Z .-NAGY [10] for the Berger–Halmos theorem (our Proposition 11.2). Moreover, we have omitted in Theorem 11.1 the condition that the spectrum of T be contained in the closed unit disc, because this turns out to be a consequence of condition (Iρ ) not only if ρ ≤ 2 (as noted originally by the authors) but for any ρ > 0. This fact was recently observed by D AVIS [1]. The fact that the class Cρ is nondecreasing as a function of ρ (0 < ρ < ∞), and nonconstant for small or for large values of ρ , was observed already in S Z .-N.–F. [6]. That it is strictly increasing (Proposition 11.3), was observed first by D URSZT [1]; he also obtained a criterion for a normal T to belong to Cρ . Propositions 11.5 and 11.6 are due to Stampfli (cf. H ALMOS [3]). As an obvious consequence we obtain the inequality (conjectured by Halmos): w(T n ) ≤ w(T )n

(n = 1, 2, . . .),

which holds for an arbitrary bounded operator T . See P EARCY [1] for a more elementary proof. The problem of unitary ρ -dilations can be generalized as follows. Given a selfadjoint operator A on a complex Hilbert space H, with positive lower and upper bounds, characterize the class CA of those operators T on H for which there exists a unitary operator U on some space K ⊃ H such that QT n Q = pr U n

(n = 1, 2, . . .), where Q = A−1/2

F URTHER RESULTS

53

(the unitary ρ -dilation corresponds to the case A = ρ I). Such a characterization is given by the condition: (Ah, h) − 2Re(z(A − I)T h, h) + |z|2((A − 2I)T h, T h) ≥ 0 (h ∈ H, |z| ≤ 1). This generalization of Theorem 11.1 was proposed by H. Langer (correspondence). ˘ ¸ ESCU [1] observed that CA ⊂ CB if A ≤ B, so that in particular CA ⊂ Cρ if I STR AT ρ ≥ kAk. From the consequences of the theorem on unitary dilations let us also mention here the result of S Z .-N.–F. [9] that to every contraction T on H we can find a unitary operator U on some space L such that the operator T ′ = T ⊕ U on M = H ⊕ L admits a “continuous scale” of invariant subspaces Mλ (0 ≤ λ ≤ 1), that is, such that T ′ Mλ ⊂ Mλ , M0 = {0}, M1 = M, Mλ ⊂ Mµ for λ < µ , and finally S T Mλ = κ<λ Mκ = µ >λ Mµ . In connection with the subject treated in this chapter, also see B ERBERIAN [1]; ´ B ERGER AND S TAMPFLI [1], [2]; E GERV ARY [1]; F URUTA [1], [2]; H OLBROOK ´ [1]; K ATO [2]; K ENDALL [1], [2]; KOR ANYI [1]; M LAK [5]–[8]; NAKANO [1]; NA˘I MARK [2], [3]; S Z .-NAGY [3]; S UCIU [1]; and T HORHAUER [1], [2].

13 Further results 1. Many of the results of dilation theory can be put in a more algebraic framework which makes it easier to study their possible extensions. Denote by B(H) the algebra of bounded linear operators on the Hilbert space H. Recall that a Banach algebra A is called a C∗ -algebra if it is isometrically isomorphic with a subalgebra of B(H), closed under taking adjoints. Such an algebra has a natural adjoint operation a → a∗ inherited from B(H). Let A be a C∗ -algebra with unit, and let B ⊂ A be a subalgebra of A , containing the unit. A representation of B is simply a unital algebra homomorphism Φ : B → B(H). Such a representation is said to be contractive if kΦ (b)k ≤ kbk for every b ∈ B. A dilation of a representation Φ : B → B(H) to the algebra A consists of a Hilbert space K ⊃ H, and a contractive representation Ψ : A → B(K) such that Ψ (a∗ ) = Ψ (a)∗ for every a ∈ A , and

Φ (b) = PΨ (b)|H (b ∈ B), where P : K → H denotes the orthogonal projection. An obvious condition for the existence of a dilation is that Φ itself be contractive. If the space H is invariant for every Ψ (b), b ∈ B, and

Φ (b) = Ψ (b)|H (b ∈ B), we say that Ψ is a lifting of Φ . The following result was first observed by S ARASON [4].

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C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

Lemma 13.1. Consider a dilation Ψ : A → B(K) of a representation Φ : B → B(H). There exist subspaces M ⊂ N ⊂ K, invariant for every Ψ (b), b ∈ B, such that N ⊖ M = H.

The spaces M, N are easily found: N is the closed linear span of {Ψ (b)h : b ∈ B, h ∈ H}, and M = N ⊖ H. This observation makes it possible to study dilations using liftings. In order to relate this algebraic framework to the material in Chap. I, denote by AC the C∗ -algebra consisting of all continuous complex functions on the unit circle C, and let B1 denote the subalgebra of AC consisting of all polynomials. A representation Φ : B1 → B(H) is entirely determined by the operator T = Φ (λ ) because Φ (p(λ )) = p(T ) for every polynomial p ∈ B1 . Theorem 4.2 can now be reformulated as follows. Theorem 13.2. Let T be a contraction on the Hilbert space H, and define Φ : B1 → B(H) by Φ (p(λ )) = p(T ), p ∈ B1 . Then Φ has a dilation to AC . Proof. Let K and U be provided by Theorem 4.2, and define Ψ : AC → B(H) by

Ψ ( f ) = f (U) ( f ∈ AC ), where the functional calculus is defined using the spectral measure of U. It is easy to verify that Ψ is indeed a dilation of Φ . The von Neumann inequality (Proposition 8.3) is then the statement that Φ is a contractive representation, provided that kT k ≤ 1. General conditions for the existence of dilations in this context were given by A RVESON [1]. To formulate the result, let us note that a matrix [Ti j ]ni, j=1 of operators on the Hilbert space H can be viewed as an operator on the direct sum H ⊕ H ⊕ · · ·⊕ H of n copies of H, and one can therefore speak of the operator norm of such a matrix. Analogously, a matrix [ai j ]ni, j=1 with entries in a C∗ -algebra A has a welldefined norm. A representation Φ : B → B(H) of a subalgebra B of a C∗ -algebra A is said to be completely contractive if k[Φ (bi j )]ni, j=1 k ≤ k[bi j ]ni, j=1 k for every positive integer n, and every matrix [bi j ]ni, j=1 with bi j ∈ B. We can now state one of the main results of A RVESON [1]. Theorem 13.3. Consider a C∗ -algebra A , a unital subalgebra B ⊂ A , and a representation Φ : B → B(H). The representation Φ has a dilation to A if and only if it is completely contractive. We see in particular that the von Neumann inequality kp(T )k ≤ kpk also extends to matrix polynomials if kT k ≤ 1. More interestingly, this gives some insight into the existence of normal dilations for operators. Thus, consider an operator T on a

F URTHER RESULTS

55

Hilbert space H, and a bounded open set Ω in the complex plane; denote by Γ the boundary of Ω . The closed set Ω is said to be spectral for T if the inequality kp(T )k ≤ sup |p(λ )| λ ∈Γ

(13.1)

holds for every rational function p with no poles in Ω . A normal operator N on a Hilbert space K ⊃ H is called a normal boundary dilation of T if σ (N) ⊂ Γ , and p(T ) = Pp(N)|H for any rational function p with no poles in Ω . As before P is the orthogonal projection onto H. The appropriate algebraic context here is given by the C∗ -algebra AΓ of continuous complex functions on Γ , and the subalgebra BΩ consisting of rational functions with no poles in Ω . The existence of a normal boundary dilation is then clearly equivalent to the existence of a dilation of the representation p → p(T ) (p ∈ BΩ ), and Theorem 13.3 implies that such dilations exist if and only if the inequality (13.1) is true for all matrix-valued rational functions. In other words, the requirement is that Ω be a complete spectral set for T . Theorem 13.2 implies that the closed unit disk is a spectral set for T if and only if it is a complete spectral set. Extensions of this theorem depend therefore on finding other sets in the plane for which this implication is true. In the positive direction, AGLER [2] proved the following result for an annulus Ω = {λ : |λ | ∈ (α , β )}, where 0 < α < β .

Theorem 13.4. Let T be an operator on the Hilbert space H such that the annulus Ω is a spectral set for T . Then T has a normal boundary dilation. Assume on the other hand that Ω has higher connectivity, for instance,

Ω = Ω (α1 , α2 ; ρ1 , ρ2 ) = {λ : |λ | < 1, |λ − α1 | > ρ1 , |λ − α2 | > ρ2 }, where α1 , α2 ∈ D, |α1 | + ρ1 < 1, |α2 | + ρ2 < 1, and ρ1 + ρ2 < |α1 − α2 |. Then D RITSCHEL AND M C C ULLOUGH [1] proved the following result. Theorem 13.5. With the above notation, there exists an operator T on some Hilbert space such that Ω (α , α ; ρ , ρ ) is a spectral set for T , but T does not have a normal boundary dilation. An explicit operator for a specific set Ω (α1 , α2 ; ρ1 , ρ2 ) had been found numerically by AGLER , H ARLAND , AND R APHAEL [1]. In this example, T is a finite matrix. 2. Fix now a positive integer n, and consider the algebra ACn of continuous complex functions on the product of n copies of C. Denote by Bn the subalgebra of ACn consisting of polynomials. A representation Φ of Bn is entirely determined by the operators T1 = Φ (λ1 ), . . . , Tn = Φ (λn ), where λ j denote the coordinate functions on Cn . These operators must commute with each other. A dilation of Φ to ACn is the same thing as a commuting unitary dilation of the n-tuple (T1 , T2 , . . . , Tn ). Theorem 6.1 can thus be reformulated as follows.

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C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

Theorem 13.6. Let T1 , T2 be two commuting contractions on a Hilbert space H, and denote by Φ : B2 → B(H) the representation defined by Φ (p) = p(T1 , T2 ) for p ∈ B2 . Then Φ is completely contractive.

For the special example in Sec. 6.3 the corresponding representation Φ of B3 can in fact be shown to be contractive. The operators T1 , T2 , T3 do not have commuting unitary dilations, thus it follows that Φ is not completely contractive. Examples of commuting contractions T1 , T2 , T3 for which Φ is not even contractive were given by C RABB AND DAVIE [1] and VAROPOULOS [1]. 3. Consider again a unital subalgebra B of a C∗ -algebra A , and a representation Φ : B → B(H). When Φ does not have a dilation to A , one can ask whether the representation b → X −1 Φ (b)X does have such a dilation for some invertible operator X. An obvious necessary condition for the existence of such an operator X is the existence of a constant k such that k[Φ (bi j )]ni, j=1 k ≤ kk[bi j ]ni, j=1 k for every positive integer n, and every matrix [bi j ]ni, j=1 with bi j ∈ B. When Φ satisfies this condition, we say that it is completely bounded. PAULSEN [1] proved that this necessary condition is sufficient as well. Theorem 13.7. Let Φ be a representation of a unital subalgebra B of a C∗ algebra A . There exists an invertible operator X , such that the representation b → X −1 Φ (b)X has a dilation to A , if and only if Φ is completely bounded. In particular, this gives a criterion for a single operator T to be similar to a contraction: the representation p → p(T ), p ∈ B1 , must be completely bounded. S Z .NAGY [11] asked whether a power-bounded operator must be similar to a contraction. After the negative answer provided by F OGUEL [1], H ALMOS [5],[6] asked whether T must be similar to a contraction if the representation p → p(T ), p ∈ B1 , is bounded. The question is therefore whether this boundedness condition implies complete boundedness. P ISIER [1] settled the problem by showing that the map may be bounded, but not completely bounded.

4. Let us return now to the case of an operator T with a spectral set Ω , where Ω is a finitely connected subset of the plane, with smooth boundary Γ . The following result was proved by D OUGLAS AND PAULSEN [1]. Theorem 13.8. Assume that Ω is a spectral set for T . Then the representation p → p(T ), p ∈ BΩ , is completely bounded. In particular, T is similar to an operator that has a normal boundary dilation. For further developments related to this material, see PAULSEN [2] and D OU PAULSEN [2].

GLAS AND

5. In order to study lifting theory for more general operators, AGLER [3] proposed the study of families of representations. Fix an algebra B with unit, and a collection F of unital representations Φ : B → B(H). Then F is called a family if the following conditions are satisfied.

F URTHER RESULTS

57

1. For each b ∈ B we have sup{kΦ (b)k : Φ ∈ F } < ∞, L 2. For any set {Φi }i∈I ⊂ F , the representation b → i∈I Φi (b) belongs to F , 3. If M ⊂ H is invariant for Φ (b), b ∈ B, then the representation b → Φ (b)|M belongs to F , 4. If Ψ is a *-representation of the C∗ -algebra generated by Φ (B), then b → Ψ (Φ (b)) belongs to F . Observe that the algebra B is not assumed to sit inside a fixed C∗ -algebra. As before, Φ is called a lifting of a representation of the form b → Φ (b)|M. Such a lifting is said to be trivial if M is reducing for Φ (b), b ∈ B. Finally, an element Φ ∈ F is said to be extremal if all its liftings are trivial. The following result is from AGLER [3]. Theorem 13.9. Every element Φ of a family F has an extremal lifting. The simplest example of a family F consists of the contractive representations of the algebra B1 considered earlier, and which correspond to contractions T on a Hilbert space. The extremal elements correspond to operators T such that T ∗ is isometric. Another interesting family is described as follows. Denote by B the algebra of all polynomials with complex coefficients in the (commuting) variables X1 , X2 , . . . , Xn , and let F consist of those representations Φ satisfying n

∑ Φ (X j )∗ Φ (X j ) ≤ I.

j=1

The result of D RURY [1] can be interpreted as a characterization of the extremal elements of F : they are the ones for which all the operators Φ (X j )∗ are isometric. For further information about this family see A RVESON [3]. VASILESCU [1] and C URTO AND VASILESCU [1,2] study another interesting example related to the polydisc. Other examples of interest can be (and have been) studied by defining appropriate families, though determining the extremal elements can be difficult. For instance, the extremals are not known for the family of contractive representations of the algebra B3 (of polynomials in three variables). 6. Isometric dilations for noncommuting operators have also been studied. The following result was proved by F RAZHO [1] for n = 2, and D URSZT AND S Z .NAGY [1], and B UNCE [1] for arbitrary (even infinite) n. See also F RAZHO [2] and P OPESCU [1],[2] for a thorough analysis of this situation, including an appropriate analogue of the Wold decomposition. Theorem 13.10. Let T1 , T2 , . . . , Tn be operators on a Hilbert space H satisfying the inequality T1 T1∗ + T2 T2∗ + · · · + Tn Tn∗ ≤ IH .

There exist isometric operators with pairwise orthogonal ranges V1 ,V2 , . . . ,Vn on a Hilbert space K ⊃ H such that V j∗ H ⊂ H and V j∗ |H = T j for j = 1, 2, . . . , n.

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C HAPTER I. C ONTRACTIONS AND T HEIR D ILATIONS

Under appropriate hypotheses, the operators V j can be chosen to be creation operators on a full Fock space. See also M UHLY AND S OLEL [1] for a dilation theory in the context of correspondences over a von Neumann algebra. 7. Theorem 8.1 was extended to bounded representations of amenable groups; see D IXMIER [2]. 8. The decomposition of a contraction into unitary and completely nonunitary parts was extended in D URSZT [3] to arbitrary operators on Hilbert space. 9. The norms of invertible operators realizing the similarity of an operator of class Cρ to a contraction are estimated in O KUBO AND A NDO [1]. These can also be estimated using complete boundedness; see PAULSEN [2].

Chapter II

Geometrical and Spectral Properties of Dilations 1 Structure of the minimal unitary dilations In the sequel we consider a contraction T on the real or complex Hilbert space H, and its minimal unitary dilation U on the Hilbert space K, real or complex, respectively (K ⊃ H). The linear manifolds L0 = (U − T )H

and L∗0 = (U ∗ − T ∗ )H

and their closures L = (U − T )H,

(⊂ K)

L∗ = (U ∗ − T ∗ )H

(1.1) (1.2)

play an important role in our investigations.

Theorem 1.1. (i) The subspaces L and L∗ are wandering subspaces for U, their dimensions being equal to the defect indices of T : dim L = dT ,

dim L∗ = dT ∗ .

(1.3)

(ii) The space K can be decomposed into the orthogonal sum K = · · · ⊕U ∗2L∗ ⊕ U ∗L∗ ⊕ L∗ ⊕ H ⊕ L ⊕ UL ⊕ U 2L ⊕ · · · .

(1.4)

Proof. It would be easy to obtain these properties from the matrix form of U constructed in Sec. I.5.2, but we prefer to give a direct proof, independent of the particular realization of U. Part (i): To prove that L and L∗ are wandering subspaces, it suffices to show that n U L0 ⊥ L0 and U ∗n L∗0 ⊥ L∗0 for n = 1, 2, . . .; by reason of symmetry it even suffices to consider one of these cases, say that of L0 . Now for h, h′ ∈ H and n = 1, 2, . . . we

B.Sz.-Nagy et al., Harmonic Analysis of Operators on Hilbert Space, Universitext, DOI 10.1007/978-1-4419-6094-8_2, © Springer Science + Business Media, LLC 2010

59

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C HAPTER II. G EOMETRICAL AND S PECTRAL P ROPERTIES OF D ILATIONS

have (U n (U − T )h, (U − T )h′ )

= (U n h, h′ ) − (U n−1T h, h′ ) − (U n+1h, T h′ ) + (U nT h, T h′ )

= (T n h, h′ ) − (T n−1 T h, h′ ) − (T n+1 h, T h′ ) + (T n T h, T h′ ) = 0.

In order to prove that dim L = dT , let us observe that for h ∈ H k(U − T )hk2 = kUhk2 − 2 Re(Uh, T h) + kT hk2 2

2

(1.5) 2

= khk − 2 Re(T h, T h) + kThk = khk − kT hk

2

= kDT hk2 .

By virtue of this relation, the transformation ϕ defined by

ϕ (U − T )h = DT h

(1.6)

maps L0 isometrically onto DT H, and consequently ϕ extends by continuity to a unitary transformation of L onto the defect space DT . This proves that dim L = dim DT = dT . The equality dim L∗ = dT ∗ can be proved analogously. Part (ii): Let us show first that the terms of the right-hand side of (1.4) are mutually orthogonal. We have already proved that L and L∗ are wandering subspaces, therefore it remains only to show that U n L ⊥ U ∗m L∗ ,

U n L ⊥ H and U ∗m L∗ ⊥ H

for m, n ≥ 0;

it even suffices to establish these relations for L0 and L∗0 instead of L and L∗ . Now we have for h, h′ ∈ H, (U n (U − T )h,U ∗m (U ∗ − T ∗ )h′ )

= (U n+m+2 h, h′ ) − (U n+m+1h, T ∗ h′ ) − (U n+m+1T h, h′ ) + (U n+m T h, T ∗ h′ )

= (T n+m+2 h, h′ ) − (T n+m+1 h, T ∗ h′ ) − (T n+m+1 T h, h′ ) + (T n+m T h, T ∗ h′ ) = 0,

(U n (U − T )h, h′ ) = (U n+1 h, h′ ) − (U nT h, h′ ) = (T n+1 h, h′ ) − (T n T h, h′ ) = 0,

and

(U ∗m (U ∗ − T ∗ )h, h′ ) = (U ∗m+1 h, h′ ) − (U ∗mT ∗ h, h′ )

= (T ∗m+1 h, h′ ) − (T ∗m T ∗ h, h′ ) = 0,

so the orthogonality relations are established. Let us denote the orthogonal sum on the right-hand side of (1.4) by K′ . Applying U term by term we obtain UK′ = · · · ⊕U ∗L∗ ⊕ L∗ ⊕ UL∗ ⊕ UH ⊕ UL ⊕ U 2L ⊕ · · · .

1. S TRUCTURE OF THE MINIMAL UNITARY DILATIONS

61

Now UK′ equals K′ , because, as we show in a moment, we have UL∗ ⊕ UH = H ⊕ L.

(1.7)

Therefore K′ is a subspace of K reducing U and containing H, and this implies by the minimality of U that K′ = K. In order to establish (1.7) it suffices to show that UL∗0 ⊕ UH = H ⊕ L0, that is, U(U ∗ − T ∗ )H ⊕ UH = H ⊕ (U − T )H.

(1.8)

Now this follows from the fact that, for an element u ∈ K, the possibility of a representation of the form u = h′ + (U − T )h′′

(h′ , h′′ ∈ H)

is equivalent to the possibility of a representation of the form u = Uh1 + U(U ∗ − T ∗ )h2

(h1 , h2 ∈ H).

Indeed, we have only to set h1 = T ∗ h′ + (I − T ∗ T )h′′ ,

h2 = h′ − T h′′

and, conversely, h′ = T h1 + (I − T T ∗ )h2 ,

h′′ = h1 − T ∗ h2 .

This completes the proof of Theorem 1.1. Theorem 1.2. In order that (a) M(L) = K or (a∗ ) M(L∗ ) = K, it is necessary and sufficient that the condition (b) T n → O (n → ∞) or (b∗ ) T ∗n → O (n → ∞) be satisfied, respectively. Thus each of conditions (b) and (b∗ ) implies that U is a bilateral shift, of multiplicity dT or dT ∗ , respectively. Proof. For h ∈ H and for n = 1, 2, . . . we have n−1

n−1

k=0

k=0

M(L) ∋ ∑ U −k−1 (U − T )T k h = ∑ (U −k T k − U −k−1 T k+1 )h = h − U −nT n h,

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and if condition (b) holds, then h = lim (h − U −nT n h) ∈ M(L). n→∞

Thus (b) implies U n H ⊂ U n M(L) = M(L)

H ⊂ M(L),

(n = 0, ±1, . . .)

and consequently K = M(L). The implication (b∗ ) ⇒ (a∗ ) can be proved similarly. We now turn to the proof of the converse implications. Condition (a) implies in particular that every element h ∈ H can be represented as the sum of an orthogonal series of the form ∞

h = ∑ U k lk , −∞

∞

where lk ∈ L, ∑ klk k2 = khk2 ; −∞

hence we deduce for n = 1, 2, . . ., ∞

T n h = PHU n h = PH ∑ U n+k lk . k=−∞

By virtue of (1.4) H is orthogonal to U m L for m ≥ 0, so we have −n−1

T n h = PH ∑ U n+k lk , k=−∞



−n−1 n+k 2 −n−1 n 2 2

kT hk ≤ ∑ U lk

= ∑ klk k , k=−∞

and hence The implication

(a∗ )

T nh → 0

⇒

(b∗ )

k=−∞

for n → ∞.

can be proved similarly.

Theorem 1.2 has in certain particular cases a converse. Proposition 1.3. If the defect index dT is finite and if the minimal unitary dilation U of T is a bilateral shift of multiplicity equal to dT , then T n → O as n → ∞. Similarly, if dT ∗ is finite and U is a bilateral shift of multiplicity dT ∗ , then T ∗n → O as n → ∞. Proof. It suffices to consider the first of the two assertions. The hypothesis that U is a bilateral shift of multiplicity dT means that there exists a wandering subspace L′ for U such that K = M(L′ ), dim L′ = dT . We also have M(L) ⊂ K, dim L = dT , and dT is finite, thus it follows from Proposition I.2.1 that M(L) = M(L′ ) = K, and hence T n → O, by the preceding proposition. Proposition 1.4. For every contraction T on H and for its minimal unitary dilation U on K, we have M(L) ∨ M(L∗ ) = K ⊖ H0 , (1.9)

2. I SOMETRIC DILATIONS. D ILATION OF COMMUTANTS

63

where H0 denotes the maximal subspace of H on which T is unitary (cf. Theorem I.3.2). In particular, if T is completely nonunitary, then M(L) ∨ M(L∗ ) = K.

(1.10)

Proof. Let f be an element of K, orthogonal to M(L) and to M(L∗ ). Because f is orthogonal, in particular, to U n L and to U ∗n L∗ for n = 0, 1, . . . , by (1.4) it is necessarily contained in H. The vector f is also orthogonal to U ∗n L (n ≥ 1), thus we have for any h ∈ H 0 = ( f ,U ∗n (U − T )h) = (U n−1 f , h) − (U n f , T h) = (T n−1 f , h) − (T n f , T h); choosing h = T n−1 f we obtain kT n−1 f k2 − kT n f k2 = 0 and thus

(n = 1, 2, . . .),

k f k = kT f k = kT 2 f k = · · · .

Similarly, from the orthogonality of f to U n L∗ (n ≥ 1) we obtain that k f k = kT ∗ f k = kT ∗2 f k = · · · . We conclude that f ∈ H0 . Conversely, for every f ∈ H0 we have U n f = T n f for n ≥ 0 and U n f = T ∗|n| f for n ≤ 0, and thus U n f ∈ H for every integer n; this implies that U n f ⊥ L, f ⊥ U −n L, and consequently f ⊥ M(L). By similar reasoning, we have f ⊥ M(L∗ ). Hence f ⊥ M(L) ∨ M(L∗ ). This concludes the proof of (1.9); (1.10) follows because for a c.n.u. T we have H0 = {0}.

2 Isometric dilations. Dilation of commutants 1. Let us begin by observing that the subspaces M(L) and M(L∗ ) reduce the operator U, and hence the same is true for the subspaces R = K ⊖ M(L∗ ) and R∗ = K ⊖ M(L).

(2.1)

R = U|R and R∗ = U|R∗

(2.2)

We call the residual part and the dual residual (or ∗-residual) part of U, respectively. These are unitary operators on R and R∗ . We investigate them more closely in the next section. Let us consider now the subspace K+ =

∞ W U nH 0

(⊂ K).

(2.3)

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C HAPTER II. G EOMETRICAL AND S PECTRAL P ROPERTIES OF D ILATIONS

This is invariant for U and contains H as a subspace; hence (2.4)

U+ = U|K+

is a minimal isometric dilation of T (cf. Theorem I.4.1). When speaking in the sequel of the minimal isometric dilation of T we always consider it as “embedded” in this way in the minimal unitary dilation. By virtue of the obvious relation U n h = T n h + (U − T )T n−1 h + U(U − T )T n−2 h + · · · + U n−1(U − T )h for h ∈ H and n ≥ 0, the space K+ is contained in H ⊕ M+ (L). On the other hand, H ⊂ K+ , thus U n (U − T )H ⊂ U n+1 H ∨U nH ⊂ K+

(n ≥ 0),

and hence H ⊕ M+(L) is contained in K+ . This proves the relation (2.5)

K+ = H ⊕ M+(L).

Because L ⊂ K+ , there is no difference in defining M+ (L) with respect to U or to U+ . Comparing (2.5) with (1.4) and (2.1) we obtain       ∞ ∞ ∞ L L L K+ = K ⊖ U ∗n L∗ = [R ⊕ M(L∗ )] ⊖ U −n L∗ = R ⊕ U n L∗ ; 0

setting

0

1

L∗ = UL∗ = U(U ∗ − T ∗ )H = (I − UT ∗ )H

(2.6)

K+ = R ⊕ M+(L∗ ).

(2.7)

L∗ ⊕ UH = H ⊕ L.

(2.8)

L ∩ L∗ = {0}.

(2.9)

H ∨UH = H ⊕ L.

(2.10)

we can also write this as

We have L∗ ⊂ H ∨UH ⊂ K+ , thus there is again no difference in defining M+ (L∗ ) with respect to U or to U+ . By virtue of (1.7) we have

We deduce hence that In fact, taking the orthogonal complements of L and of L∗ in H ⊕ L, which by virtue of (2.8) are equal to H and to UH, we see that (2.9) is equivalent to the relation

2. I SOMETRIC DILATIONS. D ILATION OF COMMUTANTS

65

This relation obviously follows from UH = [(U − T ) + T ]H ⊂ L ⊕ H and L = (U − T )H ⊂ UH ∨ H. Let us recall that R reduces U to a unitary operator R, the residual part of U. Because R ⊂ K+ , we also have R = U+ |R. Thus the decomposition (2.7) necessarily coincides with the Wold decomposition of K+ with respect to the isometric operator U+ . Consequently, L∗ = K+ ⊖ U+ K+ ,

R=

T

n≥0

U+n K+ =

T W

n≥0 k≥n

U k H.

(2.11)

By virtue of Theorem 1.2 and the definition (2.1) of R, we have R = {0} if and only if T ∗n → O (n → ∞). Decomposition (1.4) of K shows that U n L ⊥ U −m L∗ = U −(m+1) L∗ for n, m ≥ 0. Denoting by P L∗ the orthogonal projection of K onto M(L∗ )(= M(L∗ )) we have thus PL∗ M+ (L) ⊂ M+ (L∗ ).

(2.12)

If T is c.n.u., then it follows from (1.10) that (I − PL∗ )K = (I − PL∗ )M(L). With regard to (2.1) this implies (2.13)

R = (I − PL∗ )M(L). Summing up we have the following result.

Theorem 2.1. Let T be a contraction on H,U its minimal unitary dilation on K, and U+ its minimal isometric dilation on K+ (⊂ K). Then we have (cf. (2.1)), (cf. (2.5) and (2.7)),

K = M(L∗ ) ⊕ R K+ = M+ (L∗ ) ⊕ R = H ⊕ M+ (L) where L = (U − T )H

and L∗ = (I − UT ∗ )H

are subspaces of K+ , wandering for U+ (and hence for U), and R is the subspace of K+ that reduces U+ (and U) to the unitary part R of U+ . Moreover, we have L ∩ L∗ = {0} L∗

P M+ (L) ⊂ M+ (L∗ )

(cf. (2.9)), (cf. (2.12)),

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C HAPTER II. G EOMETRICAL AND S PECTRAL P ROPERTIES OF D ILATIONS

and, if T is completely nonunitary, R = (I − PL∗ )M(L)

(cf. (2.13)).

The subspace R reduces to {0} if and only if T ∗n → O as n → ∞.

2. Let T be as above, and let W be an isometry on a space G. As a first application of Theorem 2.1, more particularly of decomposition (2.5), we consider the connections between the solutions X and Y of the operator equations (a)

T X = XW

and (b)

U+Y = YW,

where X is a bounded operator from G to H, and Y is a bounded operator from G to K+ . Firstly, let us observe that every solution Y of (b) gives rise to a solution X of (a) by setting (c)

X = P+Y,

where P+ denotes the orthogonal projection of K+ into H. Indeed, this follows immediately from the relation T P+ = P+U+ ; see (I.4.2). We show that every X can be obtained this way, that is, for every X there exists a Y such that (c) holds. Actually, there can exist more than just one such Y , and as a consequence of (c) all of them obviously satisfy the inequality kX k ≤ kY k. We find a Y for which this inequality is, in fact, an equality, and for this Y it is obviously sufficient to prove that kY k ≤ kX ||. By reason of homogeneity it is possible to restrict our study to the case kX k = 1 (the case X = O is trivial: set Y = O). Then we have to find Y so that (d)

kY || ≤ 1.

Observe first that by virtue of decomposition (2.5) the general form of an operator Y from G to K+ satisfying (c) is: Y = X + B0 + U+ B1 + U+2 B2 + · · · ,

(2.14)

where each Bn is an operator from G to L. The additional condition (d) means that ∞

(kY gk2 =)kXgk2 + ∑ kBn gk2 ≤ kgk2 0

for g ∈ G.

(2.15)

2. I SOMETRIC DILATIONS. D ILATION OF COMMUTANTS

67

From (2.14) we deduce     ∞ ∞ U+Y − YW = U+ X + ∑ U+n+1 Bn − XW + ∑ U+n BnW 0

0

∞

= ∑ U+n (Bn−1 − BnW ) 0

with B−1 = U+ X − XW . Because of (a) we have B−1 = (U+ − T )X , and thus B−1 is an operator from G to L. In order that Y satisfy (b), it is therefore necessary and sufficient that the following equations hold. BnW = Bn−1 (n = 0, 1 . . .),

B−1 = (U+ − T )X.

(2.16)

To sum up: the form of an operator Y from G to K+ satisfying (b), (c), and (d) is given by (2.14), the operators Bn (from G to L) being subject to conditions (2.15) and (2.16). We construct such a sequence of operators Bn by recurrence. Suppose that, for an N ≥ 0, the operators Bn (n < N) are already determined so that they satisfy the conditions (e)N

sN (g) ≡ kXgk2 +

∑ kBn gk2 ≤ kgk2

0≤n
(g ∈ G)

and (f)N

BnW = Bn−1 (0 ≤ n < N),

B−1 = (U+ − T )X.

(For N = 0 these conditions are satisfied: s0 (g) ≡ kX gk2 ≤ kgk2 because kX k = 1, and condition (f)0 reduces to the equation defining B−1 .) Let us note that (e)N can be written in the equivalent form IG − X ∗ X − ∑ B∗n Bn ≥ O; 0≤n
let us denote the positive square root of this positive operator by DN . In order to determine the next operator in the sequence (i.e., BN ), observe first that kB−1 gk2 = k(U+ − T )Xgk2 = kX gk2 − kT X gk2 = kX gk2 − kXWgk2 , as a consequence of (a). In the case N ≥ 1 we obtain using (f)N that sN (W g) = kXW gk2 + ∑ kBnW gk2 0≤n
2

= kXW gk + kB−1gk2 +

∑

0≤m
kBm gk2;

(2.17)

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hence it follows by (2.17) and (e)N that sN (W g) = sN (g) − kBn−1gk2 ≤ kgk2 − kBN−1gk2 . The equality kgk = kW gk implies kBN−1 gk2 ≤ kW gk2 − sN (W g)

= kW gk2 − kXWgk2 −

∑ kBnW gk2 = kDN W gk2 .

0≤n
On the other hand, kXgk ≤ kgk = kW gk, therefore we obtain from (2.17) that kB−1 gk2 ≤ kW gk2 − kXWgk2 = kD0W gk2 . Thus we conclude that the inequality kBN−1 gk2 ≤ kDN W gk2

(2.18)

(g ∈ G)

holds in all cases (N ≥ 0). Hence we infer that there exists a contraction CN of DN W G into L such that BN−1 = CN DN W. (2.19) The contraction CN extends by continuity to the closure G1 of DN W G; it even extends to a contraction of the whole space G into L if one defines it, for example, to be O on the orthogonal complement G ⊖ G1 . With the contraction CN of G into L thus obtained, we define BN = CN DN . (2.20) The relation BN W = BN−1 is then obvious. Moreover, we have kBN gk2 = kCN DN gk2 ≤ kDN gk2 = kgk2 − kX gk2 −

∑ kBn gk2 ,

0≤n
and hence (2.15)N+1 holds. The operators Bn , defined in such a way by recurrence for every n ≥ 0, will satisfy (2.15) and (2.16), limit cases of (e)N and (f)N . Thus we have proved the following result. Proposition 2.2. For a contraction T on H and an isometry W on G, the general solution X of (a) is obtained by (c) from the general solution Y of (b); moreover, for given X one can choose Y so that kXk = kY k. 3. It is now easy to deduce the following more general theorem.

Theorem 2.3. Let T and T ′ be contractions on the Hilbert spaces H and H′ , and let U+ and U+′ be their minimal isometric dilations on the spaces K+ and K′+ , respectively. For every bounded operator X from H′ into H satisfying (a) T X = XT ′

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there exists a bounded operator Y from K′+ into K+ satisfying the conditions (b) (c) (c′ ) (d)

U+Y = YU+′ , X = P+Y |H′ , Y (K′+ ⊖ H′ ) ⊂ K+ ⊖ H, kXk = kY k.

Conversely, every bounded solution Y of (b) satisfying (c′ ) gives rise by (c) to a solution X of (a). Conditions (c) and (c′ ) together are equivalent to the condition (c′′ ) X ∗ = Y ∗ |H. Proof. Consider first a bounded operator Y from K′+ into K+ satisfying (b) and (c′ ). We show that the operator X to which it gives rise by (c) satisfies (a). Indeed, using relation (I.4.2) we get T X = T P+Y |H′ = P+U+Y |H′ = P+YU+′ |H′ = P+Y P+′ U+′ |H′ + P+Y (I − P+′ )U+′ |H′ .

Because P+′ U+′ |H′ = T ′ by the dilation property, and because P+Y (I − P+′ ) = O by virtue of (c′ ), we obtain (a). Consider next an arbitrary bounded solution X of (a). Multiplying in (a) by P+′ from the right and using the relation T ′ P+′ = P+′ U+′ ( cf. (I.4.2)), we obtain T X0 = X0U+′

with

X0 = X P+′ ;

X0 is an operator from K′+ into H. If we now apply Proposition 2.2 with W = U+′ , we obtain that there exists an operator Y from K′+ into K+ such that U+Y = YU+′ , P+Y = X0 ,

and kX0 k = kY k.

We obviously have X0 |H′ = X and kX0 k = kXk, thus it follows that Y satisfies (c) and (d). It also satisfies (c′ ) because P+Y (I − P+′ ) = X0 (I − P+′ ) = X P+′ (I − P+′ ) = O. Finally, the equivalence of conditions (c) and (c′ ) to (c′′ ) is straightforward. So the proof is complete. 4. Theorem 2.3 can be extended by replacing the conditions of minimality on U+ and U+′ with the weaker conditions U+∗ |H = T ∗

and U+′∗ |H′ = T ′∗ ,

respectively. Indeed, if these conditions hold then the spaces K+0 =

W

n≥0

U+n H,

K′+0 =

W

n≥0

U+′n H′

(2.21)

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are reducing U+ and U+′ respectively, moreover U+0 = U+ |K+0

′ and U+0 = U+′ |K′+0

are the minimal isometric dilations of T and T ′ (see the remark at the end of Sec. I.2). Therefore for a given operator X satisfying the property (a) above, we can apply Theorem 2.3 to obtain an operator Y0 from K′+0 into K+0 satisfying ′ U+0Y0 = Y0U+0 ,

Y0 (K′+0 ⊖ H′) ⊂ K+0 ⊖ H

and X = PHY0 |H,

kY0 k = kX k.

Defining Y = Y0 PK′ we obtain an operator from K′+ into K+ satisfying all the prop+0 erties (b), (c), (c′ ), (c′′ ), and (d) in Theorem 2.3. The proof of the fact that in the present more general case, the relations (b), (c), (c ′ ) imply (a) is identical to that given in the proof of Theorem 2.3.

3 The residual parts and quasi-similarities 1. We resume the study of the residual and ∗-residual parts of the minimal unitary dilation U of T , defined by (2.1) and (2.2). Let PH , PR , PR∗ denote the orthogonal projections of the space K onto the subspaces H, R, and R∗ , respectively. Proposition 3.1. For every h ∈ H we have PR h = lim U n T ∗n h, n→∞

PR∗ h = lim U −n T n h

(3.1)

kPR∗ hk = lim kT n hk,

(3.2)

n→∞

and consequently kPRhk = lim kT ∗n hk, n→∞

PH PR h = lim T n T ∗n h, n→∞

n→∞

PH PR∗ h = lim T ∗n T n h. n→∞

(3.3)

Proof. Relations (3.2) and (3.3) follow immediately from (3.1) because U is a unitary dilation of T . So it suffices to prove relations (3.1), or, by reason of symmetry, one of them, say the one concerning R∗ . The inequalities kT n+1 hk ≤ kT k · kT n hk ≤ kT n hk (n ≥ 0), show that the sequence {kT n hk} is nonincreasing, and hence is convergent. For 0 ≤ m ≤ n we have (U −n T n h,U −m T m h) = (U m−n T n h, T m h) = (T ∗n−m T n h, T m h) = (T ∗m T ∗n−m T n h, h) = (T ∗n T n h, h) = kT n hk2 ,

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and hence kU −nT n h − U −mT m hk2

= kU −nT n hk2 + kU −m T m hk2 − 2 Re (U −n T n h,U −m T m h) = kT n hk2 + kT m hk2 − 2kT n hk2 = kT m hk2 − kT n hk2.

Thus the convergence of the numerical sequence {kT n hk2 } implies the convergence of the sequence {U −n T n h} in K (n → ∞). Setting k = lim U −n T n h, n→∞

let us show that k = PR∗ h; that is, (a) k ⊥ M(L)

and (b) h − k ∈ M(L).

Now (a) means that k is orthogonal to U m L for every integer m, and this is indeed true because U −n T n h ⊥ U m L for n ≥ −m, as a consequence of the relation H ⊥ U m+n L (m + n ≥ 0) which follows from (1.4). As to (b), we have only to observe that h − U −nT n h

=U −1 (U − T )h + U −2(U − T )T h + · · · + U −n(U − T )T n−1 h ∈ M(L) and therefore

h − k = lim (h − U −nT n h) ∈ M(L). n→∞

This concludes the proof of Proposition 3.1. Proposition 3.2. If at least one point in the interior of the unit circle is not an eigenvalue of T , then PR H = R. (3.4) The analogous fact holds for the dual case. Proof. Assume to the contrary that (3.4) is false, so that there exists a nonzero k ∈ R, such that k ⊥ PR H, or equivalently, k ⊥ M(L∗ ) and k ⊥ H. Then by (1.4) we have k ∈ M+ (L) and hence k has an orthogonal expansion ∞

k = ∑ U n ln , n=0

∞

where ln ∈ L, ∑ kln k2 = kkk2 . n=0

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Because k 6= 0, at least one coefficient ln is nonzero; let lν be the first of these nonzero coefficients. Then we have ∞

U −ν −1 k = U −1 lν + ∑ U µ lν +µ +1 . µ =0

(3.5)

Because k belongs to R, U −ν −1 k = R−ν −1 k also belongs to R = K ⊖ M(L∗ ) and hence, in particular, U −ν −1 k ⊥ L∗ . On the other hand (1.4) implies U µ L ⊥ L∗ for µ ≥ 0, therefore we deduce from (3.5) that U −1 lν ⊥ L∗ and lν ⊥ UL∗ . Now, we obviously have lν ∈ L ⊂ H ⊕ L and, by (1.7), H ⊕ L = UL∗ ⊕ UH, so we conclude that lν ∈ UH. Thus there exists an h ∈ H such that lν = Uh; consequently PH lν = PHUh = T h. Because L ⊥ H, we have PH lν = 0; hence T h = 0. But lν 6= 0 implies h 6= 0, and this means that 0 is an eigenvalue of T . Let us now consider the contraction Ta = (T − aI)(I − aT ¯ )−1 with |a| < 1, and the analogous transform (U+ )a of U+ ; (U+ )a is the minimal isometric dilation of Ta (cf. Proposition I.4.3). An operator S is unitary if and only if its transform Sa is unitary, thus we conclude that the maximal subspace of K+ in which (U+ )a is unitary, coincides with the maximal subspace of K+ in which U+ is unitary, that is with R. Thus, applying the results already obtained to Ta instead of T , we conclude that if (3.4) does not hold then 0 is an eigenvalue of Ta and hence a is an eigenvalue of T . So if (3.4) does not hold, then every point in the interior of the unit circle is an eigenvalue of T . This concludes the proof. 2. It is convenient to introduce some further notions. (In what follows, H1 , H2 , . . . are assumed to be Hilbert spaces, but most of the notions introduced make sense for Banach spaces as well.) Definition 1. By an affinity from H1 to H2 we mean a linear, one-to-one, and bicontinuous transformation X from H1 onto H2 . Two bounded operators, say S1 on H1 and S2 on H2 , are said to be similar if there exists an affinity X from H1 to H2 such that XS1 = S2 X (and consequently X −1 S2 = S1 X −1 ). Definition 2. By a quasi-affinity from H1 to H2 we mean a linear, one-to-one, and continuous transformation X from H1 onto a dense linear manifold in H2 . (Thus X −1 exists on this dense domain, but is not necessarily continuous.1) If S1 and S2 are bounded operators, S1 on H1 and S2 on H2 , we say that S1 is a quasi-affine transform of S2 if there exists a quasi-affinity X from H1 to H2 such that X S1 = S2 X . The operators S1 and S2 are called quasi-similar if they are quasi-affine transforms of one another. Remark. Similarity is a rather strong relation, which preserves for example, the spectrum. Quasi-similarity does not have such strong implications. Nevertheless, it 1

Every one-to-one linear transformation A from H1 into H2 is invertible, the inverse transformation A−1 being a linear transformation with domain equal to the range of A. When A−1 is defined on the whole space H2 and is continuous (i.e., bounded), we say that A is boundedly invertible (or that it has a bounded inverse).

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73

will be clear from the results obtained in this book that quasi-affinity and quasisimilarity are very natural and useful concepts. Some of the first consequences of Definition 2 are listed in the following two propositions. Proposition 3.3. (1) If X is a quasi-affinity from H1 to H2 , and Y is a quasi-affinity from H2 to H3 , then Y X is a quasi-affinity from H1 to H3 . (2) If X is a quasi-affinity from H1 to H2 , then X ∗ is a quasi-affinity from H2 to H1 . (3) If X is a quasi-affinity from H1 to H2 , then |X | = (X ∗ X )1/2 is a quasi-affinity on H1 (i.e., from H1 to H1 ). Moreover, X · |X|−1 extends by continuity to a unitary transformation VX from H1 to H2 . Proof. A simple exercise left to the reader. Proposition 3.4. (1) If S1 is a quasi-affine transform of S2 and S2 is a quasi-affine transform of S3 , then S1 is a quasi-affine transform of S3 . (2) If S1 is a quasi-affine transform of S2 , then S2∗ is a quasi-affine transform of ∗ S1 . (3) If a unitary operator S1 on H1 is the quasi-affine transform of a unitary operator S2 on H2 , then S1 and S2 are unitary equivalent. Proof. (1) and (2) follow immediately from the corresponding parts of the preceding proposition. As to (3), let us observe first that (a) X S1 = S2 X (where X is a quasi-affinity from H1 to H2 ) implies, because S1 and S2 are unitary, that S2∗ X = S2−1 X = XS1−1 = XS1∗, and hence (b) X ∗ S2 = S1 X ∗ . From (a) and (b) one obtains |X|2 S1 = X ∗ XS1 = X ∗ S2 X = S1 X ∗ X = S1 |X |2 and, by iteration, |X |2n S1 = S1 |X |2n (n = 0, 1, . . .); hence p(|X|2 )S1 = S1 p(|X |2 ) for every polynomial p(x). Let {pn (x)} be a sequence of polynomials tending to |x|1/2 uniformly on the interval 0 ≤ x ≤ kXk2. Then pn (|X |2 ) converges (in the operator norm) to |X | so that we obtain as a limit the relation (c) |X|S1 = S1 |X|. From (a) and (c) it follows that S2VX |X | = S2 X = X S1 = VX |X|S1 = VX S1 |X|; because |X|H1 is dense in H1 it results that S2VX = VX S1 . By virtue of part (3) of the preceding proposition, VX is unitary, and thus S1 and S2 are unitarily equivalent. 3. Let us return to the subject of Subsec. 1 and deduce the following consequences of relations (3.1): U ∗ PR h = PR T ∗ h

and UPR∗ h = PR∗ T h

(h ∈ H).

Indeed, we have only to notice that U ∗ (limU n T ∗n h) = limU n−1T ∗n h = (limU n−1 T ∗n−1 )T ∗ h and

U(limU −n T n h) = limU −(n−1)T n h = (limU −(n−1)T n−1 )T h.

(3.6)

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Proposition 3.5. (i) If T h does not equal 0 and T ∗n h does not converge to 0 as n → ∞, for any nonzero h ∈ H, then X = PR |H is a quasi-affinity from H to R and we have R∗ X = X T ∗ ,

X ∗R = T X ∗ ,

(3.7)

so that R is a quasi-affine transform of T . (ii) If T ∗ h does not equal 0 and T n h does not converge to 0 as n → ∞, for any nonzero h ∈ H, then Y = PR∗ |H is a quasi-affinity from H to R∗ and we have R∗Y = Y T, (3.8) so that T is a quasi-affine transform of R∗ . (iii) If for no nonzero h ∈ H does either T n h or T ∗n h converge to 0 as n → ∞, then T is quasi-similar to R as well as to R∗ , which are, in this case, unitarily equivalent. Proof. Under the conditions of (i), the first of the relations (3.2) implies that X h 6= 0 for h 6= 0, and Proposition 3.2 implies XH = R. Thus X is a quasi-affinity from H to R and (3.7) follows from the first of the relations (3.6). Case (ii) is the dual of (i). In case (iii), the conditions of (i) and (ii) are simultaneously satisfied, and thus R is a quasi-affine transform of T and T is a quasi-affine transform of R∗ . By Proposition 3.4, R is then a quasi-affine transform of R∗ , and hence R∗ is unitarily equivalent to R. Hence we conclude that T is a quasi-affine transform of R too. This concludes the proof. 4. Part (iii) of Proposition 3.5 easily provides a characterization of those contractions T that are quasi-similar to a unitary operator. If quasi-similarity is replaced by the stronger relation of similarity we have the following results. Proposition 3.6. If T is a c.n.u. contraction, then the following conditions are equivalent. (a) T is similar to a unitary operator V ; that is, T = S−1V S holds with an affinity S. (b) The transformation X = PR |H from H to R is an affinity and so T is similar to R. (c) The transformation Q = PL∗ |M+ (L) is onto M+ (L∗ ) and boundedly invertible.

Furthermore, kX −1k = kQ−1 k ≤ kSkkS−1k holds.

In the proof of Proposition 3.6 we need the following lemma.

Lemma 3.7. Let M = A ⊕ B and M = X ⊕ Y

be two orthogonal decompositions of the Hilbert space M, and let PA and PB denote the orthogonal projections from M onto A and B, respectively. If PA X = A

and kPA xk ≥ ckxk (x ∈ X)

(3.9)

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75

with some positive constant c, then we also have (3.10)

PB Y = B and kPB yk ≥ ckyk (y ∈ Y). Proof. The inequality kPA xk2 ≥ c2 kxk2 = c2 [kPA xk2 + kPB xk2 ] implies C2 kPA xk2 ≥ kPB xk2

with C =

Thus the hypotheses (3.9) imply that the formula A(PA x) = PB x

p 1 − c2/c.

(x ∈ X)

defines an operator A from A into B, bounded by the constant C. The graph {a ⊕ Aa : a ∈ A} of A in M = A ⊕ B equals {PA x ⊕ PB x : x ∈ X}, and consequently X, its orthogonal complement {−A∗ b ⊕ b : b ∈ B} will be equal to Y. This implies that PB Y = B and PA y = −A∗ PB y for y ∈ Y. It follows that kPA yk ≤ CkPB yk,

kyk2 = kPA yk2 + kPB yk2 ≤ (1 + C2)kPB yk2 =

1 kPB yk2 , c2

and this concludes the proof of the relations (3.10). Proof (Proof of Proposition 3.6). Suppose T is a c.n.u. contraction on the space H, similar to a unitary operator V , so that T = S−1V S for some affinity S on H. Then T and T ∗ are boundedly invertible and we have for every integer n: T −n = S−1V −n S,

T ∗−n = S∗V n S∗−1 ,

and hence kT −n k ≤ k,

kT ∗−n k ≤ k,

with k = kSk kS−1k = kS∗ k kS∗−1k;

consequently, setting c = 1/k we obtain kT n hk ≥ ckhk,

kT ∗n hk ≥ ckhk (h ∈ H).

(3.11)

From the second inequality we deduce by (3.2) that kPRhk ≥ ckhk (h ∈ H),

(3.12)

where PR denotes the orthogonal projection from the space K (of the minimal unitary dilation U of T ) onto the subspace R (of the residual part R of U). Because 0 is not an eigenvalue of T , Proposition 3.2 asserts that PR H = R. The inequality (3.12) implies that PR H is closed, and we conclude that PR H = R.

(3.13)

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On account of (3.12) and (3.13), X = PR |H

(3.14)

is an affinity from H to R, and hence X ∗ is an affinity from R to H. Using (3.14) we deduce from Proposition 3.5 that X ∗ R = T X ∗ ; thus T is similar to R. Therefore, if a contraction is similar to a unitary operator, then it is similar in particular to the residual part of its minimal unitary dilation. Now the space K+ of the minimal isometric dilation of T admits the decompositions K+ = M+ (L∗ ) ⊕ R, K+ = M+ (L) ⊕ H. (3.15) Thus relations (3.12) and (3.13) imply, by virtue of Lemma 3.7, that

PL∗ M+ (L) = M+ (L∗ ) and kPL∗ lk ≥ cklk (l ∈ M+ (L)).

(3.16)

It follows that Q = PL∗ |M+ (L)

is a contraction from M+ (L) onto M+ (L∗ ), which is boundedly invertible with kQ−1 k ≤ 1/c. Conversely, if Q is invertible with kQ−1 k ≤ 1/c, then we infer by Lemma 3.7 that X = PR |H is invertible with kX −1k ≤ 1/c. Thus the proof is complete.

4 A classification of contractions The preceding results motivate, to some extent, the introduction of the following classes C·· of contractions. p 0 for all h 6= 0; T ∈ C0· if T n h → 0 for all h; T ∈ C1· if T n h → T ∈ C·0 if T ∗n h → 0 for all h; T ∈ C·1 if T ∗n h → p 0 for all h = 6 0. Furthermore, set Cαβ = Cα · ∩C·β

(α , β = 0, 1).

These special classes play an important role in the study of general contractions. To begin with, let us recall that, given a decomposition H = H1 ⊕ H2 ⊕ · · · ⊕ H p

(4.1)

of the Hilbert space H into the orthogonal sum of subspaces Hi (i = 1, . . . , p), to every bounded operator T in H there corresponds a matrix [Ti j ] (i, j = 1, . . . , p), whose entries Ti j are the bounded operators from H j to Hi defined by Ti j = Pi T |H j , where Pi denotes the orthogonal projection from H into Hi . It is obvious that if T is a contraction on H then Ti j is a contraction from H j into Hi . If, moreover, T is

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77

c.n.u., then all the diagonal entries Tii are also c.n.u. Indeed, if there exists in Hi a subspace M such that Tii |M is unitary, then for f ∈ M we have Tii f = Pi T f and k f k = kTii f k = kPi T f k ≤ kT f k ≤ k f k. Hence Tii f = T f so that T coincides on M with Tii ; thus T is unitary on M. This contradicts the assumption that T is c.n.u. unless M = {0}. Thus Tii is c.n.u. We say that the decomposition (4.1) generates a triangulation (or, rather, a superdiagonalization) of T if Ti j = O for i > j. This is equivalent to the set of conditions T H j ⊂ H1 ⊕ · · · ⊕ H j

for

j = 1, 2, . . . , p.

(4.2)

In this case the subspace H1 is therefore invariant for T and the matrices of the operators T n (n = 1, 2, . . .) are of the same type, with (T n )ii = (Tii )n . Theorem 4.1. Every contraction T on the space H has triangulations of the following types,     C0· ∗ C·1 ∗ (a) and (a∗ ) , O C·0 O C1·

where, in the diagonal, one has indicated the class of the respective operator only (with the operator O on the space {0} belonging to all these classes); the type of the entries denoted by ∗ is not specified. These triangulations are uniquely determined and are called the “canonical triangulations” of T . There also exists a triangulation of type   C01 ∗ ∗ ∗ ∗  O C00 ∗ ∗ ∗     (b)   O O C11 ∗ ∗  .  O O O C00 ∗  O O O O C10 Proof. Let us set

H1 = {h : h ∈ H, T n h → 0};

(4.3)

H1 is obviously a subspace of H, invariant for T . Set H2 = H ⊖ H1 . The decomposition H = H1 ⊕ H2 then yields a triangulation of T :   T ∗ T= 1 , where T1 = T |H1 , T2 = P2 T |H2 . O T2 Because T1n = T n |H1 , we have T1 ∈ C0· by the definition of H1 . We show that T2 ∈ C1· . To this end we use the second one of the relations (3.1), which asserts that Qh = lim U −n T n h n→∞

(h ∈ H),

(4.4)

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where we have written for simplification Q instead of PR∗ . It then follows for an arbitrary integer m ≥ 0 that Qh = lim U −(n+m) T n+m h n→∞

= lim (U −mU −n T n P2 T m h + U −(n+m)T n P1 T m h) n→∞

= U −m · lim U −n T n · P2 T m h = U −m QP2 T m h, n→∞

n

because lim T h1 = 0 for h1 = P1 T m h ∈ H1 . From this result we derive the inequalities kQhk ≤ kP2 T m hk (h ∈ H; m = 0, 1, . . .).

It then follows that if limm→∞ P2 T m h = 0 for an h ∈ H, then Qh = 0, and hence, by (3.2), limn→∞ T n h = 0 (i.e., h ∈ H1 ). Because T2m = P2 T m |H2 , we conclude that there exists no nonzero h ∈ H2 for which T2m h would converge to 0 as m → ∞; that is, T2 ∈ C1· . Thus the triangulation under consideration is of type (a). Now we show that if  ′  T ∗ T = 1 ′ , H = H′1 ⊕ H′2, T1′ = T |H′1 , T2′ = P2′ T |H′2 , O T2 is a triangulation of T of type (a), then it necessarily coincides with the triangulation just obtained. For this it suffices to show that H1 = H′1 .

(4.5)

Now for h ∈ H′1 we have T n h = T1′ n h → 0 because T1′ ∈ C0· ; hence it follows by the definition (4.3) of H1 that h ∈ H1 . Thus we have H′1 ⊂ H1 . Let now h ∈ H1 ⊖ H′1 . Then we have T n h → 0 because h ∈ H1 , and P2′ T n h = T2′ n h because h ∈ H′2 (P2′ denotes, of course, the orthogonal projection of H into H′2 ); thus T2′ n h → 0. As T2′ ∈ C1· , this implies h = 0. Hence H1 ⊖ H′1 = {0}, which proves (4.5). So we have proved that every contraction T on H has a unique triangulation of type (a). If we take the triangulation of type (a) for T ∗ and then interchange the order of the corresponding subspaces H1 and H2 , we obtain for T ∗ a matrix of type   C1· O ; ∗ C0· taking adjoints we obtain thus for T a matrix of type (a∗ ). Therefore the existence and uniqueness of the triangulation of type (a∗ ) follow from the same facts as for the triangulation of type (a). Let us observe now that if T ∈ C0· and if [Ti j ] is an arbitrary triangulation of T (i.e., Ti j = O for i > j), then Tii ∈ C0· for every i; in fact Tiin = Pi T n |Hi → O as n → ∞.

4. A CLASSIFICATION OF CONTRACTIONS

In particular, if

79



T ∗ T= 1 O T2 is the triangulation of T of type (a∗ ), then



T1 ∈ C0· ∩C·1 = C01 and T2 ∈ C0· ∩C·0 = C00 . On the other hand, if T ∈ C1· and if [Ti j ] is an arbitrary triangulation of T , then n h = T n h for h ∈ H. In particular, if T11 ∈ C1· , because T11   T1 ∗ T= O T2 is the triangulation of type (a∗ ), we have T1 ∈ C1· ∩C·1 = C11 and, of course, T2 ∈ C·0 . These results can be expressed by the self-explanatory formulas     C01 ∗ C11 ∗ C0· = , C1· = . (4.6) O C00 O C·0 One can derive from these the formulas   C00 ∗ C·0 = , O C10

  C0· ∗ C·1 = O C11

(4.7)

by interchanging the order of the subspaces of decomposition and by taking adjoints. Now, starting with the triangulation of type (a) and then applying formulas (4.6) and (4.7), one obtains    C01 ∗   ∗  O C00  C0· ∗   =  O C1· C11 ∗  O O C·0    C01 ∗ ∗  O C00      C11  ∗   = ,   C00 ∗  O O O C10

that is, a triangulation of type (b).

Let us recall that every contraction T of class C11 is quasi-similar to a unitary operator, indeed to the residual part R of the minimal isometric dilation of T ; see Proposition 3.5(iii). Let   T ∗ T= 1 O T2

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be a triangulation of type

  C·1 ∗ T= O C·0

corresponding to the decomposition H = H1 ⊕ H2 . It is obvious that H2 is the largest invariant subspace for T ∗ where the compression of T is of class C·0 . Proposition 4.2. With the above notation, H1 is the largest invariant subspace for T where the restriction of T is of class C·1 . Proof. Let H′ ⊂ H be invariant for T , and assume that T ′ = T |H′ is of class C·1 . There is then a decomposition H = H′ ⊕ H′1 ⊕ H′2 relative to which we have   ′ T ∗ ∗ T =  O T1′ ∗  O O T2′ with T1′ ∈ C·1 and T2′ ∈ C·0 . We show that the restriction  ′  T ∗ T ′′ = O T1′

of T to H′ ⊕H′1 is of class C·1 . This implies that H1 = H′ ⊕H′1 , and therefore H′ ⊂ H1 , thus establishing the maximality of H1 . Assume that we have limn→∞ kT ′′∗n (h′ ⊕ h′1 )k = 0 for some h′ ∈ H′ and h′1 ∈ H′1 . Writing ′ T ′′∗n (h′ ⊕ h′1 ) = kn′ ⊕ k1n

(n ≥ 1)

′ ∈ H′ , we observe first that k′ = T ′∗n h′ , and therefore h′ = 0 with kn′ ∈ H′ and k1n n 1 ′ = T ′∗n h′ , and therefore h′ = 0 because because T ′ is of class C·1 . It follows that k1n 1 1 1 T1′ is also of class C·1 .

5 Invariant subspaces and quasi-similarity 1. To begin with, let us recall the definition of the notion of invariant subspace and add the definition of some further related notions. Definition. Let T be a bounded operator on H, and let L be a subspace of H. (a) L is said to be invariant for T if T L ⊂ L. (b) L is said to be regular for T if T L = L. (c) L is said to be hyperinvariant for T if it is invariant for every bounded operator which commutes with T . From the definition it follows that if {Lα } is a system of invariant (hyperinvariant) subspaces for T , then T W Lα and Lα α

α

are also invariant (hyperinvariant) subspaces for T .

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81

It is also obvious that if U is a unitary operator on H, then L is regular for U if, and only if, it reduces U. In other words, L is regular for U if, and only if, PL (the orthogonal projection of H onto L) commutes with U. Let us prove, similarly, that L is hyperinvariant for the unitary U if, and only if, PL commutes with all the bounded operators that commute with U. In fact, if A commutes with PL , then AL = APL H = PL AH ⊂ PL H = L; thus if PL commutes with all these A then L is hyperinvariant for U. Conversely, if L is hyperinvariant for U, then APL = PL APL for every A commuting with U. Then A∗ also commutes with U (in fact, AU = UA implies U ∗ A = U ∗ AUU ∗ = U ∗UAU ∗ = AU ∗ , A∗U = UA∗ ), so we also have A∗ PL = PL A∗ PL and hence APL = PL APL = (PL A∗ PL )∗ = (A∗ PL )∗ = PL A. By virtue of these remarks, if L is hyperinvariant for U then its orthogonal complement is also hyperinvariant for U. After these preliminaries we prove the following result. Proposition 5.1. Let T be a bounded operator on the space H, quasi-similar to a unitary operator U on the space K. With every subspace L of K, which is hyperinvariant for U, we can associate a subspace q(L) of H, regular and hyperinvariant for T , so that we have: (a) q({0}) = {0},

(b) q(K) = H,

(c) q(L) ⊂ q(L′ ) if L ⊂ L′ , (e)

T α

q(Lα ) = {0} if

T α

(d) q(L) 6= q(L′ ) if L 6= L′ ,

Lα = {0},

(f)

W α

q(Lα ) = q(L) if

W α

Lα = L.

Proof. By our hypothesis there exists a quasi-affinity X from K to H and a quasiaffinity Y from H to K, such that T X = XU

and UY = Y T.

(5.1)

Let L be a subspace of K, hyperinvariant for U. We set a(L) = X L and b(L) = {h : h ∈ H,Y h ∈ L};

(5.2)

clearly, a(L) and b(L) are subspaces of H. Let A be a bounded operator on H, commuting with T . Using relations (5.1) we see that U(YAX ) = (UY )(AX) = (Y T )(AX) = Y (TA)X = Y (AT )X

(5.3)

= (YA)(T X ) = (YA)(XU) = (YAX )U; L being hyperinvariant for U, (5.3) implies (YAX)L ⊂ L.

(5.4)

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By virtue of definitions (5.2), we deduce from (5.4) that (5.5)

Aa(L) ⊂ b(L). Now we set q(L) =

W A

(5.6)

Aa(L),

where A runs over the set of bounded operators on H that commute with T . In particular, the identity operator on H belongs to this set, thus we have a(L) ⊂ q(L); on the other hand (5.5) implies that q(L) ⊂ b(L). We conclude that (5.7)

a(L) ⊂ q(L) ⊂ b(L).

By its definition (5.6), q(L) is obviously hyperinvariant for T . Moreover it is regular for T . In fact, the equality UL = L implies Ta(L) = T X L = T X L = XUL = X L = a(L)2 and, for A commuting with T , TAa(L) = ATa(L) = ATa(L) = Aa(L); hence: T q(L) =

W A

TAa(L) =

W A

Aa(L) =

W A

Aa(L) = q(L).

From q(L) it is possible to retrieve L by the formula Y q(L) = L.

(5.8)

Y XL = Y X L = Ya(L) ⊂ Y q(L) ⊂ Y b(L) ⊂ L;

(5.9)

In fact, (5.2) and (5.7) imply

on the other hand, because Y X commutes with U (apply (5.3) with A = I), Y X also commutes with PL , and hence Y XL = Y XPL K = PLY XK = PLY X K = PL K = L.

(5.10)

Thus from (5.9) and (5.10) we deduce (5.8). It remains to establish properties (a)–(f). (a) For L = {0} we have b(L) = {0}, because Y h = 0 implies h = 0. In view of (5.7) it follows that q({0}) = {0}. (b) For L = K we have a(K) = XK = H; in view of (5.7) this implies q(K) = H. 2

Here and in the sequel we make use of the following obvious fact: If A is an arbitrary set of points in a topological space M, and S is a continuous transformation of M into a topological space M, then SA = SA.

5. I NVARIANT SUBSPACES AND QUASI - SIMILARITY

(c) (d) (e) (f)

83

If L ⊂ L′ , then a(L) ⊂ a(L′ ), and hence by (5.6) also q(L) ⊂ q(L′ ). This follows T immediately from (5.8). T T T α q(Lα ) ⊂ α b(Lα ) ⊂ b( α Lα ) = b({0}) = {0} if α Lα = {0}. W Observe first that if each Lα is hyperinvariant for U then so is L = α Lα .

Now by (5.6) and (5.2), q(L) =

W A

A a(L) =

W A

AX

W α

Lα =

W

A,α

AXLα =

WW α A

A XLα =

W α

q(Lα ).

This finishes the proof of Proposition 5.1. It is to be remarked that if E(σ ) is the spectral measure corresponding to the unitary operator U (defined for the Borel subsets σ of the unit circle) then E(σ ) commutes with U and with all the bounded operators commuting with U (cf. [Func. Anal.] Sec. 109); hence the subspaces L(σ ) = E(σ )K are all hyperinvariant for U. Thus the preceding proposition has the following corollary. Corollary 5.2. If a bounded operator T on the complex Hilbert space H is quasisimilar to a unitary operator U, then there exist at least as many nontrivial subspaces of H, regular and hyperinvariant for T , as there are values different from O and I of the spectral measure E(σ ) corresponding to U. 2. The above results apply in particular to contractions of class C11 , because as remarked at the end of Sec. 4 these are quasi-similar to unitary operators. Actually, we can prove this property for a larger class of operators: Proposition 5.3. Let T be a power-bounded operator on H, that is, such that kT n k ≤ M (n = 1, 2, . . .). Suppose that for every nonzero h (∈ H) neither T n h nor T ∗n h converges to 0 as n → ∞. Then T is quasi-similar to a unitary operator. Proof. Observe first that, under the conditions stated, inf kT n hk = µ (h) > 0

n≥0

for all

h 6= 0.

In fact, µ (h) = 0 means that for every ε > 0 there exists an integer n0 = n0 (h, ε ) such that kT n0 hk < ε /M; hence we obtain kT n hk = kT n−n0 T n0 hk ≤ MkT n0 hk < ε

for n ≥ n0 .

Thus T n h → 0 as n → ∞, and for h 6= 0 this contradicts one of our hypotheses.

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We shall now use a Banach generalized limit. This is a linear functional defined on the space of bounded numerical sequences {cn }n≥0, say L{cn }, with the following properties: L{cn } ≥ 0 if cn ≥ 0 (n = 0, 1, . . .),

L{1} = 1,

L{cn+1 } = L{cn }.

For any h, k ∈ H let us set hh, ki = L{(T n h, T n k)}; this is a bilinear form on H such that ( ≥ L{µ 2 (h)} = µ 2 (h) · L{1} = µ 2 (h), hh, hi = L{kT n hk2 } ≤ L{M 2 khk2 } = M 2 khk2 · L{1} = M 2 khk2, and

hT h, T ki = L{(T n+1 h, T n+1 k)} = L{(T n h, T n k)} = hh, ki.

It follows that there exists a self-adjoint operator A on H such that hh, ki = (Ah, k)

(h, k ∈ H),

0 < (Ah, h) ≤ M 2 kh||2

(h ∈ H; h 6= 0),

(5.11)

and (AT h, T k) = (Ah, k)

(h, k ∈ H).

(5.12)

Now (5.11) implies Ah 6= 0 for h 6= 0, and the same is then true for the positive selfadjoint square root X of A. Hence X is a quasi-affinity on H. By (5.12) we have for every h ∈ H kXT hk2 = (X 2 T h, T h) = (AT h, T h) = (Ah, h) = kX hk2. In particular kXT X −1 kk = kkk for the elements k in the domain of X −1 . This domain being dense in H, XT X −1 extends by continuity to an isometry U on H such that X T = UX.

(5.13)

Now our hypotheses assure in particular that T ∗ h 6= 0 for h 6= 0, and hence T H = H. So we derive from (5.13): UH = UX H = UXH = X T H = XT H = X H = H; thus the isometry U is actually unitary on H. The hypotheses of our proposition being symmetrical in T and T ∗ , the results already obtained are also valid for T replaced by T ∗ . So we obtain that there exist a quasi-affinity Y on H and a unitary operator V on H, such that Y T ∗ = VY , and hence T Z = ZW,

(5.14)

6. S PECTRAL RELATIONS

85

where Z = Y ∗ is also a quasi-affinity and W = V ∗ is also unitary on H. From this point on, the proof proceeds similarly to that of part (iii) of Proposition 3.5. In fact, (5.13) and (5.14) imply that W is a quasi-affine transform of U, so by Proposition 3.4, U and W are unitarily equivalent. Consequently, by virtue of (5.13) and (5.14), T and U are quasi-similar. Due to Proposition 5.3, the results concerning hyperinvariant subspaces, obtained in the preceding subsection, apply in particular to the operators T just considered. Moreover, we can prove the following theorem: Theorem 5.4. Let T be a power-bounded operator on the complex Hilbert space H with dim H > 1, such that neither T n nor T ∗n converge (strongly) to O as n → ∞. Then either T = cI with |c| = 1, or there exists a nontrivial subspace of H, hyperinvariant for T . Proof. We distinguish three cases. Case 1: There exists a nonzero h0 in H such that T n h0 → 0. It is easy to see that L = {h : h ∈ H, T n h → 0} is a subspace of H, hyperinvariant for T . L contains h0 , and hence L 6= {0}. On the other hand, L 6= H, for the contrary would imply T n → O. Thus L is a nontrivial subspace of H. Case 2: There exists a nonzero h0 in H such that T ∗n h0 → 0. If we set L = H ⊖ {h : h ∈ H,

T ∗n h → 0},

then L is again a nontrivial subspace of H, hyperinvariant for T . Case 3: There is no nonzero h ∈ H for which T n h or T ∗n h converges to 0 as n → ∞. By virtue of Proposition 5.3, T is then quasi-similar to a unitary operator U on H. As dim H > 1, the spectral measure of U has values different from O and I unless U has a one-point spectrum {c}, |c| = 1. In the latter case U = cI, which implies T = cI. Thus if T is not of this form then by Corollary 5.2 there exists a nontrivial subspace of H, hyperinvariant for T . The proof is complete.

6 Spectral relations 1. To begin with let us recall that a bilateral shift has no nonzero invariant vector; see Sec. I.2. Let us consider a contraction T on the space H and its minimal unitary dilation U on the space K; let L and L∗ be the corresponding wandering subspaces defined by (1.2). Let f be an element of K invariant for U. Because M(L) and M(L∗ ) reduce U, the orthogonal projections of f to these subspaces, say f ′ and f ′′ , are also invariant for U. But these subspaces reduce U to bilateral shifts, so we must have f ′ = 0, f ′′ = 0. Thus f is orthogonal to M(L) and to M(L∗ ). Consequently, by (1.4), f belongs to H and so we have T f = PHU f = f .

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Conversely, if T h = h for an h ∈ H, then we also have Uh = h because of the relation kUh − hk2 = kUhk2 + khk2 − 2Re(Uh, h)

(6.1)

2

= 2khk − 2Re(T h, h)

= 2Re(h − T h, h) ≤ 2kh − Thk · khk, valid for every element h of H and for every isometric operator U such that T = pr U. So we have proved that T and U have the same invariant vectors. If we apply this result to cT instead of T , with |c| = 1, then, as the minimal unitary dilation of cT equals cU, we obtain the following result. Proposition 6.1. Let T be a contraction and let U be its minimal unitary dilation. Every eigenvalue of T of modulus 1 is also an eigenvalue of U, and conversely. The corresponding eigenvectors are the same for T and for U. Let us consider now any c such that T − cI is boundedly invertible. Let k be an element of the subspace R that reduces U to its residual part R (cf. Sec. 2). Owing to the relation    T n T n T L m n R= U H⊕ U K+ = U (H ⊕ M+ (L)) = U L n≥0

n≥0

n≥0

m≥n

(cf. (2.11) and (2.5)), k can be expanded for every n ≥ 0 into a series of the form ∞

k = U n hn + ∑ U m lm , m=n

where U n hn is the orthogonal projection of k into U h H, and U m lm is the orthogonal projection of k into U m L. We have ∞

kkk2 ≥ ∑ kU m lm k2 , 0

and hence

consequently

∞

∞ m 2

∑ U lm = ∑ kU m lm k2 → 0

m=n

m=n

(n → ∞);

k = lim U n hn . n→∞

This implies k(R − cIR)kk = k(U − cI)kk = lim k(U − cI)U n hn k = lim kU n (U − cI)hnk = = lim k(U − cI)hn k ≥ limkPH (U − cI)hnk = limk(T − cI)hnk ≥ ≥ C limkhn k = C limkU n hn k = Ckkk,

6. S PECTRAL RELATIONS

87

where C stands for k(T − cI)−1 k−1 . Because R is unitary, one concludes that R− cIR is boundedly invertible, and k(R − cIR)−1 k ≤ 1/C. So we have proved the following result. Proposition 6.2. For each value c such that T − cI is boundedly invertible R − cIR is also boundedly invertible and k(R − cIR)−1 k ≤ k(T − cI)−1 k; R denotes here the residual part of the minimal unitary dilation of the contraction T. If T is not itself unitary then at least one of the wandering subspaces L, L∗ is different from {0}, and hence there exists a nonzero subspace of K that reduces U to a bilateral shift, or, in other words, U contains a bilateral shift. Instead of the whole space K let us now consider the subspaces M(h) =

∞ W

n=−∞

U n h,

M+ (h) =

∞ W

n=0

U n h,

M− (h) =

∞ W

n=0

U −n h

(6.2)

generated by a nonzero element h of H. Suppose that M+ (h) and M− (h) reduce U; then M+ (h) ∩ M− (h) also reduces U. Because M+ (h) is contained in K+ = H ⊕ L ⊕ UL ⊕ · · · (cf. (2.5)) and, similarly, M− (h) is contained in K− = H ⊕ L∗ ⊕ U −1L∗ ⊕ · · · , the space M+ (h) ∩ M− (h) is contained in K+ ∩ K− = H. On the other hand, M+ (h) ∩ M− (h) contains in particular the element h. Thus, H contains a subspace reducing U and containing h; as T = pr U, T coincides on this subspace with the unitary operator U. We conclude that h belongs to the subspace H0 of H on which T is unitary. If T is c.n.u., then H0 = {0} so it cannot contain the nonzero h. Hence, in this case, M+ (h) and M− (h) cannot both reduce U so that at least one of the subspaces M+ (h) ⊖ UM+ (h) and M− (h) ⊖ U −1 M− (h) is different from {0}. But this implies that there exists in M(h) a nonzero subspace that is wandering for U. In other words, the part of U in M(h) contains a bilateral shift. Let us sum up. Proposition 6.3. For a nonunitary contraction T , the minimal unitary dilation U always contains a bilateral shift. If T is completely nonunitary, then for any nonzero element h of H the restriction of U to M(h) also contains a bilateral shift. 2. All the above considerations relate to real as well as to complex Hilbert space. For complex Hilbert space we can complete these results in the following manner. InR a complex space, every unitary operator U has a spectral representation U = 02π eit dEt by means of a spectral family {Et } (0 ≤ t ≤ 2π ). Let E(σ ) be the

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corresponding spectral measure defined for the Borel subsets σ of the unit circle C = {z : |z| = 1}; thus in particular: E((eit1 , eit2 ]) = Et2 − Et1

(0 ≤ t1 < t2 ≤ 2π ).

Let us observe that if A is a wandering subspace for U, then we have for a ∈ A ( Z 2π 0 (n 6= 0), int n e d(Et a, a) = (U a, a) = 2 kak (n = 0); 0 by the uniqueness theorem for the Fourier–Stieltjes series this implies (Et a, a) = (t/2π )kak2 and therefore kE(σ )ak2 = (E(σ )a, a) = m(σ ) · kak2,

(6.3)

where m(σ ) denotes the normalized Lebesgue measure on C. So, in particular, m(σ ) = 0 implies E(σ )a = 0 and, because E(σ ) commutes with U, also E(σ )U n a = 0 for all n. Thus m(σ ) = 0 implies E(σ ) f = 0 for all f ∈ M(A). Now let us consider a c.n.u. contraction and let U be its minimal unitary dilation, with the spectral measure E(σ ). From what precedes we obtain that m(σ ) = 0 implies E(σ ) f = 0 for all f ∈ M(L) as well as for all f ∈ M(L∗ ), hence also for all f ∈ M(L) ∨ M(L∗ ) = K. So m(σ ) = 0 implies E(σ ) = O. Conversely, E(σ ) = O implies m(σ ) = 0. It even suffices to suppose that E(σ )h = 0 for a nonzero h in H. In fact, E(σ )h = 0 implies E(σ ) f = 0 for all f belonging W n to M(h) = ∞ −∞ U h, in particular for a nonzero wandering vector a for U, the existence of which has been stated in Proposition 6.3. Thus, (6.3) implies m(σ ) = 0. So we have proved the following result. Theorem 6.4. For a completely nonunitary contraction T on the space H, the spectral measure E(σ ) of U is equivalent to the normalized Lebesgue measure m(σ ) on C, that is, if σ is a Borel set on C for which one of these measures is zero then so is the other. Moreover, the scalar-valued measures

µh (σ ) = (E(σ )h, h)

(h ∈ H, h 6= 0)

(6.4)

are also equivalent to Lebesgue measure. The fact that µh (σ ) is absolutely continuous with respect to Lebesgue measure implies that the nondecreasing function (Et h, h) is the integral of its derivative fh (t) =

d (Et h, h), dt

which exists almost everywhere. This result can be completed as follows:

6. S PECTRAL RELATIONS

89

Proposition 6.5. Under the conditions of the preceding proposition, we have log fh (t) ∈ L(0, 2π ) for every nonzero h in H. Proof. We make use of the following theorem of Szeg˝o (cf. H OFFMAN [1], p. 49): Let f (t) be a nonnegative, real valued, Lebesgue-integral function on (0, 2π ). Set d( f ) = inf p

1 2π

Z 2π 0

|1 − p(eit )|2 f (t)dt,

where p runs over the class A0 of polynomials p(z) such that p(0) = 0. Then  Z 1 2π  exp log f (t)dt if log f (t) ∈ L(0, 2π ), d( f ) = 2π 0  0 if log f (t) ∈ / L(0, 2π ).

By virtue of this theorem our proposition means that d( fh ) > 0 for all nonzero h in H. Let us suppose the contrary, thus d( fh ) = 0 for some h 6= 0. By virtue of the relation Z 2π 0

|1 − p(eit )|2 fh (t) dt =

Z 2π 0

|1 − p(eit )|2 d(Et h, h) = kh − p(U)hk2,

h can then be approximated, as closely as we wish, by elements of the form p(U)h W n with p ∈ A0 , that is, h is contained in ∞ n=1 U h = UM+ (h). From this it follows that UM+ (h) = M+ (h), and hence M+ (h) reduces U. Furthermore, Z 2π 0

it

2

|1 − p(e )| d(Et h, h) =

Z 2π 0

|1 − p˜(e−it )|2 d(Et h, h) = kh − p˜(U −1 h)k2 ,

where p˜(z) = p(¯z) ∈ A0 . W

−n −1 −1 Thus d( fh ) = 0 implies that h ∈ ∞ n=1 U h = U M− (h). Thus U M− (h) = M− (h), M− (h) = UM− (h), and consequently M− (h) also reduces U. But we have shown in the proof of Proposition 6.3 that if T is c.n.u. and if h 6= 0, then M+ (h) and M− (h) cannot both reduce U. Thus the assumption d( fh ) = 0 leads to a contradiction. It follows that d( fh ) > 0, thus log fh (t) ∈ L(0, 2π ).

Corollary 6.6. For every nonunitary contraction T the spectrum of the minimal unitary dilation U of T is the whole unit circle C. Proof. The operator T has a nontrivial c.n.u. part T (0) . It follows from Theorem 6.4 that the spectrum of the minimal unitary dilation U (0) of T (0) coincides with C. Because σ (U) ⊃ σ (U (0) ), one also has σ (U) = C. To conclude this section let us prove the following proposition.

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Proposition 6.7. Let T be a completely nonunitary contraction, and suppose that the intersection of the spectrum of T with the unit circle C has Lebesgue measure 0. Then K = M(L) = M(L∗ ) and so T is of class C00 . Proof. Let us consider the decomposition K = M(L∗ ) ⊕ R of the dilation space (cf. Sec. 2), and the corresponding decompositions of U and of the spectral measure E(σ ) of U: ′ U = U ′ ⊕ R, E U(σ ) = E U (σ ) ⊕ E R(σ ).

By virtue of Proposition 6.2 the spectrum σR of R is included in the spectrum of T , and because σR is situated on C it follows from our hypothesis that m(σR ) = 0. Now, the spectral measure of U is absolutely continuous with respect to Lebesgue measure (cf. Theorem 6.4), thus we have E U(σR ) = O and E R (σR ) = E U(σR )|R = O. On the other hand, E R (σR ) is the identity operator on R, thus we must have that R = {0} and K = M(L∗ ). The same reasoning, when applied to T ∗ instead of T , yields K = M(L). By Theorem 1.2 one then obtains that T ∈ C00 . This concludes the proof.

7 Spectral multiplicity 1. For any unitary operator U on the (real or complex) Hilbert space K, and for any nonempty subset S of K, let us denote: M(S) =

∞ W

n=−∞

U n S;

M(S) is a subspace reducing U. Thus the orthogonal projection of K onto M(S), which we denote by PS , commutes with U. This notation generalizes that already used in the particular case that S is a wandering subspace for U, or a single vector k; in the latter case we write M(k) and Pk instead of M({k}) and P{k} . Because PS commutes with U, we have (I − PS)M(S′ ) = M((I − PS )S′ ) for arbitrary S, S′ . Let S = S1 ∪ S2 . We have then M(S) = M(S1 ) ∨ M(S2 ); consequently M(S) = M(S1 ) ⊕ (I − PS1 )M(S) = M(S1 ) ⊕ M((I − PS1 )S). Hence M(S1 ∪ S2 ) = M(S1 ) ⊕ M(S′2 )

where S′2 = (I − PS1 )S2 ;

(7.1)

here we have used the fact that S1 ⊂ M(S1 ) and hence (I − PS1 )S1 = {0}. For any given G 6= {0} there exists a maximal system Σ of nonzero elements of M(S), such that M(k) ⊥ M(k′ ) if k, k′ ∈ Σ , k 6= k′ . This follows by Zorn’s lemma.

7. S PECTRAL MULTIPLICITY

91

We have then M(S) =

L

k∈Σ

M(k).

(7.2)

Indeed, otherwise one could increase the system Σ by at least one more nonzero element. Lemma 7.1. Let S be a subspace of K, of dimension d ≥ 1. Then it is possible to choose the decomposition (7.2) so that the number of terms does not exceed d. Proof. If d is infinite (countable or not), this condition is satisfied automatically. In fact, we then have d ≤ dim M(S) ≤ ℵ0 · dim S = ℵ0 · d = d, and hence d = dim M(S); on the other hand, because dim M(k) ≥ 1 for every k ∈ S, the number of terms in (7.2) cannot exceed dim M(S). This completes the proof for an infinite d. When d is finite we proceed by recurrence. For d = 1 our assertion is obvious. We suppose that it holds for d smaller than some integer N(≥ 2), and we want to show that it also holds for d = N. Let S be any subspace of K of dimension N. Choose in S a nonzero element k0 and denote by S0 the subspace of S formed by the elements orthogonal to k0 . By virtue of (7.1) we have M(S) = M({k0 } ∪ S0 ) = M(k0 ) ⊕ M(S′0 )

(7.3)

with S′0 = (I − Pk0 )S0 . We have dim S0 = N − 1, thus S′0 is a subspace of dimension ≤ N − 1. By hypothesis there exists therefore a decomposition M(S′0 ) =

r L

i=1

M(ki ),

(7.4)

where r ≤ N − 1. By virtue of (7.3) and (7.4) we have M(S) =

r L

i=0

M(ki ),

the number of terms being r + 1, and hence ≤ N. This concludes the proof of the lemma. 2. Let us suppose now that U is the minimal unitary dilation of a c.n.u. contraction T . According to (1.10) we have K = M(L) ∨ M(L∗ ) = M(L ∪ L∗ ),

(7.5)

where L and L∗ are the wandering subspaces for U defined by (1.2). By virtue of (7.1) and (7.5) we have the decomposition K = M(L) ⊕ M(S),

where

S = (I − PL )L∗ ;

note that dim L = dT and dimS ≤ dim L∗ = dT ∗ .

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If we choose in L a complete orthonormal system Σ ′ = {l}; then M(L) =

L

l∈Σ ′

M(l),

where the number of terms is equal to dT . Let us observe that the vectors l ∈ L are wandering for U (i.e., U n l ⊥ U m l if n 6= m). On the other hand, there exists, according to the lemma, a decomposition M(S) =

L

k∈Σ ′′

M(k),

where the number of terms does not exceed dT ∗ . Thus we obtain the decomposition K=

L

a∈Σ

M(a)

(Σ = Σ ′ ∪ Σ ′′ ),

where the number of terms does not exceed dT + dT ∗ , and for at least dT terms the corresponding vector a is wandering for U. As (7.5) is symmetrical in L and L∗ , we can repeat this reasoning with the roles of L and L∗ interchanged. Thus, after setting dmax = max{dT , dT ∗ }, we arrive at the following result. Proposition 7.2. If U is the minimal unitary dilation, on the space K, of a completely nonunitary contraction T , then there exists a decomposition of K of the form K=

L α

M(aα )

(aα ∈ K, aα 6= 0),

(7.6)

with at most dT + dT ∗ terms, and where at least dmax of the vectors aα are wandering for U. 3. In the rest of this section we suppose that the space H of the c.n.u. contraction T is complex; then so is the space K of the minimal unitary dilation U of T . Let {Et }0≤t≤2π be the spectral family associated with U; by Theorem 6.4, Et is an absolutely continuous function of t. For an arbitrary a ∈ K and for all integers m, n one then has (U m a,U n a) =

Z 2π 0

ei(m−n)t d(Et a, a) =

where

Z 2π 0

ei(m−n)t p(t) dt,

d (Et a, a). dt It follows that for any finite linear combination of elements of the form U n a (n = 0, ±1, ±2, . . .):

2

2 Z 2π



∑ cn U n a = ∑ cn eint p(t) dt.

n

n 0 p(t) =

7. S PECTRAL MULTIPLICITY

Thus

93

∑ cnU n a → ∑ cn eint [2π p(t)]1/2 n

n

is an isometric transformation of a linear manifold, dense in M(a), onto a linear manifold M of the space L2 (Ω ), where

Ω = {t : t ∈ (0, 2π ), p(t) > 0}, and the measure considered is always the normalized Lebesgue measure m(σ ). Now it is easy to prove that M is dense in L2 (Ω ). Consequently, the above isometric transformation extends by continuity to a unitary one, from M(a) onto L2 (Ω ); let us denote it by Φ . Let us observe that if a is a nonzero wandering vector for U, then, by virtue of (6.3), p(t) = (1/2π )kak2 so that in this case Ω = (0, 2π ). Let us also observe that the restriction of U to M(a) is transformed by Φ into the operator U ×(Ω ) on L2 (Ω ) defined by (U ×(Ω )u)(t) = eit u(t), that is, the operator of multiplication by eit on L2 (Ω ). Using the sign ∼ to indicate unitary equivalence of operators, we deduce from Proposition 7.2 the following Proposition 7.3. Let T be a completely nonunitary contraction on the complex Hilbert space H. For its minimal unitary dilation U we have U∼

L × U (Ωα ), α

(7.7)

where the sets Ωα ⊂ (0, 2π ) are measurable, the number of the terms does not exceed dT + dT ∗ , and for at least dmax terms the set Ωα coincides with the whole interval (0, 2π ). 4. Here are some more or less immediate consequences of the preceding propositions. Theorem 7.4. Let T be a completely nonunitary contraction on the complex space H, and let U be its minimal unitary dilation on K. (a) If dmax is infinite, then U is a bilateral shift of multiplicity dmax . The same is true if dim H is finite. If dim H > ℵ0 , then dmax is always infinite, and indeed, we have then dmax = dim H. (b) There always exists a (not necessarily minimal) unitary dilation U of T that is a bilateral shift of multiplicity not exceeding dT + dT ∗ . Proof. The case when dim H is finite is simple. As T is c.n.u., its spectrum lies entirely in the interior of the unit circle, and hence it follows that T n → O and T ∗n → O as n → ∞. In view of Theorem 1.2, U is then a bilateral shift of multiplicity equal to dT and to dT ∗ (hence dmax = dT = dT ∗ ).

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In general we have dmax ≤ dim H ≤ dim K ≤ dim M(L) + dim M(L∗ ) = ℵ0 · dT + ℵ0 · dT ∗ = ℵ0 · dmax ; if dim H > ℵ0 , then ℵ0 · dmax > ℵ0 , and hence dmax > ℵ0 . Consequently, dmax = ℵ0 · dmax and thus dmax = dim H. To conclude the proof of (a) it remains to show that if dmax is infinite, then U is a bilateral shift of multiplicity dmax . To this end we begin with relation (7.7). Let A denote the set of subscripts α for which Ωα coincides with the whole interval (0, 2π ), and let B be the set of the remaining subscripts α . For the cardinal numbers |A|, |B| of A and B, respectively, we have dmax ≤ |A| ≤ dT + dT ∗ ≤ 2dmax ,

|B| ≤ dT + dT ∗ ≤ 2dmax .

Because dmax is assumed infinite, we have dmax = 2 · dmax = ℵ0 · dmax , and consequently |A| = dmax = ℵ0 · dmax ≥ ℵ0 · |B|,

which implies |A| = ℵ0 · |B| + r,

(7.8)

where r is a cardinal number ≥ 0. As a consequence of this relation between the cardinal numbers |A| and |B|, we can rearrange the sum (7.7) so that each term with subscript β ∈ B is accompanied by ℵ0 terms with subscripts α ∈ A. Thus we obtain   L × L × × × U∼ [U (Ωβ ) ⊕ U (0, 2π ) ⊕ U (0, 2π ) ⊕ · · ·] ⊕ U (0, 2π ) . (7.9) r terms

β ∈B

We now make use of the obvious relation U ×(0, 2π ) ∼ U ×(Ω ) ⊕ U ×(Ω ′ ),

(7.10)

valid for any measurable subset Ω of (0, 2π ), and its complement Ω ′ with respect to this interval. In view of this relation we deduce U ×(Ωβ ) ⊕ U ×(0, 2π ) ⊕ U ×(0, 2π ) ⊕ · · ·

∼ U ×(Ωβ ) ⊕ [U ×(Ωβ′ ) ⊕ U ×(Ωβ )] ⊕ [U ×(Ωβ′ ) ⊕ U ×(Ωβ )] ⊕ · · · ∼ [U ×(Ωβ ) ⊕ U ×(Ωβ′ )] ⊕ [U ×(Ωβ ) ⊕ U ×(Ωβ′ )] ⊕ · · · ∼ U ×(0, 2π ) ⊕ U ×(0, 2π ) ⊕ · · · ,

and hence it follows from (7.9) that U is unitarily equivalent to the orthogonal sum of ℵ0 · |B| + r = |A| = dmax replicas (cf. (7.8)) of U ×(0, 2π ). Now U ×(0, 2π ) is obviously a bilateral shift of multiplicity 1; a generating subspace consists of the

8. S IMILARITY OF OPERATORS IN Cρ TO CONTRACTIONS

95

constant functions in L2 (0, 2π ). Consequently, the orthogonal sum under consideration will be a bilateral shift of multiplicity dmax . Operators unitarily equivalent to bilateral shifts are also bilateral shifts of the same multiplicity, thus it follows that U is a bilateral shift of multiplicity dmax . Part (a) of the theorem is herewith established. Part (b) follows immediately from (7.7) and (7.10). In fact, we have only to set e = U ⊕ U ′, U

where U ′ =

L × ′ U (Ωα ), α

for then we have     e ∼ L U ×(Ωα ) ⊕ L U ×(Ωα′ ) ∼ L[U ×(Ωα ) ⊕ U ×(Ωα′ )] ∼ L U ×(0, 2π ); U α

α

α

α

the last sum is a bilateral shift of multiplicity equal to the number of terms of this sum and hence not exceeding dT + dT ∗ . Corollary 7.5. If T is a contraction on the complex Hilbert space H such that kT hk < khk for every nonzero h in H, then the minimal unitary dilation of T is a bilateral shift of multiplicity equal to dim H. Proof. The operator T is obviously c.n.u., and (I − T ∗ T ) h 6= 0 for h 6= 0. Thus DT H = H, dmax = dT = dim H, and therefore we can apply part a) of Theorem 7.4.

8 Similarity of operators in Cρ to contractions In Sec I.11 we introduced the classes Cρ (ρ > 0) of operators: the operator T on the space H belongs to the class Cρ if there exists a unitary operator Uρ on some space Kρ containing H as a subspace, such that T n = ρ · pr Uρn

(n = 1, 2, . . .).

(8.1)

The class C1 consists of the contractions. The aim of this section is to prove the following result, showing that the operators belonging to any of these classes are not very far from being contractions. Theorem 8.1. Every operator belonging to a class Cρ is similar to a contraction. Proof. We essentially use Proposition I.11.3, showing that the class Cρ is an increasing function of the parameter ρ (indeed, it suffices to know that it is a nondecreasing function of ρ ). A further essential point in the proof is an appropriate modification of the method of Sec. 3 to the case of unitary ρ -dilations.

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Let us suppose, then, that T is an operator on H, of class Cr with r > 1. Then T belongs to every class Cρ with ρ ≥ r, and therefore it has, for each ρ ≥ r, a unitary ρ -dilation Uρ in some space Kρ . We set for ρ ≥ r Mρ =

W

n≥0

Uρ∗n (Uρ∗ − T ∗ )H

(⊂ Kρ )

(8.2)

and denote by PMρ the orthogonal projection of Kρ into Mρ . We also set tρ = kPMρ |Hk

(ρ ≥ r);

(8.3)

clearly tρ is the smallest value such that the inequality |(h, mρ )| ≤ tρ · khk · kmρ k

(8.4)

holds for every h ∈ H and every mρ ∈ Mρ ; it suffices to consider elements of the form mρ = ∑ Uρ∗n (Uρ∗ − T ∗ )hn (8.5) n≥0

with h0 , h1 , . . . ∈ H and hn = 0 for n large enough, because these are dense in Mρ . Relation (8.1) implies the analogous relation for the adjoints, so that we have, setting δ = 1/ρ , ( δ T ∗n+1 − δ T ∗n T ∗ = O if n ≥ 1, pr Uρ∗n (Uρ∗ − T ∗ ) = δ T ∗ − T ∗ = (δ − 1)T ∗ if n = 0. Thus we obtain for h ∈ H and for mρ defined by (8.5) that (h, mρ ) = ∑ (h,Uρ∗n (Uρ∗ − T ∗ )hn ) = (h, (δ − 1)T ∗ h0 ), n≥0

and hence (8.4) is equivalent to |(h, (δ − 1)T ∗ h0 )| ≤ tρ · khk · kmρ k, and this in turn is equivalent to (δ − 1)2kT ∗ h0 k2 ≤ tρ2 kmρ k2 . Using (8.1), we obtain by an easy calculation that (Uρ∗ j (Uρ∗ − T ∗ )h j ,Uρ∗k (Uρ∗ − T ∗ )hk )  2 ∗ 2  kh j k + (1 − 2δ )kT h j k if j = k,  (δ − 1)(T h , h ) if k − j = 1, j k = (δ − 1)(h j , T hk ) if k − j = −1,    0 in all the other cases.

(8.6)

8. S IMILARITY OF OPERATORS IN Cρ TO CONTRACTIONS

97

So we obtain for the element mρ defined by (8.5) that kmρ k2 = ∑ [kh j k2 + (1 − 2δ )kT ∗ h j k2 + 2(δ − 1) Re(h j , T ∗ h j+1 )], j≥0

and hence

ρ kmρ k2 = ρ [kT ∗ h0 k2 + ∑ kh j − T ∗ h j+1 k2 ] − 2 ∑ [kT ∗ h j k2 − Re(h j , T ∗ h j+1 )]. j≥0

j≥0

It follows that if mρ ′ (r ≤ ρ ′ ≤ ρ ) corresponds to the same sequence {hn } of vectors, then

ρ kmρ k2 − ρ ′ kmρ ′ k2 = (ρ − ρ ′ )[kT ∗ h0 k2 + ∑ kh j − T ∗ h j+1 k2 ] ≥ (ρ − ρ ′ )kT ∗ h0 k2 ; j≥0

in particular

ρ kmρ k2 − rkmr k2 ≥ (ρ − r)kT ∗ h0 k2 .

(8.7)

Obviously tr ≤ 1, thus it follows from (8.6) (for ρ = r) that 

1 −1 r

2

kT ∗ h0 k2 ≤ kmr k2 .

(8.8)

From (8.7) and (8.8) we deduce 

1 −1 ρ kmρ k ≥ (ρ − r)kT h0 k + r r 2

∗

2

2

kT ∗ h0 k2

and hence 

1 −1 ρ

2

kT ∗ h0 k2 ≤

ρ − 2 + 1/ρ kmρ k2 ρ − 2 + 1/r

(ρ ≥ r).

(8.9)

Recalling that tρ is the smallest nonnegative value for which the inequality (8.6) holds, we conclude from (8.9) that tρ2 ≤

ρ − 2 + 1/ρ ; ρ − 2 + 1/r

hence tρ < 1 if ρ > r. In the rest of the proof ρ is a fixed number larger than r; thus it is not necessary to indicate this value ρ by subscripts, so we write U, K, M, t instead of Uρ , Kρ , Mρ , tρ . Set N = K ⊖ M. The elements k of N are characterized by the equations T PHU n k = PHU n+1 k

(n = 0, 1, . . .).

(8.10)

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Indeed, k ⊥ M means that ((U ∗n+1 − U ∗nT ∗ )h, k) = 0

for all

h∈H

and n ≥ 0;

this can be reduced easily to (8.10). These equations, characterizing N, show in particular that N is invariant for U. For h ∈ H we have by (8.3) that khk2 = kPM hk2 + kPNhk2 ≤ t 2 khk2 + kPNhk2 , and hence

kPN hk2 ≥ (1 − t 2)khk2 .

(8.11)

N′ = PN H

(8.12)

Because t(= tρ ) < 1, it follows from (8.11) that the linear manifold

is closed and hence a subspace of N and, moreover, that the operator Y = PN |H is an affinity from H to N′ . Consequently, X = Y ∗ is an affinity from N′ to H. By virtue of the obvious relation (PN h, k) = (h, k) = (h, PH k),

(8.13)

valid for h ∈ H and k ∈ N (and hence in particular for k ∈ N′ ), we obtain that X = PH |N′ .

(8.14)

Let us observe that (8.13) also implies PH k = 0 for every k ∈ N which is orthogonal to PN H (i.e., to N′ ). Thus we have PH (I − PN′ )g = 0 and hence PH g = PH PN′ g for every

g ∈ N.

(8.15)

If f ∈ N′ , then we have in particular U f ∈ UN ⊂ N, and hence, setting g = U f , we obtain from (8.14), (8.15), and the case n = 0 of (8.10), that T X f = T PH f = PHU f = PH PN′ U f = XV f with

V = PN′ U|N′ .

Clearly, V is a contraction on the space N′ , and T = XV X −1 . Therefore T is similar to the contraction V . This concludes the proof. On account of Proposition I.11.2 we can formulate the following corollary.

9. N OTES

99

Corollary 8.2. Every operator T with numerical radius w(T ) ≤ 1 is similar to a contraction.

9 Notes Theorem 1.1 on the structure of the minimal unitary dilation U of a contraction T on H, appears implicitly in S Z .-NAGY [2] and explicitly in H ALPERIN [1] and S Z .-N.–F. [V]. If T ∈ C·0 then by virtue of Theorem 2.1 we have K+ = M+ (L∗ ) and hence U+ is a unilateral shift. Using relations (I.4.2) we obtain therefore that every contraction of class C0· is the restriction of the adjoint of a unilateral shift. For this explicit statement see F OIAS¸ [5] and DE B RANGES AND ROVNYAK [1]. The first general result concerning the spectral type of U was that of S CHREIBER [1] to the effect that if kT k < 1 then U is a bilateral shift, of multiplicity equal to dim H; for an alternative proof, see S Z .-NAGY [II]. (In these proofs, the space H is assumed to be complex.) The fact that the condition T n → O implies that U is a bilateral shift of multiplicity equal to the defect index dT , and the dual of this fact with T ∗ instead of T , were proved first by DE B RUIJN [1] by a matrix method valid for real and complex spaces. The result of DE B RUIJN was subsequently generalized by H ALPERIN [1] to combine the cases T n → O and T ∗n → O in the following manner. Let us suppose that there exist in H orthogonal projections, say Q1 and Q2 , such that Q2 Q1 = 0, n

Q2 T Q1 = O, ∗

(I − Q2 )T (I − Q1 ) = O,

n

(T Q1 ) → 0 and (T Q2 ) → O

(n → ∞).

Then U is a bilateral shift, with the generating subspace (U − T )Q1 H ⊕ (I − Q1 − Q2 )H ⊕ (U ∗ − T ∗ )Q2 H. The problem whether the condition kT hk < khk (for all nonzero h ∈ H) implies that U is a bilateral shift of multiplicity equal to dim H, was raised by DE B RUIJN [1]. The affirmative answer to this problem was given in S Z .-N.–F. [V] (Corollary 7.5 above) at least for complex spaces. The role of the residual and ∗-residual parts of U was indicated in S Z .-N.–F. [V], [VII]. In [3] and [VII], the authors introduced the notions of quasi-affinity and quasi-similarity, and the classes Cαβ of contractions (however, only for c.n.u. ones). In particular, they proved that every contraction of class C11 is quasi-similar both to the residual part R and the ∗-residual part R∗ of the minimal unitary dilation of U (Proposition 3.5(iii) above). This theorem has been used, first in connection with the theory of characteristic functions, and then directly, to obtain information on the invariant subspaces for T (cf. S Z .-N.–F. [IX] and [5]). In Sec. 5 above, the results of the latter paper are presented in a more developed form, giving more detailed information on the invariant (regular and hyperinvariant) subspaces, and moreover these results are extended from contractions to general power-bounded operators (The-

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orem 5.4). The same theorem (with simple invariance instead of hyperinvariance) was obtained independently by S. Parrott (correspondence). The geometric characterization of the similarity of a contraction to a unitary operator given in Proposition 3.6 is an essential step in the proof of the main result in S Z .-N.–F. [X] (presented later as Theorem VI.4.5). Let us observe that quasi-similarity preserves the commutativity of the commutant {T }′ , as well as the existence of nontrivial hyperinvariant subspaces for T . This is easily verified using the methods of Sec. 5. The important fact that if T is a c.n.u. contraction then its minimal unitary dilation U has absolutely continuous spectral measure, was proved first by the authors in [IV], where they used rather deep theorems on analytic functions. The more “geometric” proof given in Sec. 6 was found later (cf. H ALPERIN [1] and the authors [V]). The other results of Sec. 6 and Sec. 7 were obtained by the authors in [III] and [V], with the exception of Propositions 6.3 and 6.5 which are due to M LAK [1]–[3], [5]. In connection with Secs. 1 and 2.1 cf. also see H ALPERIN [3] and [5], where the structure of the spaces of minimal regular isometric and unitary dilations are studied for systems of contractions; conditions are also given for the minimal regular unitary dilation to behave in a certain sense as bilateral shifts. Proposition 2.2 and Theorem 2.3 on the dilation of commutants are due to the present authors; see S Z .-N.–F. [12] (the proof given in this book differs slightly from the original one). Some special cases of Theorem 2.3 were obtained earlier, using entirely different and more involved methods, by S ARASON [3] (T ∈ C00 , dT = dT ∗ = 1) and by S Z .-N.–F. [11] (T ∈ C00 ). It was Sarason’s paper that inspired these investigations of the authors. An alternative proof of Theorem 2.3 was found by D OUGLAS , M UHLY, AND P EARCY [1]. Let us add that Theorem 2.3 implies, and is implied by, Ando’s theorem (Theorem I.6.1), as remarked independently by PARROTT [1] also. Section 8 reproduces the paper S Z .-N.–F. [10] (with some simplification in the ′ jan). The interest in Theoˇ part following formula (8.11), suggested by Ju. L. Smul rem 8.1 is due mainly to the fact that not all power-bounded operators are similar to contractions (cf. F OGUEL [1], H ALMOS [5]; the compact ones are, cf. S Z .-NAGY [11]). However, every power-bounded operator T can be approximated in the operator norm, as closely as we wish, by operators T ′ belonging to some class Cρ (cf. H OLBROOK [3]). Indeed, this follows easily from Remark 4 to Proposition I.11.2, by choosing T ′ = cT with 1 > c → 1. Part of the structural relations treated in Sec. 1 as well as some of the spectral relations treated in Sec. 6 can be extended to unitary ρ -dilations; see D URSZT [2]. In particular, if an operator of class Cρ is c.n.u., then its minimal unitary ρ -dilation ´ has absolutely continuous spectral measure; see D URSZT [2], R ACZ [1], [2], and M LAK [10].

F URTHER RESULTS

101

10 Further results 1. The earliest form of the commutant lifting theorem appeared in S ARASON [3], and was motivated by the study of interpolation problems for bounded analytic functions in the unit disk. The general form of the theorem has also been used in a variety of interpolation problems. The solutions of the classical interpolation problems can be parametrized by fractional linear transformations applied to an arbitrary analytic function (or Schur parameter) from the unit disk to itself. Extensions of this parametrization to the solutions of commutant lifting problems have been found as well. An interesting connection was made in H ELTON [1], who showed that commutant lifting can be used in the study of control problems. Other applications arose in the study of layered media and systems theory. In this context, a new class of (nonstationary) interpolation problems has been pursued by I. Gohberg and his collaborators. In fact, I. Gohberg and W. Helton initiated an entirely new direction of research in operator theory, inspired by engineering problems. It turned out that these interpolation problems can also be connected to commutant lifting. We refer to F OIAS AND F RAZHO [1] for an exposition of the parametrization of the solutions to the commutant lifting problem and its applications, including the study of multi-layered media. The connections between commutant lifting and interpolation are explored in F OIAS , F RAZHO , G OHBERG , AND K AASHOEK [1]. ¨ ZBAY, AND TANNENBAUM [1]. For applications to control theory see F OIAS , O A new kind of interpolation problem, which is not covered in this book or the monographs just mentioned, is as follows. Given a contraction T with minimal isometric dilation U+ and an operator X commuting with T , what is the smallest possible spectral radius of a dilation Y of X in the commutant of U+ ? This problem originates in control theory, and it was first considered in B ERCOVICI , F OIAS , AND TANNENBAUM [2], where the optimal value is found. The results of this paper do not lead to an effective algorithm to estimate that optimum, or to calculate the optimal solution. In connection with this topic, see also AGLER AND YOUNG [1]–[3] and B ERCOVICI , F OIAS , AND TANNENBAUM [3]. Other variants of the commutant lifting theorem were also considered; see, for instance T REIL AND VOLBERG [1] and F OIAS AND TANNENBAUM [1]. For generalizations of the commutant lifting theorem, see also B ISWAS , F OIAS , AND F RAZHO [1] and F OIAS , F RAZHO , AND K AASHOEK [3], and for related topics, see K AFTAL , L ARSON , AND W EISS [1] and F OIAS , F RAZHO , AND L I [1]. 2. The commutant lifting theorem has been extended to the context of dilations for noncommuting contractive n-tuples of operators by P OPESCU [2]. In the commutative case, see P OPESCU [6] and B HATTACHARYYA , E SCHMEIER , AND S ARKAR [1]. In the context of the dilations studied in D RURY [1] and A RVESON [3], the corresponding commutant lifting theorem was proved in B ALL , T RENT, AND V IN NIKOV [1]. See also BALL , L I , T IMOTIN , AND T RENT [1] for a commutant lifting result in the case of a system of commuting contractions.

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3. A theorem about similarity to a contraction, inspired by systems theory, and implying Theorem 8.1, was given in H OLBROOK [3]. For a different approach to similarity problems see VAN C ASTEREN [1,2]. 4. The invariance of various properties of an operator under quasi-similarity has been studied by many authors; see, for instance, F IALKOW [1],[2]; H ERRERO [2],[3]; C LARY [1]; L. M. YANG [1]; AGLER , F RANKS AND H ERRERO [1]; M C ¨ C ARTHY [1]; C HEN , H ERRERO , AND W U [1]; M ULLER AND T OMILOV [1]; and TAKAHASHI [5].

Chapter III

Functional Calculus 1 Hardy classes. Inner and outer functions 1. In the rest of this book we consider only complex Hilbert spaces. For the contractions T on these spaces we construct, in this chapter and in the next one, a functional calculus with the aid of the minimal unitary dilation of T . Let us begin by introducing some classes of functions, holomorphic on the open unit disk D = {λ : |λ | < 1}. First a notation: for any function u on D we define its “adjoint” u˜ by u˜(λ ) = u(λ¯ );

(1.1)

the transformation u → u˜ is obviously involutive. If u is holomorphic on D then so is u˜, and for the corresponding power series we have ∞

u(λ ) = ∑ cn λ n , 0

∞

u˜(λ ) = ∑ cn λ n . 0

Let H p (0 < p ≤ ∞) be the (Hardy) class of functions u, holomorphic on D and such that the corresponding norm  h R i1/p sup 1 2π it p (0 < p < ∞), 0
is finite. We have H p ⊃ H p ⊃ H ∞ for 0 < p < p′ < ∞. Each of the classes H p is linear and invariant for the involution u → u˜; H ∞ is even an algebra. Let us recall some fundamental theorems on the Hardy classes, due to Fatou, Riesz, and Szeg˝o. For the proofs we refer to the monograph of P RIVALOV [1] or to that of H OFFMAN [1], and to the original papers cited there. ′

B.Sz.-Nagy et al., Harmonic Analysis of Operators on Hilbert Space, Universitext, DOI 10.1007/978-1-4419-6094-8_3, © Springer Science + Business Media, LLC 2010

103

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C HAPTER III. F UNCTIONAL C ALCULUS

For u ∈ H p the radial limit u(eit ) = lim u(reit )

(1.2)

r→1−0

exists almost everywhere (a.e.) on the unit circle C, and if u 6≡ 0 we have1 log |u(eit )| ∈ L1

(1.3)

and consequently u(eit ) 6= 0 a.e. Moreover, u(eit ) exists a.e. even as the nontangential limit of u(λ ), that is, for λ converging to eit in the angle formed by two chords of C issuing from the point eit . The limit function u(eit ) belongs to the Lebesgue space L p and, if 0 < p < ∞, it is also the strong limit of u(reit ): Z 2π 0

|u(eit ) − u(reit )| p dt → 0

(r → 1 − 0);

(1.4)

if p ≥ 1 this implies weak convergence, that is, Z 2π 0

f (t)u(reit ) dt →

Z 2π 0

f (t)u(eit ) dt

(r → 1 − 0)

(1.5)

for an arbitrary f ∈ Lq ((1/p) + (1/q) = 1). In particular, the Cauchy and Poisson formulas hold for the limit functions; thus for u ∈ H p (p ≥ 1) we have ( Z 1 2π int it u(0) (n = 0), e u(e ) dt = (1.6) 2π 0 0 (n = 1, 2, . . .) and

with

1 2π

Z 2π 0

P(ρ , τ − t)u(eit ) dt = u(λ ) P(ρ , τ ) =

(λ = ρ eiτ , 0 ≤ ρ < 1),

(1.7)

1 − ρ2 . 1 − 2ρ cos τ + ρ 2

Conversely, every function f ∈ L p (1 ≤ p ≤ ∞) whose Fourier series is of the type ∞

f (t) ∼ ∑ cn eint 0

generates the function ∞

u(λ ) = ∑cn λ n = 0

1 2π

Z 2π 0

P(ρ , τ − t) f (t) dt

(λ = ρ eiτ )

(1.8)

For the interval 0 ≤ t ≤ 2π , the Lebesgue spaces L p are defined with respect to the normalized measure dt/2π .

1

1. H ARDY CLASSES . I NNER AND OUTER FUNCTIONS

105

belonging to H p and such that u(eit ) = f (t)

a.e.

(1.9) p

These functions f form a subspace of L p denoted by L+ . Formulas (1.8) and (1.9) establish therefore, for a fixed p (1 ≤ p ≤ ∞), a onep to-one correspondence between the elements of H p and L+ . This correspondence is obviously linear, and it even preserves the metric structure, because h i1/p  1 R 2π |u(eit )| p dt (1 ≤ p < ∞), 0 2 π kuk p = ess sup|u(eit )| (p = ∞). p Consequently, one can identify H p with L+ . In particular, H 2 is identified with L2+ so that we have Z 1 2π u1 (eit )u2 (eit ) dt. (u1 , u2 )H 2 = 2π 0

We call an inner function every function u ∈ H ∞ such that |u(eit )| = 1

a.e. on C;

(1.10)

from the integral formula (1.7) and from (1.10) it follows that |u(λ )| ≤ 1 for every λ ∈ D also. The general form of the inner functions is u(λ ) = κB(λ )S(λ ),

(1.11)

where κ is a constant factor of modulus 1, B(λ ) is a Blaschke product: B(λ ) = ∏ and

a¯k ak − λ |ak | 1 − a¯ k λ

(|ak | < 1,

 Z S(λ ) = exp −

0

2π

∑(1 − |ak |) < ∞),

 eit + λ d µt , eit − λ

(1.12)

(1.13)

where µ is a finite nonnegative measure, singular with respect to Lebesgue measure and uniquely determined by the function u(λ ). If some ak equals 0, then the corresponding factor in the Blaschke product has to be taken equal to λ . One of the factors B(λ ) and S(λ ), or both, may be absent, that is, reduce to the constant function 1. We call an outer function every function on D that can be represented as  Z 2π it  e +λ 1 u(λ ) = κ exp log k(t) dt (λ ∈ D), (1.14) 2π 0 eit − λ where k(t) ≥ 0,

log k(t) ∈ L1 ,

(1.15)

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C HAPTER III. F UNCTIONAL C ALCULUS

and κ is a complex number of modulus 1. This function u belongs to H p (0 < p ≤ ∞) if and only if k ∈ L p ; in this case |u(eit )| = k(t) a.e.

(1.16)

It is obvious that the only functions which are at the same time inner and outer, are the constant functions of modulus 1. The class of the outer functions belonging to H p is denoted by E p . Every function u ∈ H p (0 < p ≤ ∞) such that u 6≡ 0, has a “canonical” factorization u = ui u e into the product of an inner function ui and an outer function ue , which are determined up to constant factors of modulus 1. The function ue belongs to the class E p and is given by the formula  Z 2π it  e +λ 1 it log |u(e )| dt (λ ∈ D), (1.17) ue (λ ) = κ exp 2π 0 eit − λ where |κ| = 1 (cf. (1.3)); ui and ue are called the inner factor and the outer factor of u, respectively. From (1.17) it follows easily that if u, v, and uv belong to Hardy classes and do not vanish identically, then (uv)e = ue ve and (uv)i = ui vi ; this holds in particular if u ∈ H ∞ and v ∈ H p , because then uv ∈ H p .

2. We need some characteristic properties of outer functions. It is convenient to introduce first the folowing notation: p L+0

(1 ≤ p ≤ ∞)

stands for the subspace of L p consisting of those functions f whose Fourier series is of the form ∞ f (t) ∼ ∑ cn eint . 1

p Let us observe that the only real-valued function belonging to L+0 is the function f (t) = 0 (a.e.).

Proposition 1.1. (a) For an outer function u ∈ H p (0 < p ≤ ∞, u 6≡ 0) the following implications are valid. f (t) ∈ L1

v(λ ) ∈ H

1

and u(eit ) f (t) ∈ L1+0 and u(e

it

)v(eit )

∈

L1+0

imply that

f (t) ∈ L1+0 ,

imply that v(λ ) ≡ 0.

(1.18) (1.18′)

1. H ARDY CLASSES . I NNER AND OUTER FUNCTIONS

107

(b) In order that a function u ∈ H 1 be outer, it suffices that the following implication be valid. v(λ ) ∈ H ∞

and u(eit )v(eit ) ∈ L1+0

imply v(λ ) ≡ 0.

(1.19)

Proof. Part (a). The case f (t) = 0 a.e. is trivial, so we suppose f (t) 6= 0 on a set of positive measure. Because u(eit ) 6= 0 a.e., the function g(t) = u(eit ) f (t)

(1.20)

is also 6= 0 on a set of positive measure. Moreover, g ∈ L1+0 by the hypothesis in (1.18), thus there exists a function G ∈ H 1 such that G(0) = 0, G 6≡ 0, and G(eit ) = g(t) a.e.

(1.21)

By virtue of (1.3) the functions log|G(eit )|

and

log|u(eit )|

are integrable, and hence the function log| f (t)| = log|g(t)| − log|u(eit )| is also integrable. Let us consider the outer function  Z 2π it  e +λ 1 F(λ ) = exp log| f (t)| dt ; 2π 0 eit − λ

(1.22)

(1.23)

because f ∈ L1 , we have F ∈ H 1 . Using the fact that u is outer, we deduce from (1.21)–(1.23) that F(λ ) = Ge (λ )/u(λ ),

(1.24)

where Ge is the outer factor of G; see (1.17). We have |G(λ )| = |Gi (λ )Ge (λ )| ≤ |Ge (λ )|

(λ ∈ D),

where Gi is the inner factor of G, thus (1.24) implies n G(λ ) λ (λ ∈ D; n = 0, 1, . . .). u(λ ) ≤ |F(λ )|

Because F ∈ H 1 , this inequality shows that the functions λ n G(λ )/u(λ ) (n = 0, 1, . . .) also belong to H 1 . Consequently,   Z 2π G(eit ) n G(λ ) dt = 2 eint π λ =0 (n = 0, 1, 2, . . .). (1.25) u(eit ) u(λ ) λ =0 0

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C HAPTER III. F UNCTIONAL C ALCULUS

This means that the function f (t) = g(t)/u(eit ), which by hypothesis belongs to L1 , is actually contained in L1+0 . This proves that the implication (1.18) is valid. As to (1.18′), we have only to take f (t) = v(eit ) and to observe that the conditions v(eit ) ∈ L1+0 and v(eit ) ∈ L1+ (the first of which follows from (1.18) and the second from the fact that v ∈ H 1 ) imply v(eit ) = 0 a.e. and thus v(λ ) = 0 (λ ∈ D). Part (b). The implication (1.19) is impossible for u ≡ 0, thus we have u 6≡ 0, and hence u has a canonical factorization u = ui ue . Let us set v(λ ) = 1 − ui(0)ui (λ ); clearly v ∈ H ∞ . Now u ∈ H 1 and v ∈ H ∞ imply u(eit ) ∈ L1 and v(eit ) ∈ L∞ , and hence u(eit )v(eit ) ∈ L1 . Because |ui (eit )|2 = 1 a.e., we also have Z 2π 0

eint u(eit )v(eit ) dt =

Z 2π 0

eint u(eit ) dt − ui(0)

Z 2π 0

eint ue (eit ) dt

= 2π [λ n u(λ ) − ui(0) · λ nue (λ )]λ =0 = 0

for n = 0, 1, . . .; thus u(eit )v(eit ) ∈ L1+0 . According to (1.19) this implies v ≡ 0, and hence |ui (0)|2 = 1 − v(0) = 1. Using the maximum principle one concludes that ui (λ ) ≡ κ (constant, of modulus 1). Thus u is indeed an outer function. This concludes the proof of Proposition 1.1. Proposition 1.2 (Beurling’s Theorem). Let u ∈ H 2 . In order that u be an outer function it is necessary and sufficient that the functions λ n u(λ ) (n = 0, 1, . . .), as elements of the Hilbert space H 2 , span H 2 . Proof. Let v be an element of H 2 , orthogonal to λ n u (n = 0, 1, . . .), thus Z 2π 0

eint u(eit )v(eit ) dt = 0

(n = 0, 1, . . .).

Hence v ∈ H 2 ⊂ H 1 and u(eit )v(eit ) ∈ L1+0 . If u is an outer function, we have then, by virtue of Proposition 1.1(a), v ≡ 0. Conversely, if one supposes that the functions λ n u (n = 0, 1, . . .) span H 2 , then the implication [u(eit )v(eit ) ∈ L1+0 ] ⇒ [v(λ ) ≡ 0] is valid for the functions v ∈ H 2 , and hence a fortiori for the functions v ∈ H ∞ . By Proposition 1.1(b), u is therefore outer.

3. It follows immediately from the definition (1.10) of inner functions that the product of two inner functions, and the “adjoint” u˜ of an inner function u, are also inner functions. For outer functions, it follows from the definition by (1.14)–(1.17) that the product and the quotient of two outer functions, and the adjoint of an outer function, are also outer functions. The class H ∞ is closed with respect to multiplication and

1. H ARDY CLASSES . I NNER AND OUTER FUNCTIONS

109

adjunction of its elements, therefore we conclude that the class E ∞ of the bounded outer functions is also closed with respect to these operations. We exhibit an important subclass of E ∞ that also possesses these properties. Definition. We denote by E reg the class of functions u in H ∞ that have no zeros in D and for which there exists a constant M = M(u) such that u(λ ) 0 < r < 1. u(rλ ) ≤ M for λ ∈ D,

Proposition 1.3. The class E reg is contained in E ∞ . It is closed with respect to multiplication and adjunction of its elements, and contains in particular (i) The functions in H ∞ that are continuous and different from 0 in the closed unit disk D. (ii) The functions of the form (1 − αλ )ν where |α | ≤ 1, ν ≥ 0. (We choose the branch with value 1 at λ = 0.) Proof. Let u ∈ E reg . We show that for every v ∈ H ∞ such that u(eit )v(eit ) ∈ L1+0 ,

(1.26)

we have v ≡ 0; by virtue of Proposition 1.1(b) this proves that u is outer, that is u ∈ E ∞. Now for any fixed r, 0 < r < 1, the function 1/u(rλ ) has a Taylor series expansion, uniformly convergent on D: ∞ (r) 1 = ∑ am λ m . u(rλ ) m=0

This expansion is uniformly convergent in particular on the unit circle and remains so when multiplied by the bounded function eint u(eit )v(eit ). Term-by-term integration yields, in view of (1.26), Z 2π 0

eint

∞ (r) u(eit ) v(eit ) dt = ∑ am it u(re ) m=0

Z 2π 0

ei(n+m)t u(eit )v(eit ) dt = 0

(1.27)

for n = 0, 1, . . . . The function u(eit )/u(reit ) is bounded in absolute value by the constant M, and as r → 1 − 0 it tends a.e. to 1. By virtue of Lebesgue’s dominated convergence theorem, (1.27) yields in the limit: Z 2π 0

eint v(eit ) dt = 0

(n = 0, 1, . . .),

whence it follows that v ≡ 0. So we have proved that E reg ⊂ E ∞ . The fact that E reg is closed with respect to multiplication and adjunction of its elements is obvious.

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C HAPTER III. F UNCTIONAL C ALCULUS

For a function u satisfying the hypotheses in (i), u(λ )/u(rλ ) is a continuous function of the variables λ , r, on the compact set |λ | ≤ 1, 0 ≤ r ≤ 1. Hence this function is bounded along with the function u(λ ). As to (ii), one proves without difficulty that if |α | ≤ 1 and ν ≥ 0, then (1 − αλ )ν ν (λ ∈ D, 0 < r < 1). (1 − α rλ )ν < 2 This concludes the proof of Proposition 1.3.

4. Given two functions in H ∞ , say u and v, we call v a divisor of u, and u a multiple of v, if w = u/v also belongs to H ∞ . The divisor v of u is nontrivial if neither v nor w are constant functions. The corresponding factorization u = vw is then also said to be nontrivial. As an immediate consequence of the maximum principle we see that an inner function w is constant (of modulus 1) if and only if |w(0)| = 1. It follows that w and 1/w cannot both be inner functions unless w is constant (of modulus 1). Consequently, two inner functions, which are divisors of each other, necessarily “coincide” in the sense that they are equal up to a constant factor of modulus 1. It is convenient not to distinguish two inner functions that coincide in this sense. Let us consider the parametric representation (1.11)–(1.13) of an inner function as the product of a Blaschke product B(λ ) and of a singular inner function S(λ ). By virtue of this representation, every inner function u is determined by a sequence {a1 , a2 , . . .} (finite, infinite, or empty) of complex numbers, such that |ak | < 1 and ∑(1 − |ak |) < ∞, and by a nonnegative, bounded, singular measure µ on the Borel subsets of the unit circle C (possibly µ ≡ 0); the sequence and the measure are otherwise arbitrary. In order that the function corresponding to the sequence {a′1 , a′2 , . . .} and the measure µ ′ be a divisor of the function corresponding to the sequence {a1 , a2 , . . .} and the measure µ , it is necessary and sufficient that, taking account of the multiplicities, {a′k } be a subset of {ak } and µ ′ be a minorant of µ . Let {uα } be a finite or infinite system of inner functions, uα corresponding to the sequence {aα k } and the measure µα These functions have, in an obvious sense, a largest common inner divisor u∧ ; u∧ corresponds to the intersection of the sets Aα = {aα k } (multiplicities taken into account) and to the largest minorant µ∧ of the measures µα . Similarly, if the functions uα have a common inner multiple v, corresponding to the sequence {bk } and to the measure ν , then they also have a least common inner multiple u∨ ; u∨ corresponds to the union of the sets Aα (multiplicities taken into account) and to the least common majorant µ∨ of the measures µα . Let S us note that α Aα ⊂ {bk } and that ν is a majorant of µ∨ , hence also of µ∧ ; because ν is singular, µ∨ and µ∧ are then singular too.2 Let us note that if the system {uα } is finite, a common inner multiple v is furnished by the product ∏α uα . 2

For a construction of µ∧ and µ∨ see, for example, D UNFORD AND S CHWARTZ [1], pp. 162–163.

1. H ARDY CLASSES . I NNER AND OUTER FUNCTIONS

111

Inner functions without a nonconstant common inner divisor are said to be relatively prime. From these considerations it follows that the only nonconstant inner functions without nontrivial factorizations are the functions κ

λ −a , 1 − a¯λ

where |a| < 1

and |κ| = 1.

(1.28)

5. Here are two more propositions on H ∞ that we need later. Proposition 1.4. Let {un } be a uniformly bounded sequence of functions in H ∞ , converging to 0 on D. Then Z 2π 0

un (eit ) f (t) dt → 0

(n → ∞)

(1.29)

for every function f ∈ L1 .3

Proof. The functions λ ν −1 un (λ ) (n = 1, 2, . . .; ν = 0, ±1, ±2, . . .) are holomorphic on the domain 0 < |λ | < 1, so we have Z 2π 0

un (eit )eiν t dt = lim

Z 2π

r→1 0

1 r→1 i

= lim

Z

un (reit )rν eiν t dt

|λ |=r

un (λ )λ ν −1 d λ =

1 i

Z

|λ |=1/2

un (λ )λ ν −1 d λ ;

the last integral tends to 0 as n → ∞ because un (λ ) tends boundedly to 0 on the circle |λ | = 12 . Thus (1.29) is satisfied if f (t) = eiν t and consequently also if f (t) is an arbitrary trigonometric polynomial. Every function f ∈ L1 can be approximated in the metric of L1 , as closely as we wish, by trigonometric polynomials, therefore we conclude that (1.29) is satisfied for every f ∈ L1 . Proposition 1.5. Let {uα } be a finite or infinite system of inner functions and let v be their largest common inner divisor. For every function f ∈ L1 such that uα (eit ) f (t) ∈ L1+0 we also have

for all α ,

v(eit ) f (t) ∈ L1+0 .

(1.30) (1.31)

Proof. By virtue of (1.30), there exist functions Fα ∈ H 1 such that Fα (eit ) = uα (eit ) f (t)e−it

a.e.

Let us fix one of the subscripts α , say the subscript 1. The functions dα (λ ) = F1 (λ )uα (λ ) − Fα (λ )u1 (λ ) 3 Due to the uniform boundedness principle (cf. e.g., D UNFORD AND S CHWARTZ [1], p. 66) and the formula (1.7), the converse statement also holds.

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C HAPTER III. F UNCTIONAL C ALCULUS

also belong to H 1 and their radial limits dα (eit ) are equal a.e. to 0. This implies dα ≡ 0, F1 uα = Fα u1 , or setting uα /v = wα , (1.32)

F1 wα = Fα w1 .

Omitting the trivial case that f (t) = 0 a.e., none of the functions Fα (λ ) is identically 0. Let Fα = Fα i Fα e be the canonical factorization of Fα . From (1.32) we obtain, by taking the inner factors, F1i wα = Fα i w1 . (1.33) Thus w1 is a common inner divisor of the functions F1i wα . Now, v is the largest common inner divisor of the functions uα , thus the functions wα = uα /v have no nonconstant common inner divisor. Therefore w1 must be a divisor of F1i . Consequently, we have G1 = F1 /w1 = F1e · F1i /w1 ∈ H 1 . (1.34) Because

G1 (eit ) = F1 (eit )/w1 (eit ) = u1 (eit ) f (t)e−it /w1 (eit ) = v(eit ) f (t)e−it , (1.34) implies (1.31).

2 Functional calculus: The classes H ∞ and HT∞ 1. Consider first the functions ∞

a(λ ) = ∑ ck λ k k=0

with

∞

∑ |ck | < ∞;

k=0

(2.1)

they are holomorphic on D and continuous on D. The class of these functions is denoted by A: this is obviously an algebra with respect to the usual addition and multiplication of functions, and with the involution a → a˜. If T is a contraction of the Hilbert space H, we associate to the function (2.1) the operator ∞

a(T ) = ∑ ck T k , k=0

(2.2)

this operator series converging in the operator norm. For fixed T one obtains in this way a mapping a → a(T )

of the algebra A into the algebra B(H) of the bounded operators on H; this mapping is an algebra homomorphism and we also have a(T )∗ = a˜(T ∗ ).

(2.3)

2. F UNCTIONAL CALCULUS : T HE CLASSES H ∞ AND HT∞

113

When T is a normal operator with the spectral representation Tn =

Z

σ (T )

λ n dKλ

(n = 0, 1, . . .),

(2.4)

definition (2.2) of a(T ) is equivalent to the usual definition a(T ) =

Z

σ (T )

a(λ ) dKλ ,

(2.5)

a consequence of the fact that the series (2.1) converges uniformly on D and that σ (T ) ⊂ D. Let U be the minimal unitary dilation of the contraction T . The relations T n = pr U n (n = 0, 1, . . .) imply a(T ) = pr a(U)

(2.6)

(a ∈ A),

from which the inequality of von Neumann ka(T )k ≤ sup |a(λ )|

(λ ∈ D)

(2.7)

follows in just the same way as for polynomials; see Sec. I.8. Let us observe then that for every function ϕ (λ ), holomorphic on D, the functions ϕr (λ ) = ϕ (rλ ) (0 < r < 1) belong to the class A. For u ∈ H ∞ the functions ur are uniformly bounded, |ur (λ )| ≤ kuk∞

(0 < r < 1, λ ∈ D).

(2.8)

The operators ur (T ) make sense for 0 < r < 1, so we can introduce the following definition. Definition. For a contraction T on the space H we denote by HT∞ the set of those functions u ∈ H ∞ for which ur (T ) has a limit in the strong sense as r → 1 − 0, and for u ∈ HT∞ we define u(T ) = lim ur (T ). (2.9) r→1−0

For a ∈ A this is consistent with the earlier definition. In fact, ar (T ) converges then to a(T ) even in the operator norm:

∞

∞

∞

∑ ck T k − ∑ ck rk T k ≤ ∑ |ck |(1 − rk ) → 0 as r → 1 − 0.

0

0

0

From the obvious relations (cu)r = cur ,

(u + v)r = ur + vr ,

(uv)r = ur vr

(u, v ∈ H ∞ )

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C HAPTER III. F UNCTIONAL C ALCULUS

it follows that the class HT∞ is a subalgebra of H ∞ and the mapping u → u(T ) defined above is an algebra homomorphism of HT∞ into B(H). If T = T0 ⊕ T1 then obviously a(T ) = a(T0 ) ⊕ a(T1) for a ∈ A; hence it follows readily that and, for u ∈ HT∞ ,

HT∞ = HT∞0 ∩ HT∞1

(2.10)

u(T ) = u(T0 ) ⊕ u(T1 ).

(2.11)

By taking in particular the canonical decomposition of T (cf. Theorem I.3.2) we can thus reduce the study of the mapping u → u(T ) to the two opposite cases: the unitary operators and the c.n.u. contractions. 2. Let us first consider the case of a completely nonunitary contraction T on H. Let U be the minimal unitary dilation of T on K (⊃ H). We show that in this case

and, for u ∈ H ∞ ,

HT∞ = HU∞ = H ∞

(2.12)

u(T ) = pr u(U).

(2.13)

Relation (2.6) implies ur (T ) = pr ur (U) (0 < r < 1), thus all we have to show is that if u ∈ H ∞ , then ur (U) converges strongly as r → 1 − 0. Let us recall to this end that the spectral measure E, induced on C by the spectral family {Et }02π of U, is absolutely continuous with respect to Lebesgue measure (cf. Theorem II.6.4). The limit function u(eit ), existing a.e. on C, is thus also integrable with respect to E: the integral us (U) =

Z 2π 0

u(eit ) dEt

exists (here we indicate by s functions of the unitary operator U, defined via the spectral integral). On the other hand, (2.5) implies that ur (U) = usr (U). Now for every f ∈ K, k[us (U) − usr (U)] f k2 =

Z 2π 0

|u(eit ) − ur (eit )|2 d(Et f , f ) → 0

(r → 1 − 0),

because the integrand converges boundedly to 0 as t → 1 − 0, a.e. with respect to Lebesgue measure and hence also a.e. with respect to E. Thus we have proved the existence of u(U) and that u(U) = us (U). (2.14)

2. F UNCTIONAL CALCULUS : T HE CLASSES H ∞ AND HT∞

115

Because kus (U)k ≤ sup ess|u(eit )| = kuk∞, (2.13) and (2.14) imply ku(T )k ≤ kuk∞

for every

u ∈ H ∞.

(2.15)

Let us apply relation (2.3) to the function a = ur with 0 < r < 1. By the obvious relation (ur )˜ = (u˜)r we obtain ur (T )∗ = (u˜)r (T ∗ ).

(2.16)

Now T ∗ being c.n.u. along with T and u˜ belonging to H ∞ along with u, we conclude that the right-hand side of (2.16) converges strongly to u˜(T ∗ ) as r → 1 − 0. Thus ur (T )∗ converges to u(T )∗ not only weakly (this follows immediately from the convergence ur (T ) → (T )), but also strongly, and we have u(T )∗ = u˜(T ∗ ). Let us turn now to the problems regarding the continuity of the mapping u → u(T ). Inequality (2.15) implies at once that if un (λ ) tends to u(λ ) uniformly on D, then un (T ) tends to u(T ) in operator norm: un (T ) ⇒ u(T ). We show that for the strong convergence un (T ) → u(T ) it suffices that the functions un (eit ) tend boundedly to u(eit ) a.e. on C: kun k∞ ≤ K

un (eit ) → u(eit )

(n = 1, 2, . . .),

a.e. on C.

In fact, setting vn = un − u it follows from (2.13) and (2.14) that for all h ∈ H kvn (T )hk2 ≤ kvn (U)hk2 =

Z 2π 0

|vn (eit )|2 d(Et h, h),

(2.17)

and this integral tends to 0 as n → ∞, by Lebesgue’s theorem. Here we have again used the fact that the spectral measure E of U is absolutely continuous and therefore d(Et h, h) = ϕh (t)dt,

ϕh ∈ L1 .

Finally, for the weak convergence un (T ) ⇀ u(T ) it suffices that the functions un converge boundedly to u on D. For a proof one starts again with (2.13) and (2.14) for vn = un − u; hence it results (vn (T )h, h′ ) = (vn (U)h, h′ ) =

Z 2π 0

vn (eit ) d(Et h, h′ )

(h, h′ ∈ H).

This integral tends to 0 as n → ∞ on account of Proposition 1.4, because d(Et h, h′ ) = ϕh,h′ (t)dt and ϕh,h′ ∈ L1 . Let us note that bounded convergence a.e. on C implies bounded convergence on D by virtue of the Poisson formula (1.7); the converse is not true: consider, for example, the functions λ n (n = 1, 2, . . .).

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For a normal c.n.u. contraction T , our definition of u(T ) for u ∈ H ∞ is compatible with the usual definition via the spectral integral. In fact, this being true for the functions ur (∈ A), we have only to show that Z

σ (T )

ur (λ ) dKλ →

Z

σ (T )

u(λ ) dKλ

(r → 1 − 0).

Now this follows again from Lebesgue’s convergence theorem, because ur (λ ) → u(λ ) boundedly on D, and σ (T )\D has O measure with respect to the spectral measure K (otherwise T could not be c.n.u.). Let us return to the case of an arbitrary c.n.u. contraction T . Let v ∈ H ∞ be such that |v(λ )| < 1 on D. By virtue of (2.15), T ′ = v(T )

(2.18)

is then also a contraction. Let us show that T ′ is also c.n.u. To this end let us consider the function w(λ ) =

v(λ ) − v(0)

1 − v(0)v(λ )

;

we have w ∈ H ∞ and |w(λ )| < 1 on D. Consequently, w(T ) is also a contraction and, obviously, w(T ) = [v(T ) − v(0)I][I − v(0)v(T )]−1 .

We recall that if V is a unitary operator on a Hilbert space, then so is (V − aI)(I − aV ¯ )−1 for |a| < 1, and conversely; see Sec. I.4.3. Hence it follows that the space H′0 of the unitary part of T ′ = v(T ) is the same as that of the unitary part of w(T ). Because |w(λ )| < 1 on D and w(0) = 0, it follows from Schwarz’s lemma that w(λ ) = λ · z(λ ) with |z(λ )| ≤ 1 on D; z(T ) is therefore a contraction and we have w(T ) = T · z(T ) = z(T ) · T. For h ∈ H′0 we have khk = kw(T )n hk = kz(T )n T n hk ≤ kT n hk

khk = kw(T )∗n hk = kz(T )∗n T ∗n hk ≤ kT ∗n hk for n = 1, 2, . . . ; thus kT n hk = khk = kT ∗n hk because T is a contraction, and therefore h = 0 because T is c.n.u. We conclude that H′0 = {0} and hence T ′ = v(T ) is c.n.u. So u(T ′ ) = u(v(T )) makes sense for all u ∈ H ∞ . From (2.18) it follows at once that p(T ′ ) = (p ◦ v)(T ) for all polynomials p; here we used the following notation for the composite function, ( f ◦ g)(λ ) = f (g(λ )).

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117

Applying this relation to the partial sums pn of the power series of a function a ∈ A, and letting n → ∞, we see that a(T ′ ) = (a ◦ v)(T ) because pn → a and pn ◦ v → a ◦ v uniformly on D. Hence we have in particular ur (T ′ ) = (ur ◦ v)(T ) for all u ∈ H ∞ and for 0 < r < 1. If r → 1 − 0, ur (T ′ ) tends to u(T ′ ) by definition and (ur ◦ v)(T ) tends to (u ◦ v)(T ) weakly, because ur ◦ v tends to u ◦ v boundedly on D. Thus we have u(v(T )) = (u ◦ v)(T ) for all u ∈ H ∞ . Summing up, we have the following theorem. Theorem 2.1. For a completely nonunitary contraction T on H and for its minimal unitary dilation U we have HT∞ = HU∞ = H ∞ . The mapping u → u(T ) of H ∞ into the algebra B(H), defined by ∞

u(T ) = lim ∑ rk ck T k r→1−0 k=0

∞

for u(λ ) = ∑ ck λ k ∈ H ∞ , k=0

is an algebra homomorphism of H ∞ into B(H), with the following properties: ( I if u(λ ) = 1, (a) u(T ) = T if u(λ ) = λ , (b) ku(T )k ≤ kuk∞, (c) un (T ) ⇒ u(T ) if un tends to u uniformly on D, (c′ ) un (T ) → u(T ) if un tends boundedly to u almost everywhere on C, (c′′ ) un (T ) ⇀ u(T ) if un tends boundedly to u on D, (d) u(T )∗ = u˜(T ∗ ), (e) For v ∈ H ∞ such that |v(λ )| < 1 on D, T ′ = v(T ) is a completely nonunitary contraction, and we have u(T ′ ) = (u ◦ v)(T ) for every u ∈ H ∞ ; (f) If T is normal, u(T ) is equal to the integral us (T ) of u(λ ) with respect to the spectral measure corresponding to T ; (g) u(T ) = pr u(U). Remark 1. The mapping u → u(T ) defined above is the only algebra homomorphism of H ∞ into B(H) having properties (a) and (c′ ). Indeed, (a) implies the uniqueness for polynomials. For the classes A and H ∞ , uniqueness follows successively, on account of the fact that the partial sums pn of the power series of a function a ∈ A, as well as the functions ur derived from a function u ∈ H ∞ , converge boundedly a.e. on C (pn → a, ur → u). Remark 2. By virtue of the one-to-one correspondence between the elements of H ∞ ∞ and L∞ + , discussed in Sec. 1.1, the algebraic homomorphism u → u(T ) of H into ∞ B(H) can also be considered as an algebra homomorphism of L+ into B(H). This homomorphism f → f (T ) has in particular the following properties.  I if f (t) = 1 a.e., (a′ ) f (T ) = T if f (t) = eit a.e. ′ (b ) k f (T )k ≤ k f k∞ .

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Proposition 2.2. Let T be a c.n.u. contraction on H (6= {0}). Let M be a subalgebra of the algebra L∞ containing L∞ + and let us suppose that there exists an algebra homomorphism f → f (T ) of M into B(H) with properties (a′ ) and (b′ ). Then M = L∞ +. Proof. On account of (b′ ) we can extend the homomorphism under consideration to the closure M of M in L∞ . Now it is known that every closed subalgebra of L∞ , containing L∞ + as a proper subalgebra, contains the function f1 (t) = e−it (cf. H OFFMAN [1], Chap. 10. Maximality). Hence if M 6= L∞ + , then f 1 (T ) makes sense, and in view of the relations eit f1 (t) = f1 (t)eit = 1, (a′ ) gives T f1 (T ) = f1 (T )T = I; that is, f1 (T ) = T −1 . On the other hand, (b′ ) implies kT −1 k = k f1 (T )k ≤ k f1 k∞ = 1. Now kT k ≤ 1 and kT −1 k ≤ 1 obviously imply that T is unitary, and this contradicts the hypothesis that T is c.n.u. on H 6= {0}. This concludes the proof of Proposition 2.2. By virtue of the preceding two remarks and of Proposition 2.2, our functional calculus for the c.n.u. contractions is unique and maximal. 3. For u ∈ H ∞ we denote by u(eit ) the nontangential limit of u(λ ) at the point z = eit of C, if it exists. Let us note that this nontangential limit can exist without being the limit of u(λ ) at the point z in the sense that u(λ ) should tend to the value u(z) as λ converges in D to z arbitrarily. We introduce the following sets of points z on C, associated with a function u ∈ H ∞. Cu = {z : Cu = {z : Cu0 = {z : C0u = {z :

u(λ ) does not have a nontangential limit at z}. u(λ ) does not have a limit at z}. u(λ ) does not have a nonzero nontangential limit at z}4 . u(λ ) does not have a nonzero limit at z}.

All these sets are Borel sets, Cu and (if u(λ ) 6≡ 0) Cu0 are of Lebesgue measure 0 (cf. Sec. 1), Cu ⊂ Cu and Cu0 ⊂ C0u . Let us consider then a unitary operator V on a Hilbert space H, and let EV be the corresponding spectral measure on C. It follows from (2.5) that a(V ) = as (V ) = 4

Z 2π 0

a(eit ) dEV,t

In other words, the nontangential limit either does not exist, or it equals zero.

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119

for every a ∈ A, and in particular for the functions ur (0 < r < 1), where u ∈ H ∞ . Let us note that |ur (eit )| ≤ kuk∞ (0 < r < 1) and that ur (eit ) → u(eit ) (r → 1 − 0), with the exception of the points eit ∈ Cu . If (2.19)

EV (Cu ) = O

then usr (V ) therefore converges strongly to us (V ) as r → 1−0; indeed, by Lebesgue’s dominated convergence theorem we have then kusr (V )h − us(V )hk2 =

Z 2π 0

|ur (eit ) − u(eit )|2 d(EV,t h, h) → 0

(h ∈ H).

Hence (2.19) ensures that u ∈ HV∞ and u(V ) = us (V ). Let us examine the validity of the properties of our functional calculus, established in Theorem 2.1 for c.n.u. contractions T , in the actual case T = V . We note first that in this case U = V , so property (g) is trivial; property (a) is obvious, and property (f) has just been established, at least under the condition (2.19). Property (b) follows from the inequality (2.7), valid for the class A, if one applies it to the function ur and if one observes that kur k∞ ≤ kuk∞ and ur (V ) → u(V ) (strongly) for all u ∈ HV∞ . As to the convergence criteria, (c) follows from (b); (c ′ ) subsists if the term “almost everywhere on C” is meant with respect to the spectral measure EV (and if it is understood that un (eit ) and u(eit ) exist a.e. with respect to EV (i.e. EV (Cun ) = EV (Cu ) = O)), and (c′′ ) is no longer valid: a counterexample is furnished by V = I and un (λ ) = λ n . Property (d) remains valid, with the addition that HV∞∗ consists exactly of the adjoints u˜ of the functions u ∈ HV∞ ; that is, HV∞∗ = (HV∞ )˜. In fact, V being unitary its polynomials and their strong limits are normal operators; thus in particular ur (V ) is a normal operator for u ∈ HV∞ and for 0 < r < 1. We just have then to recall relation (2.3) and the fact that, for a sequence of commuting normal operators Nn , Nn → N strongly if and only if Nn∗ → N ∗ strongly. Let us also note that for this reason u(V ) is a normal operator for all u ∈ HV∞ . As to property (e), it follows from (b) that if v ∈ HV∞ and if |v(λ )| < 1 on D, then N = v(V ) is a contraction. The relation p(N) = (p ◦ v)(V ) for polynomials p follows because the functional calculus is an algebra homomorphism, the relation a(N) = (a ◦ v)(V ) for a ∈ A follows hence by the use of (c); in fact, the partial sums pn of the power series of a yield a sequence of polynomials such that pn → a and pn ◦ v → a ◦ v uniformly on D. In particular, ur (N) = (ur ◦ v)(V )

(2.20)

for u ∈ H ∞ , 0 < r < 1. Let us make the additional hypotheses that the sets Cv

and v−1 (Cu ) = {eit : v(eit ) ∈ Cu }

(2.21)

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are of O measure with respect to EV . These hypotheses imply that the limits v(eit ) = lim v(reit ) and (u ◦ v)(eit ) = lim u(v(reit )) r→1

exist and

r→1

(u ◦ v)(eit ) = lim u(rv(eit )) r→1

holds EV -almost everywhere on C. This implies that u ◦ v ∈ HV∞ and, by the convergence criterion (c′ ), (ur ◦ v)(V ) → (u ◦ v)(V)

(r → 1 − 0).

In view of (2.20) one concludes that u(N) = limr→1−0 ur (N) exists and equals (u ◦ v)(V ). Hence all the properties (a)–(g) of the functional calculus, valid by Theorem 2.1 for c.n.u. contractions, can be extended with some precaution to the case of a unitary T = V , except property (c′′ ). It is easy now to treat the general case. Let T be an arbitrary contraction, with the unitary part T0 and the c.n.u. part T1 . Then T = T0 ⊕ T1

and UT = UT0 ⊕ UT1 ,

where UT0 = T0 .

Recalling (2.11) and combining Theorem 2.1 with the results just obtained, we arrive at the following theorem. Theorem 2.3. Let T be a contraction on H and let U be its minimal unitary dilation. Then we have HT∞ = HU∞ , and HT∞ contains in particular the functions u ∈ H ∞ for which the set Cu has O measure with respect to the spectral measure ET corresponding to the unitary part of T . The mapping u → u(T ) of HT∞ into B(H) defined by ∞

u(T ) = lim ∑ rk ck T k r→1−0 k=0

∞

for u(λ ) = ∑ ck λ k ∈ HT∞ k=0

is an algebra homomorphism with the following properties.  I if u(λ ) = 1, (a) u(T ) = T if u(λ ) = λ . (b) ku(T )k ≤ kuk∞ . (c) un (T ) ⇒ u(T ) if un tends to u uniformly on D. (c′ ) un (T ) → u(T ) if un tends boundedly to u almost everywhere as well as ET almost everywhere on C. (d) HT∞∗ = (HT∞ )˜, and u(T )∗ = u˜(T ∗ ) for u ∈ HT∞ . (e) If u, v are such that |v(λ )| < 1 on D, and the sets Cv and v−1 (Cu ) have measure O with respect to ET , then we have v ∈ HT∞ , u ◦ v ∈ HT∞ , moreover T ′ = v(T ) is a contraction, u belongs to HT∞′ , and u(T ′ ) = (u ◦ v)(T ).

3. T HE ROLE OF OUTER FUNCTIONS

121

(f) For a normal contraction T and for u ∈ H ∞ such that ET (Cu ) = O, u(T ) exists and equals the integral us (T ) of the function u(λ ) with respect to the spectral measure corresponding to T . (g) u(T ) = pr u(U) for every u ∈ HT∞ .

3 The role of outer functions 1. The first step towards an extension of our functional calculus to unbounded functions is to study the conditions under which the operator u(T ) has a (not necessarily bounded) inverse. The detailed study of this extension is found in the next chapter. We consider the case of c.n.u. contractions. Proposition 3.1. For every completely nonunitary contraction T on H, and for every outer function u ∈ H ∞ (i.e., u ∈ E ∞ ), the operator u(T ) has an inverse with dense domain in H (i.e., u(T ) is a quasi-affinity on H). Proof. Let u ∈ E ∞ and h ∈ H be such that u(T )h = 0. If U is the minimal unitary dilation of T and if E is the spectral measure corresponding to U, then 0 = (T n u(T )h, h) = (U n u(U)h, h) =

Z 2π 0

eint u(eit )ϕh (t) dt

(n = 0, 1, . . .),

where ϕh (t) = d(Et h, h)/dt ∈ L1 and ϕh (t) ≥ 0 a.e. Hence u(eit )ϕh (t) ∈ L1+0 and by virtue of Proposition 1.1 we have ϕh (t) ∈ L1+0 . Because ϕh (t) is real-valued, this implies that ϕh (t) = 0 a.e. Consequently, khk2 =

Z 2π 0

d(Et h, h) =

Z 2π 0

ϕh (t) dt = 0.

Therefore u(T )h = 0 implies that h = 0, and hence u(T ) is invertible. Now the operator T ∗ is also a c.n.u. contraction and u˜ also belongs to E ∞ . Thus u˜(T ∗ ) is invertible and because u˜(T ∗ ) = u(T )∗ , this implies that the range of u(T ), that is, the domain of u(T )−1 , is dense in H. This concludes the proof. The condition on the function u to be outer turns out to be also necessary in the following sense. Proposition 3.2. For every nonouter function u ∈ H ∞ there exists a c.n.u. contraction T on a space H 6= {0} such that u(T ) = O. Proof. We can assume that u(λ ) 6≡ 0. Let u = ui ue be the canonical factorization of u. By hypothesis, the inner factor ui is nontrivial. If we find a c.n.u. contraction T for which ui (T ) = O, then we also have u(T ) = ui (T )ue (T ) = O. Let ui H 2 denote the set of the functions ui v where v runs over H 2 . (These products belong to H 2 because ui ∈ H ∞ .) Because |ui (eit )| = 1 a.e., the mapping v → ui v is linear and isometric in the Hilbert space metric of H 2 . Consequently, ui H 2 is a

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subspace of H 2 . Because ui is not outer, it follows from Proposition 1.2 (Beurling’s theorem) that ui H 2 does not coincide with H 2 . Hence, setting H = H 2 ⊖ ui H 2 , we have H 6= {0}. Let V denote multiplication on H 2 by the variable λ ;V is a unilateral shift on H 2 , and consequently V ∗n → O as n → ∞. The explicit form of V ∗ is 1 V ∗ v = [v(λ ) − v(0)] for v ∈ H 2 . λ Let us note that the subspace ui H 2 is invariant for V , and hence its orthogonal complement in H is invariant for V ∗ . Setting T = (V ∗ |H)∗

(3.1)

we thus obtain a contraction T on H such that T ∗n = V ∗n |H → O

(n → ∞).

(3.2)

Consequently, V and T are c.n.u. Moreover, (3.2) implies T n = PV n |H

(n = 0, 1, . . .),

(3.3)

where P denotes the orthogonal projection of H 2 onto H. It follows that w(T ) = Pw(V )|H for any w ∈ H ∞ , w(V ) being the operator of multiplication by w(λ ) on H 2 . In particular, we have ui (T )h = Pui (V )h = Pui (λ )h(λ ) for all

h ∈ H.

Because ui h ∈ ui H 2 and hence ui h ⊥ H for every h ∈ H 2 , we obtain that ui (T ) = O. Thus the operator T has all the desired properties: it is a c.n.u. contraction on a space H 6= {0}, such that ui (T ) = O. It should be added that every other nonzero function w ∈ H ∞ for which w(T ) = O, is a multiple of ui in H ∞ . To show this, let us first observe that the function h0 (λ ) = 1 − ui(0)ui (λ ) belongs to H; indeed we have for every v ∈ H 2 : Z

1 2π ui (eit )v(eit )[1 − ui(0)ui (eit )] dt 2π 0 Z 1 2π it v(e )[ui (eit ) − ui(0)] dt = 0. = 2π 0

(ui v, h0 )H =

Thus w(T ) = O implies in particular that 0 = w(T )h0 = Pwh0

3. T HE ROLE OF OUTER FUNCTIONS

123

and hence wh0 ∈ ui H 2 . It follows that wh0 = ui v for some v ∈ H 2 and therefore we have that w = ui v1 with v1 = v + ui (0)w ∈ H 2 . This implies |v1 (eit )| = |ui (eit )||v1 (eit )| = |w(eit )| ≤ kwk∞ a.e. on C. By virtue of Poisson’s formula connecting the values of the function v1 on D with its values (i.e., radial limits) on C, we have |v1 (λ )| ≤ kwk∞ for every λ ∈ D. Hence v1 ∈ H ∞ , and thus w is a multiple of ui in H ∞ . From Propositions 3.1 and 3.2 we obtain the following result immediately. Proposition 3.3. Let u ∈ H ∞ . In order that u(T ) be invertible for every completely nonunitary contraction T on H, it is necessary and sufficient that u be an outer function, that is, u ∈ E ∞ . For such a function, u(T )−1 exists and has dense domain in H. Corollary. Let v ∈ H ∞ be such that |v(λ )| < 1 for λ ∈ D. Then u ∈ E ∞ implies u ◦ v ∈ E ∞. Proof. By virtue of Theorem 2.1, if T is any c.n.u. contraction then so is T ′ = v(T ) and we have u(T ′ ) = (u ◦ v)(T ).

Now, for u ∈ E ∞ , u(T ′ ) is invertible and so is (u ◦ v)(T ), and hence u ◦ v ∈ E ∞ . 2. Now we make the following definition.

Definition. For an arbitrary contraction T let KT∞ denote the class of functions u ∈ HT∞ for which u(T )−1 exists and has dense domain, that is, u(T ) is a quasi-affinity. For u ∈ KT∞ we have that

u˜(T ∗ ) = u(T )∗

(3.4)

is a quasi-affinity, hence u˜ ∈ KT∞∗ . By reason of symmetry we have therefore KT∞∗ = (KT∞ )˜.

(3.5)

Similarly, if u, v are in KT∞ then so is uv, because (uv)(T ) = u(T )v(T ). Thus the class KT∞ is multiplicative. By virtue of Proposition 3.3 we have for a c.n.u. T : E ∞ ⊂ KT∞ . Let T be unitary, T = V , and let u ∈ H ∞ . We know that if u(eit ) exists EV -a.e. (condition (2.19)) then u ∈ HV∞ and u(V ) = us (V ). In order that u(V ) be invertible, it is necessary and sufficient that u(eit ) exist and be different from 0 EV -a.e., that is, EV (Cu0 ) = O.

(3.6)

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Under this condition, according to the theory of spectral integrals, us (V )−1 = (1/u)s (V ), and hence u(V )−1 =

Z 2π 0

[1/u(eit )] dEV,t ,

and this operator has dense domain. Condition (3.6) thus ensures that u ∈ KV∞ and that u(V )−1 = (1/u)s (V ). Let u, v ∈ H ∞ such that |v(λ )| < 1 on D and that the sets Cv

and v−1 (C0u )

are of EV -measure O. These conditions ensure that (u ◦ v)(eit ) exists and is not zero EV -a.e., and this implies by what precedes that u ◦ v ∈ KV∞ . General contractions T can be dealt with by using the canonical decomposition T = T0 ⊕ T1 and applying Proposition 3.3, its corollary, and the remarks which we have just made in the case of a unitary operator. So we arrive at the following result. Theorem 3.4. (i) For a contraction T of general kind, the class KT∞ is multiplicative and we have KT∞∗ = (KT∞ )˜. (ii) KT∞ contains in particular the functions u ∈ E ∞ for which the set Cu0 is of O measure with respect to the spectral measure ET corresponding to the unitary part of T . (iii) For functions u ∈ E ∞ , v ∈ H ∞ such that |v(λ )| < 1 on D and both Cv , v−1 (C0u ) are of ET -measure O, we have u ◦ v ∈ KT∞ .

In the present chapter we apply our functional calculus to study a new class of contractions, called class C0 , and the continuous one-parameter semigroups of contractions. In these investigations we only use a part of our results, in particular those established in Theorems 2.1 and 2.3. We further exploit these results in the next chapter when we extend our functional calculus to some classes of unbounded analytic functions.

4 Contractions of class C0 1. Let T be a contraction on the space H and let U be its minimal unitary dilation. We know that the limit Lh = lim U −n T n h (4.1) n→∞

exists for every h ∈ H (cf. Proposition II.3.1, where we have also proved that Lh is the orthogonal projection of h into the subspace R∗ of the dilation space). From (4.1) we obtain U −m LT m h = lim U −m−n T m+n h = Lh, n→∞

and hence

LT m h = U m Lh

for m = 0, 1, . . . .

(4.2)

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125

If we assume that T is c.n.u. then (4.2) implies that Lu(T )h = u(U)Lh

(4.3)

for all u ∈ H ∞ and all h ∈ H. Choose u and h so that u 6≡ 0 and u(T )h = 0;

(4.4)

u(U)Lh = 0.

(4.5)

by virtue of (4.3) we have Now u(eit ) exists and is nonzero a.e. with respect to Lebesgue measure, and hence also with respect to the spectral measure corresponding to U (recall that this spectral measure is absolutely continuous). This implies that u(U) is invertible: u(U)−1 = (1/u)s (U). So we deduce from (4.5) that Lh = 0, and hence lim kT n hk = lim kU −nT n hk = kLhk = 0.

n→∞

n→∞

We can formulate our result as follows. Proposition 4.1. Let T be a completely nonunitary contraction on the space H, and let u be a nonzero function in H ∞ . For every h ∈ H such that u(T )h = 0 we have then T n h → 0 as n → ∞. If u(T )h = 0 for all h ∈ H (i.e., if u(T ) = O), then u˜(T ∗ ) = u(T )∗ = O, so we can apply the preceding result to u˜ and T ∗ , and thus obtain the following proposition. Proposition 4.2. Let T be a completely nonunitary contraction such that u(T ) = O for some nonzero function u ∈ H ∞ . Then we have T n → O and T ∗n → O, i.e. T ∈ C00 .

The class of these contractions merits further investigation. First we make a definition.

Definition. We call C0 the class of those completely nonunitary contractions T for which there exists a nonzero function u ∈ H ∞ such that u(T ) = O. Proposition 4.2 can then be expressed by the formula C0 ⊂ C00 . Let us observe that for T ∈ C0 , the function u in the definition can always be taken to be inner. Indeed, if u = ue ui is the canonical factorization of u, then ue (T )ui (T ) = u(T ) = O, and hence ui (T ) = O, because ue (T ) is invertible by Proposition 3.1. It is obvious that if u(T ) = O for some u ∈ H ∞ (u 6≡ 0) then also v(T ) = O for the multiples v of u in H ∞ . There arises the question of whether for every contraction T ∈ C0 there exists an inner function u with u(T ) = O, such that every other function v ∈ H ∞ with v(T ) = O is a multiple of u. Such a function, if it exists, is called a minimal function for T and denoted by mT ; this function is then determined up to a

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constant factor of modulus 1. This follows from the fact that if two inner functions are divisors of each other, then they necessarily coincide; see Sec. 1. An important example for these notions was constructed in the proof of Proposition 3.2 (see also the discussion following that proof). Let us formulate it as Part (a) of the following proposition. Proposition 4.3. (a) For every nonconstant inner function m, the space H = H 2 ⊖ mH 2 is different from {0}; the operator T defined on H by T v = PH [λ v(λ )],

T ∗v =

1 [v(λ ) − v(0)] (v ∈ H) λ

belongs to the class C0 and its minimal function is m. (b) A subspace H1 of H is invariant for T if and only if it has the form H1 = m2 (H 2 ⊖ m1 H 2 ), where m = m1 m2 is a factorization of m into a product of inner factors m1 and m2 . Proof. It is enough to prove (b). Let m = m1 m2 be a factorization of m where m1 and m2 are inner functions. If m1 or m2 is constant then H1 = {0} or H1 = H, respectively, is trivially invariant. Thus we can assume that neither m1 nor m2 is constant. We have H = (H 2 ⊖ m2 H 2 ) ⊕ m2(H 2 ⊖ m1H 2 ), so that H2 = H 2 ⊖ m2 H 2 ⊂ H and the restriction of T ∗ to H2 is precisely the analogue of the operator T ∗ for the case when m is replaced by m2 . Therefore, by virtue of Part (a), T ∗ H2 ⊂ H2 and consequently T H1 ⊂ H1 . Conversely, let H1 be an invariant subspace of T . The cases H1 = {0} or H1 = H correspond to the trivial factorizations m = 1 · m or m = m · 1, respectively, therefore we assume that {0} = 6 H1 6= H. The space H2 = H ⊖ H1 is invariant for T ∗ and (with the notation as in the proof of Proposition 3.2) also for V ∗ because T ∗ = V ∗ |H. For T2 = (T ∗ |H2 )∗ we also have T2∗ = V ∗ |H2 . This implies that PH2 V = T2 PH2 which in turn yields that V is an isometric dilation of T2 . If V were nonminimal, then V would be reducible, by virtue of the remark following the proof of Theorem I.4.1. This contradicts the fact that V is obviously of multiplicity 1. For the same reason (V − T2 )H2 6= {0}, because in the opposite case H2 is reducing V . But by virtue of Theorem II.2.1, L = (V − T2 )H2 is a wandering space of V . From Proposition I.2.1 we can now infer that dim L = 1, hence L is formed by the scalar multiples of some m2 ∈ H 2 , km2 k = 1. But L is a wandering space for V , thus we have 0 = (V n m2 , m2 ) =

1 2π

Z 2π 0

eint |m2 (eit )|2 dt

(n = 1, 2, . . .)

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127

whence |m2 (eit )|2 = 1 a.e.; that is, m2 (λ ) is an inner function. Now from Theorem II.2.1 it follows that H 2 ⊖ H2 = M+ (L) = m2 H 2 . But H 2 ⊖ H2 = H1 ⊕ mH 2 , so mH 2 ⊂ m2 H 2 . Consequently m = m2 m1 where m1 ∈ H 2 . Because m and m2 are inner so must be m1 . We have thus obtained that H1 = m2 H 2 ⊖ mH 2 = m2 (H 2 ⊖ m1 H 2 ), concluding the proof. Further important examples of contractions of class C0 are studied later, particularly in Chaps. VIII and X. 2. We now state an important result. Proposition 4.4. For every contraction T of class C0 there exists a minimal function mT . Proof. By hypothesis, the class J of inner functions u with u(T ) = O is not empty. From the lemma below it follows as a special case that v(T ) = O for the largest common inner divisor v of the functions u in J . Obviously, mT = v. The lemma in question reads as follows. Lemma 4.5. Let uα (α ∈ A) be inner functions, and let v be their largest common inner divisor. Suppose that the equation uα (T )h = 0 holds for a completely nonunitary contraction T on the space H, for an element h of H, and for every α ∈ A. Then we have also v(T )h = 0. Proof. Let U be the minimal unitary dilation of T and let {Et } (0 ≤ t ≤ 2π ) be the spectral family of U; Et is an absolutely continuous function of t. For an arbitrary element g of the space H we have

ϕh,g (t) = d(Et h, g)/dt ∈ L1 . As uα (T )h = 0 for every α ∈ A, we have for ν = 0, 1, . . ., 0 = (T ν uα (T )h, g) = (U ν uα (U)h, g) = that is,

uα (eit )ϕh,g (t) ∈ L1+0

By virtue of Proposition 1.5 this implies

Z 2π 0

eiν t uα (eit )ϕh,g (t) dt;

(α ∈ A).

v(eit )ϕh,g (t) ∈ L1+0 , and hence (v(T )h, g) =

Z 2π 0

v(eit )ϕh,g (t) dt = 0.

Because g is arbitrary, this implies v(T )h = 0.

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3. The minimal function of a contraction of class C0 plays a role analogous in many respects to the well-known role of the minimal polynomials of finite matrices in linear algebra. We know, for instance, that similar matrices have the same minimal polynomial. We prove an analogous fact for the minimal functions of contractions, even under a less stringent condition. Proposition 4.6. Let T1 , T2 be two completely nonunitary contractions on the spaces H1 , H2 , respectively, and suppose that T2 is a quasi-affine transform of T1 (cf. Sec. II.3). If one of these contractions is of class C0 then so is the other and their minimal functions coincide. Proof. Let X be a quasi-affinity from H2 to H1 such that X T2 = T1 X. Then X T2n = T1n X (n = 0, 1, . . .), hence also X u(T2 ) = u(T1 )X for every function u ∈ H ∞ . If T1 ∈ C0 we have thus XmT1 (T2 ) = mT1 (T1 )X = O. We conclude that mT1 (T2 ) = O because X is invertible, thus T2 ∈ C0 and mT2 is a divisor of mT1 . Conversely, T2 ∈ C0 implies mT2 (T1 )X = XmT2 (T2 ) = O; because XH2 = H1 , it follows that mT2 (T1 ) = O, thus T1 ∈ C0 and mT1 is a divisor of mT2 . Thus we see that if one of the contractions belongs to C0 then so does the other, and their minimal functions are divisors of each other, and hence they coincide. This result shows in particular that quasi-similar contractions of class C0 have the same minimal function. 4. We add the following result. Proposition 4.7. (a) If T ∈ C0 then T ∗ ∈ C0 also, and mT ∗ = mT˜ . (b) Let T ∈ C0 and let u be an inner function. We have5 u ∈ KT∞ if and only if u and mT have no nonconstant common inner divisor. Proof. Part (a) is obvious from the relation u˜(T ∗ ) = u(T )∗ , holding for every u ∈ HT∞ . To prove part (b), set p = u ∧ mT (the sign ∧ indicating the largest common inner divisor). Because mT (T ) = O, we deduce from Lemma 4.5 that u(T )h = 0 implies p(T )h = 0; the converse implication holds trivially. Thus u(T ) is invertible if and only if p(T ) is invertible. Now we have mT = pq with some inner function q and hence p(T )q(T ) = mT (T ) = O. If p(T ) is invertible this implies q(T ) = O and hence, by virtue of the minimality property of mT , we have q = mT and p = 1. As 1(T ) = I is invertible, we conclude that u(T ) is invertible if and only if p = 1. If p = 1 then u˜ ∧ mT˜ = p˜ = 1 and hence, using (a) also, we conclude from the above result (applied to u˜ instead of u) that u(T )∗ (= u˜(T ∗ )) is invertible, which implies that u(T )−1 has dense domain; thus in this case we have u ∈ KT∞ . This concludes the proof. 5

The definition of KT∞ was given in Sec. 3.2.

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129

5 Minimal function and spectrum A well-known fact in linear algebra is that the zeros of the minimal polynomial of a matrix are exactly the characteristic values of the matrix. We prove an analogous fact for the class C0 . Theorem 5.1. Let mT be the minimal function and σ (T ) the spectrum of the contraction T of class C0 . Let sT be the set consisting of the zeros of mT in the open unit disk D and of the complement, in the unit circle C, of the union of the arcs of C on which mT is analytic (i.e., through which it can be continued analytically). Then

σ (T ) = sT . Proof. Let α be a point of the closed disk D not belonging to sT . Then mT (α ) 6= 0; indeed, if α ∈ D this follows immediately from the definition of sT , and if α ∈ C from the fact that mT being an inner function we have |mT (eit )| = 1 a.e. and thus in particular at the points of C where mT is analytic. Let u(λ ) =

1 [mT (α ) − mT (λ )]. α −λ

It is obvious that u ∈ H ∞ ; by our functional calculus for contractions we have therefore (α I − T )u(T ) = u(T )(α I − T ) = mT (α )I − mT (T ) = mT (α )I.

This shows that α I − T is boundedly invertible, with (α I − T )−1 =

1 u(T ); mT (α )

hence α does not belong to σ (T ). So we have proved the inclusion

σ (T ) ⊂ sT .

(5.1)

Let now α ∈ D be such that mT (α ) = 0. Then we have mT (λ ) =

λ −α n (λ ) ¯ α 1 − αλ

with some inner function nα . By virtue of the functional calculus this implies (T − α I)nα (T ) = (I − α¯ T )mT (T ) = O; because nα is not a multiple of mT , so we have nα (T ) 6= O, and therefore T − α I is not invertible. Hence α ∈ σ (T ). Thus we have sT ∩ D ⊂ σ (T ).

(5.2)

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By virtue of (5.1) and (5.2) it only remains to prove

or, equivalently,

sT ∩C ⊂ σ (T ),

(5.3)

C\sT ⊃ ρ (T ) ∩C,

(5.4)

ρ (T ) denoting the resolvent set of T . We start to this end with the factorization

mT (λ ) = κB(λ )S(λ )

(|κ| = 1),

(5.5)

where B(λ ) is a Blaschke product and S(λ ) is of the form   Z 2π it e +λ , d µ S(λ ) = exp − t eit − λ 0

µ being a nonnegative, finite singular measure (with respect to Lebesgue measure); see (1.11)–(1.13). In order to prove (5.4) we have to show that if an open arc ω of C is contained in ρ (T ), then the function mT is analytic on ω . Every point of ω is at a positive distance from σ (T ), therefore it follows from (5.2) that the zeros of mT cannot accumulate to a point of ω . This ensures that the function B(λ ) is analytic on ω . Hence it remains to consider S(λ ). This function is certainly analytic on ω if µ (ω ) = 0, so it suffices to prove this equation. Now if µ (ω ) > 0 then there exists a closed subarc ω1 of ω , with µ (ω1 ) > 0. The function6   Z eit + λ m1 (λ ) = exp − d µt it (ω1 ) e − λ

is then an inner divisor of mT (λ ); as |m1 (0)| = exp[−µ (ω1 )] < 1, the function m1 (λ ) is not constant. The function m2 = mT /m1 is also inner, and we have m1 m2 = mT ; hence m1 (T )m2 (T ) = mT (T ) = O. (5.6) If the operator m1 (T ) were invertible, (5.6) would imply m2 (T ) = O, and hence mT would be a divisor of m2 , which is impossible. Thus m1 (T ) is not invertible: the subspace H1 = {h : h ∈ H, m1 (T )h = 0}

does not reduce to {0}. H1 is obviously invariant (even hyperinvariant) for T . Let T1 = T |H1 . Because m1 (T1 ) = m1 (T )|H1 , the minimal function mT1 of T1 is a divisor of m1 , so it must be of the form   Z 2π it e +λ mT1 (λ ) = exp − d µ1t eit − λ 0 6

For any subset β of C we denote by (β ) the set of points t in [0, 2π ) for which eit ∈ β .

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131

with a nonnegative, singular measure µ1 , having the majorant µ on ω1 and vanishing elsewhere. This shows that the set sT1 corresponding to the function mT1 is included in the arc ω1 . Relation (5.1), when applied to T1 , then yields σ (T1 ) ⊂ sT1 ⊂ ω1 . Let |ν | > 1. By virtue of the reciprocal relations g = (ν I − T )h,

∞

h = ∑ ν −n−1 T n g, n=0

any invariant subspace for T is transformed by ν I − T onto itself. Hence it follows that (ν I1 − T1)−1 = (ν I − T )−1 |H1 .

Thus if ν tends to a point ξ of C belonging to ρ (T ), (ν I1 − T1 )−1 remains bounded, and this shows that ξ also belongs to ρ (T1 ). In particular, we have ω1 ⊂ ρ (T1 ). When combined with the preceding result σ (T1 ) ⊂ ω1 this gives σ (T1 ) ⊂ ρ (T1 ), an obvious absurdity. This contradiction followed from the hypothesis that µ (ω ) > 0. Thus we must have µ (ω ) = 0, and so the proof of the theorem is complete. Corollary 5.2. Let T be a contraction of class C0 and let λ1 , λ2 , . . . be its different eigenvalues in D. Then ∑(1 − |λn|) < ∞. In fact, the values λn being the zeros of mT (λ ), the convergence of the series in question follows from a theorem of Blaschke on the zeros of a function in H ∞ (cf. H OFFMAN [1], p. 64).

Corollary 5.3. There exists a T ∈ C0 such that σ (T ) = C.

In fact, let µ be a nonnegative, finite, singular measure whose support is the entire unit circle C (e.g., choose a countable dense subset of points of C and assign to each of these points a positive mass so that their sum is finite). The corresponding function   Z 2π it e +λ m(λ ) = exp − d µt , eit − λ 0 being inner, generates by virtue of Proposition 4.3 a contraction T ∈ C0 such that mT = m. By Theorem 5.1 we have σ (T ) = sT , and in this example sT equals C.

6 Minimal function and invariant subspaces 1. The problem of constructing invariant subspaces for an arbitrary (not necessarily normal or compact) operator, and in this way reducing the study of the operator to the study of its “parts” in these subspaces, has been resolved only in certain cases; one such case has been considered in Sec. II.5. We deal with this problem in this book repeatedly and from various aspects; in the present section we show that for contractions T of class C0 an approach to this problem is offered by the factorizations of the minimal function mT .

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Proposition 6.1. Let T be a contraction of class C0 on H (6= {0}), and let H1 be a nontrivial subspace of H, invariant for T . Let   T ∗ T= 1 O T2 be the triangulation of T corresponding to the decomposition H = H1 ⊕ H2 . Then T1 and T2 are also of class C0 , their minimal functions mT1 and mT2 are divisors of mT , and mT is a divisor of mT1 mT2 . Proof. We have obviously T1n = T n |H1

and T2n = P2 T n |H2

(n = 0, 1, . . .),

(6.1)

where P2 denotes the orthogonal projection of H into H2 . Because T is c.n.u. (indeed, T ∈ C0 ), so are T1 and T2 , and hence u(T1 ) and u(T2 ) exist for every u ∈ H ∞ ; from (6.1) we obtain easily that u(T1 ) = u(T )|H1 ,

u(T2 ) = P2 u(T )|H2 .

(6.2)

Choosing u = mT , this yields mT (T1 ) = O and mT (T2 ) = O; hence T1 and T2 belong to C0 and their minimal functions mT1 , mT2 are divisors of mT . From (6.2) it also follows that (mT1 mT2 )(T )h1 = (mT1 mT2 )(T1 )h1 = mT1 (T1 )mT2 (T1 )h1 = 0 for h1 ∈ H1 and P2 mT2 (T )h2 = mT2 (T2 )h2 = 0 for h2 ∈ H2 .

By the second relation, h = mT2 (T )h2 is orthogonal to H2 and so it belongs to H1 . Hence: (mT1 mT2 )(T )h2 = mT1 (T )mT2 (T )h2 = mT1 (T )h = mT1 (T1 )h = 0. Thus we have (mT1 mT2 )(T )h = 0 for h ∈ H1 as well as for h ∈ H2 , and hence for every h ∈ H (i.e., (mT1 mT2 )(T ) = O). Consequently, mT is a divisor of mT1 mT2 . This concludes the proof. Proposition 6.2. Let T be a contraction of Wclass C0 on H (6= {0}), and let Hα (α ∈ A) be invariant subspaces for T . H∨ = α ∈A Hα is then also invariant for T , and the minimal function of T∨ = T |H∨ is the least common inner multiple of the minimal functions of the contractions Tα = T |Hα (α ∈ A). Proof. The invariance of H∨ is obvious; the fact that Tα and T∨ are of class C0 follows from the preceding proposition. Because Tα can also be considered as the restriction of T∨ to Hα , mT∨ is divisible by mTα for each α ∈ A, and hence mT∨ is

6. M INIMAL FUNCTION AND INVARIANT SUBSPACES

133

also divisible by the least common inner multiple m∨ of the functions mTα (α ∈ A). On the other hand we have m∨ (T∨ )hα = m∨ (T )hα = m∨ (Tα )hα = 0

for hα ∈ Hα

by (6.2) and because m∨ is a multiple of mTα . The equation m∨ (T∨ )h = 0 holds then for the elements h of the span of the subspaces Hα , that is, for h ∈ H∨ ; hence m∨ (T∨ ) = O. Consequently, mT∨ is a divisor of m∨ . Being divisors of each other, the (inner) functions mT∨ and m∨ coincide. 2. It is convenient also to admit the operator T = O on the trivial space {0} as belonging to the class C0 ; its minimal function is then obviously the constant function 1. It is obvious that except for this trivial case no minimal function is constant. The following theorem establishes a mutual correspondence between the inner divisors of the minimal function of T and some of the invariant subspaces for T . Theorem 6.3. Let T be a contraction on H, of class C0 , with minimal function mT . To each inner divisor m of mT we let correspond the subspace Hm = {h : h ∈ H, m(T )h = 0};

(6.3)

Hm is hyperinvariant for T . Set H′m = H ⊖ Hm . The contractions Tm (= T |Hm ) and Tm′ appearing in the triangulation   T X T = m m′ , H = Hm ⊕ H′m O Tm have their minimal functions equal to m and m′ = mT /m, respectively. Moreover, we have  {0} if m = 1, (i) Hm = H if m = mT . (ii) Hm1 ⊂ Hm2 if, and only if, m1 is a divisor of m2 . (iii) If {mα } (α ∈ A) is a finite or infinite system of inner divisors of mT , with the largest common inner divisor m∧ and the least common inner multiple m∨ , then W T Hmα = Hm∧ and Hmα = Hm∨ . α

α

Proof. The hyperinvariance of Hm for T follows immediately from the fact that if a bounded operator S commutes with T then it also commutes with m(T ). By virtue of (6.2) and of the definition (6.3) of Hm we have m(Tm ) = m(T )|Hm = O. Consequently, mTm is a divisor of m, and hence m = mTm p for some inner function p.

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On the other hand, for every h ∈ H we have m(T ) m′ (T )h = mT (T )h = 0, and hence h1 = m′ (T )h belongs to Hm . Applying (6.2) one obtains for h ∈ H′m m′ (Tm′ )h = P′ m′ (T )h = P′ h1 = 0, where P′ is the orthogonal projection of H onto H′m . So we have m′ (Tm′ ) = O, and hence mTm′ is a divisor of m′ ; that is, m′ = mTm′ q for some inner function q. These results imply mT = mm′ = mTm p · mTm′ q = mTm mTm′ · pq. On the other hand it follows from Proposition 6.1 that mT is a divisor of mTm mTm′ . Consequently, pq = 1, and hence p = 1, q = 1, m = mTm , m′ = mTm′ . Assertion (i) is obvious. As to (ii), it follows readily from the definition (6.3) that if m1 is a divisor of m2 , then Hm1 ⊂ Hm2 . Conversely, Hm1 ⊂ Hm2 implies Tm1 = Tm2 |Hm1 and hence m2 (Tm1 ) = m2 (Tm2 )|Hm1 = O, because m2 (T2 ) = O by the definition of Tm2 . Hence we conclude that the minimal function of Tm1 , that is, m1 , is a divisor of m2 . Finally, as to (iii), let us observe first that because m∧ is a divisor of mα we have Hm∧ ⊂ Hmα (α ∈ A); consequently Hm∧ ⊂

T α

Hmα .

(6.4)

Let h be an element of this intersection, that is, such that mα (T )h = 0 for all α ∈ A. By virtue of Lemma 4.5 we have then m∧ (T )h = 0, thus h ∈ Hm∧ . Thus the opposite of the inclusion (6.4) is also valid, and so there is equality in (6.4). Let us turn now to m∨ . Because m∨ is a multiple of mα , we have Hmα ⊂ Hm∨ for all α ∈ A; hence also W Hmα ⊂ Hm∨ . (6.5) α

H+

denote the left-hand side of (6.5): this is an invariant subspace for T . Let Let H0 = Hm∨ ⊖ H+ , and consider the triangulation  +  T X Tm∨ = O T0

corresponding to the decomposition Hm∨ = H+ ⊕ H0 . Setting mα′ = m∨ /mα we have

(α ∈ A)

m∨ (T ) = mα (T )m′α (T );

(6.6)

6. M INIMAL FUNCTION AND INVARIANT SUBSPACES

hence

135

mα′ (T )Hm∨ ⊂ Hmα

and consequently

m′α (T 0 )H0 = P0 m′α (Tm∨ )H0 ⊂ P0 m′α (T )Hm∨ ⊂ P0 Hmα ⊂ P0 H+ = {0}; that is, mα′ (T 0 ) = O. As this is true for every α ∈ A, we have by Lemma 4.5 that m′∧ (T 0 ) = O, where m′∧ denotes the largest common inner divisor of the functions m′α (α ∈ A). Now by definition (6.6) the functions m′α are relatively prime (i.e., m′∧ = 1), and this implies that H0 = {0}, Hm∨ = H+ ; thus there is equality in (6.5). This completes the proof of Theorem 6.3. 3. One of the consequences of this theorem is that if m1 and m2 are relatively prime inner divisors of mT , then Hm1 ∩ Hm2 = {0} and Hm1 ∨ Hm2 = Hm1 m2 .

(6.7)

This is the case in particular if there exist u1 , u2 ∈ H ∞ such that m1 u1 + m2 u2 = 1.

(6.8)

Hm1 ∔ Hm2 = Hm1 m2 ,

(6.9)

Then we have even the sign ∔ denoting direct (not necessarily orthogonal) sum. Indeed, (6.8) implies m1 (T )u1 (T )h + m2(T )u2 (T )h = h for every h ∈ H; now h1 = m2 (T )u2 (T )h ∈ Hm1 for h ∈ Hm1 m2 , because m1 (T )h1 = u2 (T )m1 (T )m2 (T )h = u2 (T ) · (m1 m2 )(T )h = 0, and by analogous reason h2 = m1 (T )u1 (T )h ∈ Hm2 . Thus we have proved the following result. Proposition 6.4. Let m1 , m2 be inner divisors of mT such that there exist u1 , u2 ∈ H ∞ satisfying the equation m1 u1 + m2 u2 = 1. Then we have Hm1 m2 = Hm1 ∔ Hm2 . Another consequence of Theorem 6.3 is that if m is a nontrivial inner divisor of mT then the corresponding subspace Hm of H is also nontrivial. In fact, none of the subspaces Hm and H′m = H ⊖ Hm equals {0}, because the operators Tm and Tm′ appearing in the corresponding triangulation of T have the nonconstant minimal functions m and m′ = mT /m. Now we know that every nonconstant inner function has a nontrivial inner divisor, except the functions κ

λ −a 1 − a¯λ

(|κ| = 1, |a| < 1);

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¯ − a¯λ )mT (λ ), and see Sec. 1.4. If mT is of this exceptional form, then λ − a = κ(1 hence T − aI = κ(I ¯ − aT ¯ )mT (T ) = O, T = aI;

in this case every subspace of H is invariant for T , thus if dim H > 1 there are nontrivial invariant subspaces. We conclude that if dim H > 1, every contraction T of class C0 on H has a nontrivial invariant subspace. This result can be generalized as follows. Let H′ and H′′ be two subspaces of H, invariant for T , and such that H′ ⊃ H′′

and

dim(H′ ⊖ H′′ ) > 1.

By virtue of Proposition 6.1 we have T ′ = T |H′ ∈ C0 . Let   ′′ T ∗ ′ T = O T ′′′ be the triangulation of T ′ corresponding to the decomposition H′ = H′′ ⊕ H′′′ ; then T ′′′ ∈ C0 , and as dim H′′′ > 1, there exists in H′′′ a nontrivial subspace H′′′ 1 , invariant ′ , invariant for T and properly contained for T ′′′ . H′′ ⊕ H′′′ is then a subspace of H 1 between H′′ and H′ . Let us state this result. Theorem 6.5. Let H′ and H′′ be invariant subspaces for the contraction T ∈ C0 , such that H′ ⊃ H′′ and dim(H′ ⊖ H′′ ) > 1. Then there exists a subspace, invariant for T , and properly contained between H′ and H′′ .

7 Characteristic vectors and unicellularity 1. Let T ∈ C0 and let a be a point of the spectrum of T in D, that is, a zero of the minimal function mT (cf. Theorem 5.1). Setting ba (λ ) = we have then

λ −a 1 − a¯λ

mT (λ ) = bka (λ ) · ma (λ ),

(7.1)

(7.2)

where k ≥ 1 and ma is an inner function such that ma (a) 6= 0. The factors in (7.2) are relatively prime. We show in fact that there exist u, v ∈ H ∞ such that bka · u + ma · v = 1. By a homography we can reduce this assertion to the case a = 0. Then m0 (0) 6= 0 so that 1/m0 (λ ) has a Taylor series expansion around the point 0; let v(λ ) be the

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137

(k − 1)th partial sum of this series. Then we have 1 = m0 (λ )

1 = m0 (λ )v(λ ) + λ k u(λ ) m0 (λ )

with a function u(λ ), which is holomorphic in some neighborhood of 0 and whose definition extends by means of the relation λ k u(λ ) = 1 − m0 (λ )v(λ ) to the whole of D. The function u(λ ) thus obtained belongs obviously to H ∞ . So we have

λ k u(λ ) + m0(λ )v(λ ) = 1

(λ ∈ D)

with u, v ∈ H ∞ . Let Ha , H′a be the hyperinvariant subspaces corresponding to the factors in (7.2), that is, Ha = {h : h ∈ H, bka (T )h = 0},

H′a = {h : h ∈ H, ma (T )h = 0}.

By virtue of Proposition 6.4 we have H = Ha ∔ H′a . Then, by Theorem 6.3, the minimal functions of Ta = T |Ha and Ta′ = T |H′a are equal to bka and ma , respectively. Now the conditions bna (T )h = 0 and (T − aI)nh = 0 for an h ∈ H are obviously equivalent. Hence a is a characteristic value of T of index k, and Ha consists precisely of the characteristic vectors of T associated with the characteristic value a (and of 0).7 As to Ta′ , ma (a) 6= 0 implies by Theorem 5.1 that a does not belong to the spectrum of Ta′ . So we have proved the following proposition. Proposition 7.1. For a contraction T of class C0 on H, the points a of the spectrum in the interior of the unit circle are eigenvalues of T . As a characteristic value of T, a has finite index, equal to its multiplicity as zero of the minimal function mT (λ ). Furthermore, for every such a, H decomposes into the (not necessarily orthogonal) direct sum of two subspaces, hyperinvariant for T , say Ha and H′a , so that Ha consists of the characteristic vectors of T associated with the value a (and of the vector 0), whereas Ta′ = T |H′a has a in its resolvent set. There is some interest in the following result.

7 A complex number a is called a characteristic value of T if there exists a vector h 6= 0 such that (T − aI)n h = 0 for n large enough; h is called a characteristic vector associated with the value a. If (T − aI)n h = 0, but h0 = (T − aI)n−1 h 6= 0, then h0 is obviously an eigenvector of T corresponding to the eigenvalue a; thus every characteristic value is also an eigenvalue. The characteristic value a is said to have a (finite) index k if (T − aI)k h = 0 for every characteristic vector h associated with a, but (T − aI)k−1 h0 6= 0 for at least one characteristic vector h0 associated with a.

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Proposition 7.2. Let T ∈ C0 . In order that the characteristic vectors of T associated with the points of the spectrum of T in D span the whole space H, it is necessary and sufficient that the minimal function mT be a Blaschke product. W

Proof. Let us suppose that H = a Ha , a running over the points of the spectrum of T in D, and Ha being the subspace formed by the characteristic vectors of T associated with the value a (and by the vector 0). Let B denote the Blaschke factor in the factorization of type (1.11) of the inner function mT . If k = k(a) is the index of a as a characteristic value of T , then B is divisible by bka so that for h ∈ Ha we have B(T )h = (B/bka )(T ) · bka(T )h = 0; W

this implies B(T ) = O because H = a Ha . Then mT must be a divisor of B; hence mT = B. Conversely, if we suppose that mT is a Blaschke product, then mT is obviously equal to the least common inner multiple of its factors bka , so that by Theorem 6.3 we have W Ha = HmT = H. This concludes the proof of Proposition 7.2.

2. A bounded operator T on H is said to be unicellular if its invariant subspaces are totally ordered by inclusion, that is, if for any two of these subspaces, say M and N, we have M ⊂ N or M ⊃ N. Observe that if T is unicellular then so is T ∗ , a consequence of the fact that if {M} is the collection of invariant subspaces for T then {M⊥ } is the collection of invariant subspaces for T ∗ . We consider the unicellular contractions T of class C0 . Proposition 7.3. Let T be a contraction on H of class C0 and unicellular. Then mT (λ ) is of the form   λ −α n (7.3) (|α | < 1; n a positive integer), ¯ 1 − αλ or of the form   λ +α exp s λ −α

(|α | = 1; s a positive real number),

(7.4)

accordingly as dim H = n or dim H = ∞. Proof. Let us observe first that the minimal function mT cannot have relatively prime nontrivial inner divisors, for in that case there would exist nontrivial invariant subspace H1 , H2 for T such that H1 ∩ H2 = {0} (cf. Theorem 6.3 (iii)), which is impossible on account of the unicellularity of T . From the general representation (1.11)–(1.13) of the inner functions we deduce then that mT is either a Blaschke product all factors of which correspond to the same zero α in D, or an inner function S generated by a nonnegative, finite, singular measure on C, whose support is

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a one-point set {α } (α ∈ C). Thus the only functions to be considered are those given by (7.3) and (7.4). It remains to show that the case (7.3) occurs precisely if dim H = n. As a unicellular operator T can have at most one eigenvector (up to a scalar factor), it follows in the case dim H = n that T has, with respect to an appropriately chosen basis in H, a matrix   α 1    α 1        .. ..  n  . .     α 1     α

consisting of just one Jordan cell (whence the term “unicellular”). If T ∈ C0 , then T n → O, and hence |α | < 1. From this representation of T we obtain that the subspaces Hν = {h : h ∈ H, (T − α I)ν h = 0} (ν = 0, . . . , n) (7.5) satisfy the relations

{0} = H0 ⊂ . . . ⊂ Hn = H

and

dim Hν = ν ,

(7.6)

and hence in particular Hn−1 6= Hn . This is obviously equivalent to the fact that ¯ ). Because the specbαn (T ) = O and bαn−1 (T ) 6= O, where bα (λ ) = (λ − α )/(1 − αλ trum of T is the one-point set {α }, mT (λ ) is equal to a power of bα (λ ), so we have necessarily mT (λ ) = bnα (λ ). Let us now show that, conversely, if a unicellular contraction T (∈ C0 ) on a space H is such that mT = bαn with |α | < 1 and a natural number n, then dim H < ∞. In fact, bαn (T ) = O implies that (T − α I)n = O and hence   n−1 n n T =− ∑ (−α )n−k T k . k=0 k It follows that for every h ∈ H the subspace M(h) spanned by h, T h, . . . , T n−1 h is invariant for T ; clearly dim M(h) ≤ n. Let h0 be chosen so that M(h0 ) have maximal dimension; we show that then M(h0 ) = H. In the contrary case there would exist an h1 ∈ H not belonging to M(h0 ). As T is unicellular, M(h0 ) is necessarily included in M(h1 ), and hence dim M(h1 ) > dim M(h0 ), which is impossible. This proves that dim H ≤ n. (Actually we have here an equality: this follows from what was proved above for the case of spaces of finite dimension.) From these results we conclude that the case (7.4) occurs precisely if dim H = ∞. This concludes the proof. Corollary. For a unicellular contraction of class C0 on H, the spectrum σ (T ) consists of a single point α . We have |α | < 1 or |α | = 1, accordingly as dim H is finite or infinite.

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We add a further remark. Proposition 7.4. For a unicellular contraction T of class C0 , on a space H with dim H = n < ∞, the subspaces Hν defined by (7.5) (where {α } = σ (T )) are the only invariant subspaces for T . Proof. The invariance (even hyperinvariance) of Hν for T is obvious. Let M be an arbitrary invariant subspace. From the unicellularity of T and from the inclusions (7.6) it follows that Hµ ⊂ M ⊂ Hµ +1 for some µ . Because dim Hν = ν for every ν , we have µ ≤ dim M ≤ µ + 1 and hence M coincides with Hµ or Hµ +1 . This concludes the proof. Let us remark that in the definition of Hν the condition (T − α I)ν h = 0 can be replaced by the equivalent one bνα (T )h = 0, and that the functions bνα (ν = 0, . . . , n) are the only inner divisors of mT (= bnα ). Analogous facts can be established in the case of infinite-dimensional spaces. Indeed, consider a contraction T of class C0 with σ (T ) = {α }, |α | = 1. Replacing T by α¯ T we can reduce our study to the case σ (T ) = {1}. The minimal function of T then has the form (7.4) with α = 1. These functions play an important role in the sequel, therefore we introduce the shorter notation   λ +1 es (λ ) ≡ exp s (s ≥ 0). (7.7) λ −1 These functions are inner, and the only inner divisors of ea are the functions es with 0 ≤ s ≤ a. Proposition 7.5. Let T be a contraction of class C0 on H, such that mT (λ ) = ea (λ ) with

a = aT > 0.

The subspaces Hs = {h : h ∈ H, es (T )h = 0}

(0 ≤ s ≤ a)

(7.8)

are hyperinvariant for T , and we have H0 = {0}; Ha = H; Hs1 $ Hs2 Hs =

Hs =

T

x>s

S

x<s

for 0 ≤ s1 < s2 ≤ a;

(7.9)

Hx

for 0 ≤ s < a;

(7.10)

Hx

for 0 < s ≤ a.

(7.11)

Thus the corresponding orthogonal projections Es form a strictly increasing, continuous spectral family in the interval [0, a]. If, moreover, T is unicellular then the subspaces Hs are the only invariant subspaces for T .

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Proof. Let us set Ts = T |Hs (0 ≤ s ≤ a). By Theorem 6.3 we have mTs = es ; hence it follows in particular that if s1 6= s2 then Ts1 6= Ts2 and consequently Hs1 6= Hs2 . The other assertions in (7.9) are obvious. Relations (7.10) and (7.11) follow from Theorem 6.3(iii) by the obvious fact that the function es is the largest common inner divisor of the functions ex (x > s), and the least common inner multiple of the functions ex (x < s). It remains only to consider the case of a unicellular T . Let M be a nontrivial invariant subspace for T and set s = sup{x : Hx ⊂ M}. By (7.9) and (7.11) we also have then Hs ⊂ M; consequently s < a. If s < x < a, Hx is not included in M, therefore M must be included in Hx . On account of (7.10) we then have T M⊂ Hx = Hs x>s

also. So we have simultaneously Hs ⊂ M and M ⊂ Hs , and hence M = Hs . This completes the proof of Proposition 7.5.

An example of an operator T of the type considered in Proposition 7.5 is the operator defined in Proposition 4.3(a) with m = ea . This operator is unicellular. Indeed, by Proposition 4.3(b) any invariant subspace H1 of this operator T is given by the formula H1 = m2 (H 2 ⊖ m1 H 2 ) = m2 H 2 ⊖ ea H 2 , where ea = m1 m2 and m1 , m2 are inner functions. Therefore in the definition of H1 we can take m1 = es , m2 = ea−s for some 0 ≤ s ≤ a. So the invariant subspaces of T are of the form ea−s H 2 ⊖ ea H 2 which clearly increase with s. This concludes the proof of the unicellularity of T .

3. We conclude this section by establishing two properties of arbitrary unicellular operators on a (complex) Hilbert or Banach space H. The second property concerns “cyclic” vectors, whose definition is as follow. Definition. For an operator T on H a vector h is called cyclic if the vectors T n h (n = 0, 1, . . .) span H. Proposition 7.6. For every bounded unicellular operator T , the invariant subspaces are hyperinvariant. Proof. Let L be an invariant subspace for T and let X be a bounded operator commuting with T . Then for any complex λ the subspace Lλ = (λ I − X )L is also invariant for T , and hence we have either Lλ ⊂ L or Lλ ⊃ L, depending on λ . If the first case occurs for at least one λ , then XL ⊂ L. If the second case occurs for every

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λ , then we have in particular L ⊃ (λ I − X)−1 L for |λ | > kX k and hence, by virtue of the relation X=

1 2π i

I

|λ |=ρ

λ (λ I − X)−1 d λ

(ρ > kX k), 8

we also have L ⊃ X L. Thus XL ⊂ L holds in every case. This proves that L is hyperinvariant for T . Proposition 7.7. If T is a unicellular operator on a separable space H9 then the set S of noncyclic vectors for T is a union of at most countably many proper10 invariant subspaces of H. Proof. If {Mα } is the family of all proper invariant subspaces of T then clearly S = ∪Mα . Let {xi } be a countable dense subset of S. For each i choose an Mαi such that xi is in Mαi , and set S′ = ∪Mαi . If S′ = S then S is exhibited as a countable union of subspaces of H. If S′ is a proper subset of S then there exists an Mβ that is contained in none of the Mαi and hence contains all the Mαi . But then Mβ contains {xi } and therefore S; thus we have S = Mβ . The proof is now complete. Because a proper subspace is nowhere dense in H we can apply the Baire category theorem (cf., e.g., D UNFORD AND S CHWARTZ [1] p. 20) to obtain from Proposition 7.7 the following corollary. Corollary 7.8. For any countable set {Ti } of unicellular operators on a separable space H there exist vectors which are cyclic for each Ti . In particular, if T is unicellular on a Hilbert space then there exist vectors which are cyclic both for T and T ∗.

8 One parameter semigroups 1. We now indicate how our functional calculus can be applied to the study of continuous one-parameter semigroups {T (s)}s≥0 of contractions on the space H; see Secs. I.8.2 and I.10.2. Let us recall that for such a semigroup the (infinitesimal) generator A, defined by 1 Ah = lim [T (s) − I]h (8.1) s→+0 s whenever this limit exists, is a closed operator with domain D(A) dense in H, A−I is boundedly invertible and, moreover, A determines the semigroup {T (s)} uniquely. (Theorem of Hille and Yosida, cf. H ILLE [1] p. 238, or [Func. Anal.] Secs. 142 and 143.) For |λ | > kXk we have λ (λ I − X)−1 = I + λ −1 X + λ −2 X 2 + · · ·. It is easy to see that if H is a nonseparable Hilbert space then no operator on H can be unicellular. 10 That is, different from H.

8

9

8. O NE PARAMETER SEMIGROUPS

143

Let us note that kT (s)k ≤ 1 implies Re(Ah, h) ≤ 0

(8.2)

for h ∈ D(A); see (I.10.6). This implies further k(A + I)hk2 − k(A − I)hk2 = 4 Re(Ah, h) ≤ 0 for h ∈ D(A); hence we obtain that the operator T defined by T = (A + I)(A − I)−1

(8.3)

is a contraction (its domain of definition is the whole of H, for so is the domain of (A − I)−1). We call this operator T the cogenerator of the semigroup {T (s)}; this is justified by the fact that T determines A, and hence also {T (s)}, uniquely. In fact, (8.3) implies that T − I = (A + I)(A − I)−1 − I = 2(A − I)−1; (8.4) hence we see that (T − I)−1 exists and equals 21 (A − I), and therefore A = (T + I)(T − I)−1.

(8.5)

The existence of the (generally unbounded) operator (T − I)−1 simply means that 1 is not an eigenvalue of T . By virtue of what we said above, the cogenerator T can replace the generator A in the study of the semigroup {T (s)}; the advantage of using T instead of A is obvious: T is a bounded operator (indeed, a contraction), whereas A is not bounded in general. We show that several properties of the semigroup are reflected in a striking way by analogous properties of the cogenerator. The following problems should be addressed first: (1) to characterize the contractions T that are cogenerators of some continuous one-parameter semigroups {T (s)} of contractions; and (2) to make explicit the relations between T and {T (s)} without referring to the infinitesimal generator A. Before formulating the solution of these problems, let us remark that if 1 is not an eigenvalue of a contraction T , then it is not an eigenvalue of the unitary part of T , either. Thus, if ET is the spectral measure on C corresponding to the unitary part of T , then ET ({1}) = O. Consequently, every function u ∈ H ∞ that is defined and continuous on D\{1}, belongs to the class HT∞ . This is in particular the case for the functions es (s ≥ 0) (cf. (7.7)), which are holomorphic on the whole complex plane except the point 1, and satisfy |es (λ )| ≤ 1 on D

and |es (λ )| = 1 on C\{1}.

Theorem 8.1. Let T be a contraction on H. In order that there exist a continuous semigroup {T (s)}s≥0 of contractions whose cogenerator equals T , it is necessary and sufficient that 1 not be an eigenvalue of T . If this is the case, then T and {T (s)}

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determine each other by the relations T (s) = es (T ) and

(s ≥ 0)

(8.6)

T = lim ϕs (T (s)),

(8.7)

s→+0

where

ϕs (λ ) =

λ −1+s 1−s λn 2s ∞ = − ∈ A. ∑ λ − 1 − s 1 + s 1 + s n=1 (1 + s)n

(8.8)

Proof. We have already observed that cogenerators are contractions for which 1 is not an eigenvalue. Let T be an arbitrary contraction not having the eigenvalue 1. Applying Theorem 2.3 we deduce from the obvious relations e0 (λ ) = 1, and

es (λ )et (λ ) = es+t (λ ); lim es (λ ) = 1

s→+0

|es (λ )| ≤ 1 for λ ∈ D

for λ ∈ D\{1},

that the operators es (T ) (s ≥ 0) form a continuous semigroup of contractions. Let us denote its generator by A′ and its cogenerator by T ′ . We show that T ′ = T . To this end we consider the functions us (λ ) = (ϕs ◦ es )(λ ) = [es (λ ) − 1 + s][es(λ ) − 1 − s]−1

(s > 0).

(8.9)

It is easy to show that they are holomorphic and bounded by 1 on D, continuous on D\{1}, and such that lim us (λ ) = λ for λ 6= 1. s→+0

Applying again Theorem 2.3 we obtain that us ∈ HT∞ , and that kus (T )k ≤ 1 (s ≥ 0) and

lim us (T ) = T.

s→+0

(8.10)

On the other hand, (8.9) implies us (T ) = (ϕs ◦ es )(T ) = ϕs [es (T )].

(8.11)

By virtue of (8.9) we also have   1 1 us (T ) (es (T ) − I) − I = (es (T ) − I) + I; s s applying both sides to an arbitrary element h of the domain of A′ and letting s → +0 we obtain, in view of (8.10), that T (A′ − I)h = (A′ + I)h;

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thus T (A′ − I) = A′ + I and T = (A′ + I)(A′ − I)−1 ; that is, T = T ′ . Equation (8.6) follows from the fact that a semigroup is determined uniquely by its cogenerator. Remark. It follows from Proposition I.3.1 that if 1 is not an eigenvalue of the contraction T then it is not an eigenvalue of T ∗ , either. Moreover, as e˜s = es and hence es (T )∗ = es (T ∗ ), we conclude that if T is the cogenerator of the semigroup {T (s)}s≥0 then T ∗ is the cogenerator of the adjoint semigroup {T (s)∗ }s≥0 . Consequently, the strong convergence (8.7) remains valid with T replaced by T ∗ and T (s) replaced by T (s)∗ . We use this fact in the following proof. Proposition 8.2. A continuous semigroup of contractions {T (s)}s≥0 consists of normal, self-adjoint, or unitary operators, if and only if its cogenerator T is normal, self-adjoint, or unitary, respectively. Proof. If T (s) is normal then so is ϕs [T (s)]. Now

ϕs [T (s)] → T

and ϕs [T (s)∗ ] → T ∗

for s → +0

by virtue of (8.7) and the last remark. As ϕs (T (s)∗ ) = ϕs (T (s))∗ (because ϕ ˜s = ϕs ), it follows that T is also normal. If T (s) is self-adjoint then so is ϕs (T (s)) as well as T = lims→+0 ϕs (T (s)). If T (s) is unitary then ϕs [T (s)] is the integral of the function ϕs (λ ) on the unit circle with respect to the spectral measure corresponding to the unitary operator T (s). Now one shows (e.g., by using the Apollonius circles) that 1 ≥ |ϕs (λ )| ≥

2−s 2+s

for |λ | = 1 and 0 < s ≤ 1,

which implies khk ≥ kϕs [T (s)]hk ≥

2−s khk for h ∈ H, 2+s

0 < s ≤ 1.

Letting s → +0 this yields kT hk = lim kϕs [T (s)]hk = khk. s→+0

Thus T is isometric, and because it is also normal, it is unitary. We turn now to the converse implications. Let us suppose that T is normal, with the spectral representation Z T=

λ dKλ .

From Theorem 2.3 (f) we derive that

T (s) = es (T ) =

Z

es (λ ) dKλ ,

(8.12)

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and hence T (s) is also normal. As it suffices to restrict the domain of integration to the spectrum of T , and as es (λ ) is real-valued on the real axis and of modulus 1 on the unit circle, it results from (8.12) that if T is self-adjoint, or unitary, then so is T (s) for every s ≥ 0, respectively. This concludes the proof. 2. It is of interest to note the following particular consequences of (8.12). If T (s) (and T ) are unitary, (8.12) takes the form   Z λ +1 T (s) = dKλ exp s λ −1 |λ |=1 and this representation reduces to Stone’s theorem T (s) =

Z ∞

−∞

eisx dEx

(8.13)

by means of the mapping

λ → x = −i

λ +1 λ −1

(|λ | = 1, λ 6= 1)

of the unit circle, except the point λ = 1, onto the real axis.11 Similarly, if T (s) (and T ) are self-adjoint, we deduce from (8.12) the spectral representation Z T (s) =

∞

0

by means of the mapping

λ →y=

e−sy dEy

(8.14)

1+λ 1−λ

of the interval −1 ≤ λ < 1 onto the semiaxis 0 ≤ y < ∞.12

3. Let T be the cogenerator of the semigroup {T (s)} on H, and let H = H0 ⊕ H1 be the decomposition of the space corresponding to the unitary part T0 = T |H0 and the c.n.u. part T1 = T |H1 of T : (8.15)

T = T0 ⊕ T1 .

Then we have u(T ) = u(T0 ) ⊕ u(T1 ) for every u ∈ HT∞ and in particular for the functions es ; so we obtain T (s) = T0 (s) ⊕ T1 (s) 11

(s ≥ 0)

(8.16)

This way of obtaining Stone’s theorem is close to the one followed by VON N EUMANN [2]. This is a particular case of a theorem of Sz.-Nagy and Hille (cf. [Func. Anal.] Sec. 141). The general case can be reduced to this one, because it can be proved directly that if {N(s)}s≥0 is a strongly continuous semigroup of normal operators then there exists a real number α such that T (s) = e−sα N(s) is a contraction for every s ≥ 0.

12

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147

for the corresponding semigroups, that is, for T0 (s) = es (T0 ) and T1 (s) = es (T1 ).

(8.17)

As T0 is unitary, T0 (s) is also unitary for every s ≥ 0; see Proposition 8.2. On the other hand, there is no subspace H′1 of H1 reducing all the operators T1 (s) to unitary ones except the trivial one H′1 = {0}. Indeed, in the contrary case the operators T1′ (s) = T1 (s)|H′1 would form a unitary semigroup on H′1 , so the corresponding cogenerator T1′ = T1 |H1′ would be unitary too (cf. Proposition 8.2), which in the case H′1 6= {0} would contradict the fact that T1 is c.n.u. It is natural to call a semigroup of contractions {T (s)}s≥0 on H completely nonunitary, if none of the subspaces H′ 6= {0} of H reduces all the operators T (s) to unitary ones. We can then formulate our result as follows. Proposition 8.3. For every continuous one-parameter semigroup of contractions, the canonical decomposition (8.15) of the cogenerator induces by means of (8.16) and (8.17) a decomposition of the semigroup into the orthogonal sum of a unitary semigroup and of a completely nonunitary semigroup. The uniqueness of this decomposition follows from the uniqueness of the canonical decomposition of the cogenerator.

9 Unitary dilation of semigroups 1. Let {T (s)}s≥0 be a continuous one-parameter semigroup of contractions on H, let T be its cogenerator, and let U be the minimal unitary dilation of T acting on the space K, W K= U n H. (9.1) −∞
As 1 is not an eigenvalue of T , it is not an eigenvalue of U either (cf. Proposition II.6.1). Consequently, U is the cogenerator of some continuous semigroup {U(s)}s≥0 of unitary operators, or what amounts to the same thing by setting U(−s) = U(s)−1 , of a continuous group of unitary operator {U(s)}−∞<s<∞. By virtue of Theorem 8.1 we have T (s) = es (t), and

T = lim ϕs [T (s)],

U(s) = es (U) (s ≥ 0) U = lim ϕs [U(s)]

Relations (9.2) imply by Theorem 2.3 (g) that T (s) = pr U(s)

(s ≥ 0).

(s → +0).

(9.2) (9.3)

(9.4)

So we have obtained another proof of Theorem I.8.1 on the existence of a unitary dilation {U(s)} of a continuous semigroup of contractions {T (s)}; this proof

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derives the existence of the unitary dilation of the semigroup from the existence of the unitary dilation of the cogenerator. Moreover, the fact that U is the minimal unitary dilation of T implies that {U(s)} is the minimal unitary dilation of {T (s)}, that is, K′ =

W

−∞<s<∞

U(s)H

equals K. Indeed, K′ reduces each operator U(t), hence also ϕt [U(t)] as well as its limit as t → +0; that is, K′ reduces U. Because H is contained in K′ , U n H is also contained in K′ for n = 0, ±1, . . . . On account of (9.1) this implies K ⊂ K′ . Thus K′ = K. We also have W n W U H= U(s)H (9.5) n≥0

s≥0

and, generally,

W

n≥0

U n K1 =

W

s≥0

for an arbitrary subset

U(s)K1

K1 of K.

(9.6)

In fact, if we use the Taylor series expansions of the functions es (λ ) ∈ H ∞ and [ϕs (λ )]n ∈ A (s > 0, n > 0), say ∞

es (λ ) = ∑ ck (s)λ k , k=0

∞

[ϕs (λ )]n = ∑ dk (s, n)λ k , k=0

(9.7)

as well as relations (8.6) and (8.7), we obtain ∞

U(s) = es (U) = lim ∑ rk ck (s)U k r→1−0 k=0

and

(s > 0)

∞

U n = lim ϕsn [U(s)] = lim ∑ dk (s, n)U(ks) s→+0

s→+0 k=0

(n > 0);

(9.8)

(9.9)

hence (9.6) follows in an obvious manner. The dual relation W

n≥0

U ∗n K1 =

W

s≥0

U(s)∗ K1

(9.10)

is derived in the same way, considering the adjoint semigroup {U(s)∗ }. The following relations can be proved similarly. W

(U n − T n )H1 =

n≥0

and

W

(U ∗n − T ∗n )H1 =

n≥0

W

(U(s) − T (s))H1

(9.11)

W

(U(s)∗ − T (s)∗ )H1 ,

(9.12)

s≥0

s≥0

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149

where H1 is an arbitrary subset of H. In fact, (9.11) results from ∞

∑ rk ck (s)(U k − T k ) r→1−0

U(s) − T (s) = es (U) − es (T ) = lim

(s > 0)

k=0

and ∞

U n − T n = lim ϕsn [U(s)] − lim ϕsn [T (s)] = lim ∑ dk (n, s)(U(ks) − T (ks)), s→+0

s→+0

s→+0 k=0

and (9.12) follows by considering the adjoint semigroups. As a combination of the relations (9.10) and (9.11), and of the relations (9.6) and (9.12), we obtain W

m,n≥0

and

W

m,n≥0

U ∗m (U n − T n )H1 =

U m (U ∗n − T ∗n )H1 =

W

U(t)∗ (U(s) − T (s))H1

(9.13)

W

U(t)(U(s)∗ − T (s)∗ )H1

(9.14)

t,s≥0

t,s≥0

for an arbitrary subset H1 of H. These relations are interesting mainly because they indicate once again the symmetry between semigroups and their cogenerators. But they also have an interest all their own. In fact, the space (9.5) is the space of the minimal isometric dilation of T (cf. Sec. I.4), and the spaces (9.11)–(9.14) are, for H1 = H, equal in this order to the spaces ∞ L U n L, 0

where

∞ L U ∗n L∗ , 0

M(L) =

L = (U − T )H

∞ L

−∞

U n L,

and M(L∗ ) =

∞ L

−∞

U n L∗ ,

(9.15)

and L∗ = (U ∗ − T ∗ )H;

see Sec. II.1. To this end we just have to observe that n

U n+1 − T n+1 = ∑ U k (U − T )T n−k k=0

(n = 0, 1, . . .),

and conversely, U n (U − T ) = (U n+1 − T n+1 ) − (U n − T n )T

(n = 0, 1, . . .).

Owing to the importance of the spaces (9.15) in the study of a contraction (cf. Chaps. II and VI), the interest of the above relations is apparent. The following application is connected with Sec. II.3, and deserves particular attention. There we proved by a simple calculation that kU ∗n T n h − U ∗mT m hk2 = kT m hk2 − kT n hk2

(0 ≤ m < n)

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C HAPTER III. F UNCTIONAL C ALCULUS

for h ∈ H, and we have derived from this that the limit ah = lim U ∗n T n h

(9.16)

n→∞

exists. Now we obtain by an analogous calculation kU(s)∗ T (s)h − U(t)∗T (t)hk2 = kT (t)hk2 − kT (s)hk2

(0 ≤ t < s)

from which follows the existence of the limit a′h = lim U(s)∗ T (s)h.

(9.16′)

s→∞

If H1 consists of a single element h, let us denote the spaces appearing on the lefthand side and the right-hand side of (9.13) by Ah and A′h , respectively. From (9.16) and (9.16′) we deduce h − ah = lim U ∗n (U n − T n )h ∈ Ah , n→∞

h − a′h = lim U(s)∗ (U(s) − T (s))h ∈ A′h . s→∞

Furthermore we have for any nonnegative integers p, q (U ∗n T n h,U ∗p(U q − T q )h) = (T n h,U n−p(U q − T q )h)

= (T n h, T n−p+qh − T n−p T q h) = 0

whenever n ≥ p, and hence ah ⊥ Ah . An analogous reasoning yields a′h ⊥ A′h . We conclude that h − ah and h − a′h are the orthogonal projections of h into Ah and A′h , respectively. By virtue of (9.13), Ah = A′h , therefore ah = a′h . This proves one-part of the following proposition; the other part can be proved similarly. Proposition 9.1. Let {T (s)}s≥0 be a continuous one-parameter semigroup of contractions and let T be its cogenerator. Let {U(s)} and U be the corresponding minimal unitary dilations: U(s) = es (U). Then we have lim U ∗n T n h = lim U(s)∗ T (s)h,

n→∞

s→∞

lim U n T ∗n h = lim U(s)T (s)∗ h

n→∞

s→∞

(9.17)

and consequently lim kT n hk = lim kT (s)hk,

n→∞

s→∞

lim kT ∗n hk = lim kT (s)∗ hk.

n→∞

s→∞

(9.18)

2. Let us introduce, for continuous semigroups of contractions {T (s)}s≥0 , the classes C0· , C1· , and so on, in analogy to the corresponding classes for a single contraction T ; see Sec. II.3. For example, the semigroup is said to be of class C0· if, for every h ∈ H, T (s)h tends to 0 as s → +∞. Relations (9.18) imply the following result.

9. U NITARY DILATION OF SEMIGROUPS

151

Corollary. In order that {T (s)}s≥0 belong to one of the classes C0· , C1· , and so on, of semigroups, it is necessary and sufficient that its cogenerator T belong to the corresponding class of single contractions. Let us recall Theorem II.4.1, which asserts the existence of certain triangulations of a contraction, in connection with the classes considered. When applied to the cogenerator T of a semigroup of contractions {T (s)}, these triangulations generate triangulations of analogous types of the semigroup. To this end, we simply observe that if an arbitrary contraction T has a triangulation   T1 ∗   T =  ... , O

Tr

then each function u ∈ HT∞ also belongs to HT∞i (i = 1, . . . , r), and we have   u(T1 ) ∗   .. u(T ) =  . . O u(Tr )

3. Let us return to the first relation (9.18). As kT n hk and kT (s)hk are nonincreasing functions of n and s, respectively (n = 0, 1, . . .; 0 ≤ s < ∞), their common limit cannot equal khk unless kT n hk = khk = kT (s)hk for every n ≥ 0 and s ≥ 0. Hence we obtain the following complement of Proposition 8.2. Proposition 9.2. A continuous one-parameter semigroup of contractions consists of isometries if and only if its cogenerator is an isometry. Let {V (s)}s≥0 be a continuous semigroup of isometries on the space H and let V be its cogenerator. Because V is isometric, it induces a Wold decomposition (cf. Sec. I.1) H = H0 ⊕ H1 (9.19) with

H0 =

T

n≥0

V n H and H1 =

∞ L V n A, 0

A = H ⊖ V H;

(9.20)

H0 reduces V to a unitary operator V0 , and H1 reduces V to a unilateral shift V1 . (One of the subspaces H0 , H1 may be equal to the trivial subspace {0}.) In the corresponding decomposition V (s) = V0 (s) ⊕ V1 (s),

where V0 (s) = es (V0 ) and V1 (s) = es (V1 ),

{V0 (s)} is a unitary semigroup (because V0 is unitary), and {V1 (s)} is a c.n.u. semigroup (because V1 is c.n.u.). Now set I1 =

T

s≥a

V1 (s)H1

(a ≥ 0).

(9.21)

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C HAPTER III. F UNCTIONAL C ALCULUS

The subspace V1 (s)H1 is a nonincreasing function of s, that is, V1 (t)H1 = V1 (s)V1 (t − s)H1 ⊂ V1 (s)H1

for 0 ≤ s < t,

thus definition (9.21) does not depend on a. So we have for t ≥ 0: V1 (t)I1 = V1 (t)

T

s≥0

V1 (s)H1 =

T

s≥0

V1 (t + s)H1 =

T

s≥t

V1 (s)H1 = I1 ;

hence it follows that the subspace I1 reduces the isometries V1 (t) (t ≥ 0) to unitary operators (cf. Sec. I.1) and consequently V1 |I1 is also unitary. As V1 is c.n.u. this implies I1 = {0}. On the other hand, as we have obviously V0 (s)H0 = H0 for every s ≥ 0, it follows that T

s≥0

V (s)H =

T

s≥0

V0 (s)H0 ⊕

T

s≥0

V1 (s)H1 = H0 .

Thus for H0 the following two representations are valid, H0 =

T

n≥0

V nH =

T

s≥0

(9.22)

V (s)H,

one more relation where a semigroup and its cogenerator play a symmetric role. Let us consider the case H1 6= {0}, that is, d = dim A ≥ 1; d is the multiplicity of the unilateral shift V1 and we also call it the multiplicity of the c.n.u. semigroup of isometries {V1 (s)}. The cardinal number d determines V1 — hence also {V1 (s)}—up to unitary equivalence. An example of a completely nonunitary continuous semigroup of isometries is the continuous unilateral shift {v(s)}s≥0 on the space L2 (0, ∞; N) of functions f (x) with values in a Hilbert space N, defined by v(s) f (x) = f (x − s) for s ≥ x,

and f (x) = 0

for 0 ≤ x < s.

We show that the multiplicity of this semigroup of isometries equals dim N. The fact that {v(s)}s≥0 is a continuous semigroup of isometries, is obvious. Moreover, we have v(s)∗ f (x) = f (x + s) (s ≥ 0). Hence it follows that

∗

2

kv(s) f k =

Z ∞ s

k f (ξ )k2N d ξ → 0 as s → ∞;

this shows that the semigroup {v(s)} is c.n.u. Direct calculation gives that the generator a of {v(s)} is defined by a f (x) = −(d/dx) f (x)

9. U NITARY DILATION OF SEMIGROUPS

153

for the functions f ∈ L2 (0, ∞; N) which are absolutely continuous and such that f (0) = 0 and (d/dx) f (x) ∈ L2 (0, ∞; N). It then follows that the cogenerator v = (a + I)(a − I)−1 and its adjoint v∗ are given explicitly by the formulas (v f )(x) = f (x) − 2e−x

Z x 0

f (ξ )eξ d ξ ,

(v∗ f )(x) = f (x) − 2ex

Z ∞ x

f (ξ )e−ξ d ξ .

Because L2 (0, ∞; N) ⊖ vL2 (0, ∞; N) = { f : f ∈ L2 (0, ∞; N), v∗ f = 0} and equation v∗ f = 0 obviously has the only solutions f (x) = e−x h (h ∈ N), we conclude that, indeed, the multiplicity of the semigroup {v(s)} equals dim N. We have proved the following result. Theorem 9.3. Every continuous semigroup {V (s)}s≥0 of isometries on the space H is the orthogonal sum of a continuous semigroup of unitary operators and of a completely nonunitary semigroup of isometries. The latter one is unitarily equivalent to the continuous unilateral shift on L2 (0, ∞; N), where N is any Hilbert space whose dimension equals that of H ⊖ V H, V denoting the cogenerator of the semigroup {V (s)}.

It is understood that one of the components may be absent: the corresponding subspace may reduce to {0}. This theorem has an easy consequence for continuous groups {U(s)}−∞<s<∞ of unitary operators on a space H, for which there exists an “outgoing” subspace, that is, a subspace H0 such that (i) U(s)H 0 ⊂ H0 for all T (ii) U(s)H0 = {0},

(iii)

s>0 W s<0

s > 0,

U(s)H0 = H.

The prototype of a group with these properties is the continuous bilateral shift {u(s)}−∞<x<∞ on the space L2 (−∞, ∞; N) of functions f (x) with values in a Hilbert space N, defined by u(s) f (x) = f (x − s)

(−∞ < s < ∞).

Here we have the outgoing space L2 (0, ∞; N) (embedded in the natural way as a subspace in L2 (−∞, ∞; N)). Proposition 9.4. Every continuous group {U(s)}−∞<s<∞ of unitary operators on H, for which there is an outgoing subspace H0 , is unitarily equivalent to the continuous bilateral shift {u(s)}−∞<s<∞ on L2 (−∞, ∞; N), where N is any Hilbert space whose dimension equals that of H0 ⊖ V H0 , V denoting the cogenerator of the semigroup

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C HAPTER III. F UNCTIONAL C ALCULUS

formed by the restrictions V (s) = U(s)|H0 (s ≥ 0). Moreover, the unitary operator effectuating this unitary equivalence can be chosen so that it maps H0 onto the subspace L2 (0, ∞; N) of L2 (−∞, ∞; N). Proof. By virtue of (i) and (ii), {V (s)}s≥0 is a c.n.u. semigroup of isometries on H0 ; thus by Theorem 9.3 it is unitarily equivalent to the continuous unilateral shift {v(s)}s ≥ 0 on L2 (0, ∞; N). Let τ be the map of H0 onto L2 (0, ∞; N), which effectuates this unitary equivalence. For any f ∈ H such that f ∈ U(−s)H0 for some s > 0, let us define τ f = u(−s) · τ (U(s) f ); by the unitarity of U(s) and u(s) (for all real s) it follows that τ is thereby extended in a unique way to an isometric map of a dense subset of H (use property (iii)) onto a dense subset of L2 (−∞, ∞; N). By taking closures, we arrive at an extension of τ (denoted by the same letter) that maps H unitarily on L2 (−∞, ∞; N) and satisfies the condition τ ·U(s) = u(s) · τ for all real s.

10 Notes The results on the Hardy classes of scalar valued functions dealt with in Sec. 1 are mostly classical, (see, e.g., H OFFMAN [1]). Proposition 1.1, generalizing the theorem of Beurling, is new; also see S Z .-N.–F. [VI], Th´eor`eme 2. Propositions 1.3–1.5 appear in S Z .-N.–F. [VI] and [VII]. It has been apparent since the paper of S Z .-NAGY [I] that the existence of a unitary dilation for every contraction T provided a possibility for an extended functional calculus on the basis of the well-known functional calculus for the unitary operators. A detailed study of such a calculus was given first by S CHREIBER [2] and then by S Z .-N.–F. [III]. It is in the last-mentioned paper that the functional relations between a continuous one-parameter semigroup of contractions and its cogenerator first appear; these relations form the subject of Sec. 8 of the present chapter. At this stage the theory had a limited range, due to the fact that one did not know general criteria in order that the minimal unitary dilation U of T has an absolutely continuous spectrum, except the rather strong condition kT k < 1 found by S CHREIBER [1]. This limitation was removed by the discovery (cf. S Z .-N.–F. [IV]) of the fact that U has an absolutely continuous spectrum for every completely nonunitary T . This made it possible to construct the functional calculus for contractions such as presented in Secs. 2–3 above; see S Z .-N.–F. [VI]. It should be mentioned that in the particular case of an isometry (or, what amounts to the same by using Cayley transforms, in the case of a maximal symmetric operator), a functional calculus somewhat related to ours was proposed earlier by P LESSNER [1]–[3]. The essential difference between the Riesz–Dunford functional calculus (cf. [Func. Anal.] Chap. XI) and our functional calculus for contractions T is that the analytic functions which we admit may not be regular at some points of the boundary of the spectrum of T , for example at a finite number of points of C not belonging to the point spectrum of T . A functional calculus of similar type was proposed by

10. N OTES

155

F OIAS¸ [2]; this is founded upon the notion of spectral sets (introduced by VON N EU MANN, cf. [Func. Anal.] Chap. XI), and allows analytic functions that may have a finite number of singular points at the boundary of the spectrum, but not belonging to the point spectrum. An application of this functional calculus to the theory of contraction semigroups has also been given in F OIAS¸ [2], [3]. Proposition 2.2 on the maximality of our functional calculus is new; it generalizes a former result of F OIAS¸ [6]. For our functional calculus the “spectral mapping theorem” is also valid, at least if T is a c.n.u. contraction and u(λ ) is a function belonging to H ∞ and also continuous at the points of σ (T ) ∩C. Then we have the relation σ (u(T )) = u(σ (T )); see F OIAS¸ AND M LAK [1]. The contractions of class C0 and their minimal functions were introduced in S Z .N.–F. [VII], where most of the results of Secs. 4–7 also appeared. Proposition 7.5 was proved first in S Z .-N.–F. [XI], whereas the results concerning the continuous semigroups of contractions appeared in S Z .-N.–F. [III], and those concerning continuous semigroups of isometries (Theorem 9.3) in S Z .-NAGY [8]. Propositions 7.6 and 7.7 are due to ROSENTHAL [1]; the existence of cyclic vectors for unicellular operators on Hilbert space was also proved in G OHBERG AND K RE˘I N [7], pp. 52– 53. Proposition 9.4 was obtained by S INA˘I [1] as a consequence of the following theorem of VON N EUMANN [3]. Theorem. Let {U(s)} and {V (s)} be two one-parameter continuous groups of unitary operators on H, satisfying the Weyl commutation relation (∗)

U(s)V (t) = e−ist V (t)U(s)

(−∞ < s,t < ∞).

Then there exist a Hilbert space N and a unitary operator from H to L2 (−∞, ∞; N) transforming {U(s),V (s)} to {u(s), v(s)} defined by (u(s) f )(x) = f (x − s),

(v(s) f )(x) = eisx f (x),

the (auxiliary) Hilbert space N being determined up to unitary equivalence. Conversely, this theorem can beR obtained from Proposition 9.4 (cf. L AX AND ∞ iλ s P HILLIPS [2]). Indeed, if V (s) = −∞ e dEλ is the Stone representation of the group {V (s)}, then it follows from (∗) readily that U(s)Eλ = Eλ +sU(s) Setting H0 = (I − E0)H we have therefore ∗
U(s)H0 = (I − Es )H ∗

(−∞ < s, λ < ∞).

(⊂ H0 for s ≥ 0),

and this makes it apparent that H0 is an outgoing subspace for the group {U(s)}. So we have, up to a unitary equivalence, H = L2 (−∞, ∞; N),

H0 = L2 (0, ∞; N),

(U(s) f )(x) = f (x − s),

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C HAPTER III. F UNCTIONAL C ALCULUS

and therefore U(s)H0 = L2 (s, ∞; N). Now the projection from L2 (−∞, ∞; N) onto its subspace L2 (s, ∞; N) is the multiplication by 1 − χs (x), where
χs (x) is the characteristic function of the interval −∞ < x < s. So we get from ∗∗ that (Es f )(x) = χs (x) f (x), and hence we conclude that (V (s) f )(x) =

Z ∞

−∞

eiλ s d χλ (x) · f (x) = eisx f (x).

In connection with this chapter also see C OOPER [1]; F OIAS¸ [1]; F OIAS¸ , G EH E´ R , AND S Z .-NAGY [1]; H ELSON [1]; L AX AND P HILLIPS [1], [2]; M ASANI [3], [4]; M LAK [5], [7]; P HILLIPS [3]; S CHREIBER [4]; and S Z .-NAGY [3], [5]–[7], [13].

11 Further results 1. There has been much progress in the study of the class C0 , and Chap. X is dedicated to some of this material. Here we only mention that analogues of this class have been found with multiply connected regions in the plane in place of the disk D; see BALL [3]. Much of the theory developed for the disk was transferred to this context in Z UCCHI [1]; see also PATA AND Z UCCHI [1]. Some results in the context of the bidisk D2 are found in R. YANG [3], based on the theory started in D OUGLAS AND YANG [1] and R. YANG [1,2]. Operators of class C0 with finite defect indices also have been studied from the point of view of their polar decompositions in W U [7] and their singular unitary dilations in W U AND TAKAHASHI [2]. Some of the techniques developed for the study of the class C0 were extended to operators of class C·0 , at least when one of the defect indices is finite; see S Z .-N.–F. [24] for operators with both defect indices finite, and S Z .-NAGY [15] for operators with one finite defect index. 2. As we have seen, the spectral mapping theorem σ (u(T )) = u(σ (T )) is valid if T is a completely nonunitary contraction and u ∈ H ∞ extends continuously to σ (T ) ∩C. If this continuity hypothesis is relaxed, spectral mapping can fail spectacularly, as shown, for instance, in F OIAS AND P EARCY [1] and B ERCOVICI , F OIAS , AND P EARCY [2]. These results are based on the characterization given in S Z .-N.– F. [22] of invertible operators of the form u(T ). 3. A remarkable connection between the functional calculus developed in this chapter and the existence of invariant subspaces has been discovered after the seminal work in B ROWN [1]. For this discussion, we fix a contraction T on H, whose unitary part has absolutely continuous spectral measure relative to arclength on C. We have therefore HT∞ = H ∞ . Assume for the moment that, for some λ ∈ D, there exist vectors h, k ∈ H satisfying the identity (u(T )h, k) = u(λ )

(11.1)

for every polynomial u (and, by continuity of the functional calculus, for every u ∈ H ∞ ). Denote by Hh and HT h the invariant subspaces for T generated by h and

11. F URTHER RESULTS

157

T h, respectively. Clearly then k ⊥ HT h , and (h, k) = 1. It follows that one of these invariant subspaces is proper. Note that (11.1) can be rewritten as (u(T )h, k) =

1 2π

Z 2π 0

u(eit ) f (eit ) dt,

u ∈ H ∞,

(11.2)

where the function f ∈ L1 is defined by f (eit ) = 1/(1 − λ e−it ). The existence of invariant subspaces can thus be deduced from the existence, for an arbitrary f ∈ L1 , of vectors h, k ∈ H satisfying (11.2). This method is used in B ROWN , C HEVREAU , AND P EARCY [1],[2] and B ROWN [2] to prove the following remarkable result. Theorem 11.1. Assume that T is a contraction such that σ (T ) ⊃ C. Then T has nontrivial invariant subspaces. A complete exposition of the proof appears in B ERCOVICI [5]. The proof proceeds by a reduction to the case when (11.2) can indeed be solved for every f ∈ L1 . It is easy to see that (11.1) implies the inequality ku(T )k ≥ |u(λ )|,

u ∈ H ∞.

Thus, if (11.2) can be solved for every f ∈ L1 , we must have ku(T )k = kuk∞ ,

u ∈ H ∞,

so the functional calculus must be an isometry. The following converse was proved in B ERCOVICI [4] and C HEVREAU [1]. Theorem 11.2. Assume that T is a contraction for which the functional calculus is isometric on H ∞ . Then the equation (11.2) can be solved for every f ∈ L1 .

¨ [1] to the case of Theorem 11.1 has been extended by A MBROZIE AND M ULLER polynomially bounded operators on a Banach space. This is an improvement even in the Hilbert space situation, as seen from the examples in P ISIER [1]. The techniques used in these papers generally yield a wealth of information about the structure of the invariant subspace lattice of T . The following result is in B ROWN AND C HEVREAU [1]. Theorem 11.3. If the functional calculus for T is isometric on H ∞ , then T is a reflexive operator. See Sec. IX.3 for a discussion of reflexivity, as well as a special case of this result. The monograph by B ERCOVICI , F OIAS , AND P EARCY [1] contains further information on the rich dilation theory arising from these methods. 4. For contractions T such that X T = SX, where S is the unilateral shift and X has dense range, it was shown in TAKAHASHI [3] that {T }′′ = {u(T ) : u ∈ H ∞ }. Here {T }′′ denotes the double commutant of T . See also Theorem X.4.2 (3) for a related result in the case of operators of class C0 ; this result is from S Z .-N.–F. [26]. In S Z .-N.–F. [28] it is shown that {T }′ = {u(T ) : u ∈ H ∞ } provided that T is of class C10 , it has a cyclic vector, and I − T ∗ T has finite trace.

Chapter IV

Extended Functional Calculus 1 Calculation rules 1. We extend our functional calculus for a contraction T on H so that certain unbounded functions are also allowed. Let us recall the definitions of the classes HT∞ and KT∞ as given in Secs. 2 and 3 of the preceding chapter: HT∞ consists of the functions u ∈ H ∞ for which the strong operator limit u(T ) = limr→1−0 ur (T ) exists, and KT∞ consists of those functions u ∈ HT∞ for which u(T )−1 exists and is densely defined in H. The class HT∞ is an algebra, and the class KT∞ is multiplicative. Definition. Let NT be the class of the functions ϕ (meromorphic on the open unit disc) that admit a representation of the form

ϕ=

u v

with

u ∈ HT∞

and v ∈ KT∞

(1.1)

(note that v 6≡ 0). For such a function ϕ we define

ϕ (T ) = v(T )−1 u(T ).

(1.2)

As HT∞ is an algebra and KT∞ is multiplicative, it follows from the relations

α

u αu = , v v

u1 u2 u 1 v2 + u 2 v1 + = , v1 v2 v1 v2

u1 u2 u1 u2 · = v1 v2 v1 v2

(1.3)

that the class NT is also an algebra. Theorem 1.1. (i) The definition (1.2) of ϕ (T ) does not depend upon the particular choice of the functions u, v in the representation (1.1). The operator ϕ (T ) is closed, with dense domain in H, and commutes with T as well as with every bounded operator S commuting with T , that is, T S = ST

implies ϕ (T )S ⊃ Sϕ (T ).

B.Sz.-Nagy et al., Harmonic Analysis of Operators on Hilbert Space, Universitext, DOI 10.1007/978-1-4419-6094-8_4, © Springer Science + Business Media, LLC 2010

159

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C HAPTER IV. E XTENDED F UNCTIONAL C ALCULUS

In particular we have v(T )−1 u(T ) ⊃ u(T )v(T )−1

for u ∈ HT∞ , v ∈ KT∞ .

(1.4)

Furthermore, we have (ii) (cϕ )(T ) = cϕ (T ) for all constants c. (iii) (ϕ1 + ϕ2 )(T ) ⊃ ϕ1 (T ) + ϕ2 (T ), with equality, for example, if D[ϕ2 (T )] ⊃ D[(ϕ1 + ϕ2 )(T )], 1 and hence in particular if ϕ2 ∈ HT∞ ; (iv) (ϕ1 ϕ2 )(T ) ⊃ ϕ1 (T )ϕ2 (T ), with equality, for example, if D[ϕ2 (T )] ⊃ D[(ϕ1 ϕ2 )(T )], and hence in particular if ϕ2 ∈ HT∞ . We have always ϕ1 (T )ϕ2 (T ) = (ϕ1 ϕ2 )(T ) [D[ϕ2 (T )] ∩ D[(ϕ1 ϕ2 )(T )]];

(v) (vi) (vii) (viii)

(1.5)

in particular we have (1/ϕ )(T ) = ϕ (T )−1 if ϕ , 1/ϕ ∈ NT . If ϕ ∈ NT then ϕ ˜ ∈ NT ∗ and ϕ ˜(T ∗ ) ⊃ ϕ (T )∗ . If ϕ ∈ NT and ϕ ∈ H ∞ then ϕ ∈ HT∞ and therefore kϕ (T )k ≤ kϕ k∞ . If ϕ ∈ NT and 1/ϕ ∈ H ∞ then 1/ϕ ∈ NT . Let ϕ = u/v with u ∈ H ∞ , v ∈ E ∞ , and let w ∈ H ∞ , |w(λ )| < 1 on D. Let us suppose, furthermore, that the sets Cw ,

w−1 (Cu ),

w−1 (C0v )

(cf. Secs. III.2 and 3)

(1.6)

have O measure with respect to the spectral measure ET corresponding to the unitary part of T . Then w ∈ HT∞ , and T ′ = w(T ) is a contraction; furthermore we have ϕ ◦ w ∈ NT , ϕ ∈ NT ′ , and ϕ (T ′ ) = (ϕ ◦ w)(T ). (ix) If the contraction T is normal, with the corresponding spectral measure K = KT , and if the function ϕ = u/v is such that u ∈ H ∞,

v ∈ E ∞,

ET (Cu ) = O,

ET (Cv0 ) = O, 2

then ϕ ∈ NT and ϕ (T ) equals the integral ϕ s (T ) of ϕ with respect to K. Proof. Part (i): Let ϕ = u/v and ϕ = u′ /v′ be two representations of type (1.1) of the same function ϕ . The relation uv′ = vu′ implies u(T )v′ (T ) = v(T )u′ (T ), 1 2

If L is a linear transformation, D(L) denotes its domain of definition. If T is c.n.u., ET (ω ) is equal for every ω to the operator O on the trivial space {0}.

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and hence v(T )−1 u(T ) = v(T )−1 v′ (T )−1 v′ (T )u(T ) = v′ (T )−1 v(T )−1 u(T )v′ (T ) = = v′ (T )−1 v(T )−1 v(T )u′ (T ) = v′ (T )−1 u′ (T ). Thus by formula (1.2), ϕ (T ) is uniquely determined by ϕ . Let us set u(T ) = U, v(T ) = V . The operator

ϕ (T ) = V −1U is closed, for the hypotheses hn ∈ D[ϕ (T )],

hn → h,

ϕ (T )hn = V −1Uhn → g

(n → ∞)

imply Uhn → Uh,Uhn = V ϕ (T )hn → V g, and thus Uh = V g, which shows that h ∈ D[ϕ (T )] and ϕ (T )h = V −1Uh = g. Relation (1.4) follows from the fact that U and V are bounded and commuting. Indeed, we have V −1U ⊃ V −1UVV −1 = V −1VUV −1 = UV −1 . By virtue of the definition of the class KT∞ , V −1 has domain dense in H, and hence so does ϕ (T ). If S is a bounded operator commuting with T , then S also commutes with the functions of T of class HT∞ , for these are limits of polynomials of T . Consequently, we have

ϕ (T )S = V −1US = V −1 SU ⊃ V −1 SVV −1U = V −1V SV −1U = = SV −1U = Sϕ (T ),

that is, S commutes with ϕ (T ). Parts (ii)–(iv): (ii) is obvious. To obtain (iii) and (iv), let us consider the functions

ϕk = uk /vk

(k = 1, 2),

with

uk ∈ HT∞ ,

vk ∈ KT∞ ,

and set Uk = uk (T ) and Vk = vk (T ). On account of (1.3) we have, by definition, (ϕ1 + ϕ2 )(T ) = V1−1V2−1 (U1V2 + U2V1 ),

(ϕ1 ϕ2 )(T ) = V1−1V2−1U1U2 .

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Using (1.4), we obtain (ϕ1 + ϕ2 )(T ) ⊃ V1−1V2−1U1V2 + V2−1V1−1U2V1

⊃ V1−1U1V2−1V2 + V2−1U2V1−1V1 = = ϕ1 (T ) + ϕ2 (T ).

For the same reason,

ϕ1 (T ) + ϕ2 (T ) = [(ϕ1 + ϕ2 ) − ϕ2 ](T ) + ϕ2 (T ) ⊃ [(ϕ1 + ϕ2 )(T ) − ϕ2 (T )] + ϕ2 (T ). If D[ϕ2 (T )] ⊃ D[(ϕ1 + ϕ2 )(T )] this implies

ϕ1 (T ) + ϕ2 (T ) ⊃ (ϕ1 + ϕ2 )(T ); as the opposite inclusion is always valid we conclude that in this case

ϕ1 (T ) + ϕ2 (T ) = (ϕ1 + ϕ2 )(T ). From (1.4) it also follows that (ϕ1 ϕ2 )(T ) = V1−1V2−1U1U2 ⊃ V1−1U1V2−1U2 = ϕ1 (T )ϕ2 (T ). This clearly implies D[ϕ1 (T )ϕ2 (T )] ⊂ D[(ϕ1 ϕ2 )(T )] ∩ D[ϕ2 (T )]. In order to prove (1.5) it only remains to show that the opposite inclusion also holds; that is, h ∈ D[(ϕ1 ϕ2 )(T )] ∩ D[ϕ2 (T )]

implies h ∈ D[ϕ1 (T )ϕ2 (T )].

Now for such h we have V1 (ϕ1 ϕ2 )(T )h = V1V1−1V2−1U1U2 h = V2−1U1U2 h = V2−1U1V2 ϕ2 (T )h = V2−1V2U1 ϕ2 (T )h = U1 ϕ2 (T )h, and this relation implies

ϕ2 (T )h ∈ D[ϕ1 (T )] and ϕ1 (T )ϕ2 (T )h = (ϕ1 ϕ2 )(T )h. The last assertion in (iv) follows by applying (1.5) to the functions ϕ , 1/ϕ in both orders, and by observing that (ϕ · 1/ϕ )(T ) = (1/ϕ · ϕ )(T ) = 1(T ) = I.

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Part (v): If ϕ = u/v with u ∈ HT∞ and v ∈ KT∞ , then ϕ ˜ = u˜/v˜ with u˜ ∈ HT∞∗ and v˜ ∈ KT∞∗ , and

ϕ ˜(T ∗ ) = v˜(T ∗ )−1 u˜(T ∗ ) = [v(T )∗ ]−1 u(T )∗ = [v(T )−1 ]∗ u(T )∗ = [u(T )v(T )−1 ]∗ ⊃ [v(T )−1 u(T )]∗ = ϕ (T )∗ , where we have used (1.4) and the fact that A ⊃ B implies B∗ ⊃ A∗ . Part (vi): If ϕ = u/v (u ∈ HT∞ , v ∈ KT∞ ) and |ϕ (λ )| ≤ M on D, then the functions ur , vr , ϕr (0 ≤ r < 1) belong to the class A, and satisfy ur = ϕr · vr and |ϕr (λ )| ≤ M on D. According to Sec. III.2.1 we have ur (T ) = ϕr (T ) · vr (T ) and kϕr (T )k ≤ M. This implies that ϕr (T )v(T ) = ur (T ) − ϕr (T )[vr (T ) − v(T )] → u(T ) as r → 1 − 0. The range of v(T ) is dense in H, thus we conclude that ϕr (T )h converges for every h ∈ H and therefore we have ϕ ∈ HT∞ . The inequality kϕ (T )k ≤ kϕ k∞ follows by Theorem III.2.1(b). Part (vii): If ϕ = u/v (u ∈ HT∞ , v ∈ KT∞ ) and |1/ϕ (λ )| ≤ M, then it follows by a reasoning similar to the above, that kv(T )hk ≤ Mku(T )hk for every h ∈ H. As v(T ) is invertible this inequality implies that u(T ) is also invertible. Applying this result to the function ϕ ˜ = u˜/v˜ yields that u˜(T ∗ ) is also invertible. Because u˜(T ∗ ) = u(T )∗ , this means that u(T ) has dense range in H. Thus u ∈ KT∞ and therefore 1/ϕ ∈ NT . Part (viii): By virtue of Theorem III.2.3, T ′ = w(T ) is a contraction and we have that u(T ′ ) = (u ◦ w)(T ), v(T ′ ) = (v ◦ w)(T )

(the operators indicated exist) and v ∈ HT ′ . Now the third condition (1.6) implies that v(T ′ ) is a quasi-affinity, therefore v ∈ KT∞′ and ϕ ∈ NT ′ . Moreover, by Theorem III. 3.4 we have v ◦ w ∈ KT∞ . Since ϕ = u/v and ϕ ◦ w = (u ◦ w)/(v ◦ w), it follows that ϕ ◦ w ∈ NT , and

ϕ (T ′ ) = v(T ′ )−1 u(T ′ ) = [(v ◦ w)(T )]−1 (u ◦ w)(T ) = (ϕ ◦ w)(T ). Part (ix): Under the conditions stated for u and v we deduce from Theorems III.2.3 and III.3.4, by the usual rules of calculation with spectral integrals, that

ϕ (T ) = vs (T )−1 us (T ) = (u/v)s (T ) = ϕ s (T ). This concludes the proof of Theorem 1.1. The next proposition follows readily from properties (vi) and (vii). Proposition 1.2. If ϕ ∈ NT , the spectrum of ϕ (T ) is contained in the closure of the set of the values of ϕ (λ ) on D:

σ [ϕ (T )] ⊂ ϕ (D). Proof. One just has to observe that if ζ is at distance d > 0 from the set ϕ (D), then |[ζ − ϕ (λ )]−1 | ≤ 1/d on D, and consequently ζ I − ϕ (T ) is boundedly invertible, with kζ I − ϕ (T )k ≤ 1/d.

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2. The above reasoning makes it clear that it is of considerable interest to know under which conditions there is equality in (1.4). Here is an answer to this question. Proposition 1.3. (i) Suppose u, v satisfy the following condition.  u, v are continuous on D, holomorphic on D, and (Q) have no common zeros in D. Moreover, let v ∈ KT∞ , where T is a contraction on H.3 Then and, for ϕ = u/v,

of

v(T )−1 u(T ) = u(T )v(T )−1

(1.7)

ϕ (T )∗ = ϕ ˜(T ∗ ).

(1.8)

(ii) Let ϕk = uk /vk , where uk ∈ HT∞ , vk ∈ KT∞ (k = 1, 2), and suppose that the pair functions u1 and v2 satisfies condition (Q). Then (ϕ1 , ϕ2 )(T ) = ϕ1 (T )ϕ2 (T ).

Proof. We know (cf. H OFFMAN [1] p. 88) that if u, v satisfy condition (Q), then there exist functions a, b, continuous on D and holomorphic on D, such that au + bv = 1

on D.

Let us set a(T ) = A, b(T ) = B, u(T ) = U, and v(T ) = V . Then we have AU + BV = I. Hence from the relation valid by virtue of (1.4), we obtain

AV −1 ⊂ V −1 A,

V −1U = (AU + BV )V −1U = UAV −1U + BVV −1U ⊂ UV −1 AU + BU = U(V −1 AU + B) = UV −1 (AU + V B) = UV −1 ; combined with the relation UV −1 ⊂ V −1U this yields

(cf. (1.4))

V −1U = UV −1 ,

that is, (1.7). To obtain (1.8) we just have to repeat the argument in the proof of assertion (v) of Theorem 1.1, using now (1.7) instead of (1.4). 3 Because u, v are continuous on D and holomorphic on D, we have u, v ∈ H ∞ for every contraction T T ; see Theorem III.2.3.

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165

This proves (i). To obtain (ii) we observe that, on account of (1.7), v2 (T )−1 u1 (T ) = u1 (T )v2 (T )−1 ; it follows that (ϕ1 ϕ2 )(T ) = v1 (T )−1 v2 (T )−1 u1 (T )u2 (T ) = v1 (T )−1 u1 (T )v2 (T )−1 u2 (T ) = ϕ1 (T )ϕ2 (T ). 3. If for some function ϕ ∈ NT the operator ϕ (T ) happens to be bounded, then we can ask whether there exists a function w ∈ HT∞ such that ϕ (T ) = w(T ). The following example shows that this is not always the case. Let T be an operator with kT k < 1 and with nondiscrete spectrum. Then T is c.n.u., and as a consequence of Theorem III.5.1, T does not belong to the class C0 . Choose a number a such that kT k < a < 1, and set ϕ (λ ) = 1/(λ − a). Clearly ϕ ∈ NT , and ϕ (T ) = (T − aI)−1 is a bounded operator. Suppose there exists w ∈ H ∞ such that ϕ (T ) = w(T ). Then we have (T − aI) w(T ) = I, and consequently u(T ) = O for u(λ ) = (λ − a) w(λ ) − 1. Because T does not belong to C0 this implies that u(λ ) ≡ 0, which contradicts the equation u(a) = −1. Thus there exists no w ∈ HT∞ with w(T ) = ϕ (T ).

2 Representation of ϕ (T ) as a limit of ϕr (T ) In Sec. III.2 we defined the functions of the contraction T of class HT∞ as limits of functions of T of class A, that is, by u(T ) = lim ur (T ). r→1−0

Now for an arbitrary holomorphic function ϕ on D, the functions ϕr (0 ≤ r < 1) belong to the class A, so ϕr (T ) makes sense and one can ask whether ϕr (T ) tends to ϕ (T ) as r → 1, if ϕ belongs to the class NT or at least to an appropriate subclass of NT . Theorem 2.1. (i) Let T be a contraction on H, and let ϕ be a function belonging to the class NT and holomorphic on D. Then every vector h ∈ H such that sup kϕr (T )hk < ∞,

0
(2.1)

belongs to the domain of definition of ϕ (T ) and the weak convergence relation holds: ϕr (T )h ⇀ ϕ (T )h (r → 1 − 0). (2.2)

(ii) Suppose the functions u, v satisfy condition (Q) of Proposition 1.3, v belongs to the class E reg (cf. Sec. III.1.3) and does not vanish on C except at the points of a set of ET -measure O. Then ϕ = u/v belongs to the class NT and is holomorphic on

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D, condition (2.1) characterizes the vectors h for which ϕ (T )h is defined, and for every such vector h the strong convergence relation holds:

ϕr (T )h → ϕ (T )h

(r → 1 − 0).

(2.3)

Proof. Part (i): Let ϕ = u/v (u ∈ HT∞ , v ∈ KT∞ ) and let us consider a sequence {rn } tending to 1 − 0. From (2.1) it follows that there exists a subsequence {ρn } for which ϕρn (T )h converges weakly, say to g:

ϕρn (T )h ⇀ g.

(2.4)

Because vr (T )∗ f → v(T )∗ f (as r → 1 − 0) for every f ∈ H, we deduce from (2.4) that (uρn (T )h, f ) = (vρn (T )ϕρn (T )h, f ) = (ϕρn (T )h, vρn (T ∗ ) f ) → (g, v(T )∗ f ) = (v(T )g, f ). On the other hand, we have ur (T )h → u(T )h (r → 1 − 0), and hence in particular (uρn (T )h, f ) → (u(T )h, f ). Comparing these results we obtain (v(T )g, f ) = (u(T )h, f ) for every f ∈ H, and hence v(T )g = u(T )h. This shows that ϕ (T )h exists and equals g. So we have proved that (2.1) implies that ϕ (T )h exists and that every sequence rn → 1 − 0 contains a subsequence ρn → 1 − 0 for which

ϕρn (T )h ⇀ ϕ (T )h (n → ∞).

(2.5)

From this we infer that

ϕr (T )h ⇀ ϕ (T )h

(r → 1 − 0).

Indeed, in the contrary case there would exist an f ∈ H and a sequence rn → 1 − 0 such that |(ϕrn (T )h − ϕ (T )h, f )| ≥ ε > 0 (n = 1, 2, . . .),

and this contradicts the existence of a subsequence satisfying (2.5). This concludes the proof of (i). Part (ii): According to our hypotheses the function v is continuous on D, holomorphic on D, and vanishes at most at the points of a subset of ET -measure O of C; furthermore the function v(λ ) w(r; λ ) = v(rλ )

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167

is bounded on D by a constant M independent of r (0 ≤ r < 1). These conditions imply that v ∈ KT∞ (cf. Proposition III.1.3 and Theorem III.3.4), |w(r; eit )| ≤ M, and

lim w(r; eit ) = 1 a.e. and ET - a.e. on C.

r→1−0

From Theorem III.2.3 it follows that vr (T )−1 v(T ) = w(r; T ) → I

(r → 1 − 0).

On the other hand, in view of the condition (Q) for u and v, we have by Proposition 1.3 ϕ (T ) = u(T )v(T )−1 , and hence every element h of the domain of ϕ (T ) is of the form h = v(T )g. Consequently, we have

ϕr (T )h = vr (T )−1 ur (T )h = vr (T )−1 ur (T )v(T )g = vr (T )−1 v(T )ur (T )g = w(r; T )ur (T )g → u(T )g = u(T )v(T )−1 h = ϕ (T )h as r → 1 − 0. This concludes the proof.

3 Functions limited by a sector In view of some later applications we consider in this section functions ϕ whose values lie in some sector of the plane of complex numbers, whose angle does not exceed π . We begin with functions belonging to the simple class A. Proposition 3.1. Let ϕ ∈ A be such that | arg ϕ (λ )| ≤ α

π 2

for λ ∈ D

(3.1)

with 0 ≤ α ≤ 1. Then we have for every contraction T on H and for every h ∈ H:

π | arg(ϕ (T )h, h)| ≤ α . 2

If (3.1) holds with 0 ≤ α <

(i) (ii) (iii) (iv) (v)

1 2

(3.2)

then the following inequalities are valid.

Re (ϕ (T )h, ϕ (T )∗ h) ≥ cos απ · max{kϕ (T )hk2 , kϕ (T )∗ hk2 }. kϕ (T )hk ≥ cos απ · kϕ (T )∗ hk, kϕ (T )∗ hk ≥ cos απ · kϕ (T )hk. Re(ϕ (T )h, [Reϕ (T )]h) ≥ cos2 (απ /2) · kϕ (T )hk2 , (cos2 (απ /2)/cos απ ) · kϕ (T )hk ≥ k[Reϕ (T )]hk ≥ cos2 (απ /2) · kϕ (T)hk, kImϕ (T )hk ≤ tan(απ /2) · kReϕ (T )hk, where Reϕ (T ) = [ϕ (T ) + ϕ (T )∗ ]/2, and Imϕ (T ) = [ϕ (T ) − ϕ (T )∗ ]/2i.

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Proof. By virtue of (III.2.6) we have ϕ (T ) = pr ϕ (U),U being the minimal unitary dilation of T . If E is the spectral measure corresponding to the unitary operator U, then we have for every h ∈ H, Z 2π | arg(ϕ (T )h, h)| = | arg(ϕ (U)h, h)| = arg ϕ (eit ) d(Et h, h) ϕ π ≤ max | arg(ϕ eit )| ≤ α , t 2 that is, (3.2). If (3.1) holds with 0 ≤ α <

1 2

then

| arg[ϕ (λ )]2 | ≤ απ < Hence and consequently

π . 2

Re[ϕ (λ )]2 ≥ cos απ · |ϕ (λ )|2

Re(ϕ (T )h, ϕ (T )∗ h) = Re(ϕ (T )2 h, h) = Re(ϕ (U)2 h, h) =

Z 2π 0

Re[ϕ (eit )]2 d(Et h, h) Z 2π

|ϕ (eit )|2 d(Et h, h) ≥ cos απ · 0 ( cos απ · kϕ (U)hk2 ≥ cos απ · kϕ (T )hk2 , = cos απ · kϕ (U)∗ hk2 ≥ cos απ · kϕ (T )∗ hk2 ; here we have also used the relations

ϕ (T )2 = pr ϕ (U)2 ,

ϕ (T )∗ = pr ϕ (U)∗ ;

see Sec. III.2.1. So we have proved (i). The constant cos απ is the best one, as it can be seen by the example of the constant function ϕ (λ ) ≡ exp(iαπ /2). The other inequalities derive from (i) as follows. Part (ii) is implied by the Schwarz inequality: Re(ϕ (T )h, ϕ (T )∗ h) ≤ kϕ (T )hk · kϕ (T )∗ hk. Inequality (iii) follows immediately: 1 1 Re(ϕ (T )h, [Re ϕ (T )]h) = kϕ (T )hk2 + Re(ϕ (T )h, ϕ (T )∗ h) 2 2 1 απ · kϕ (T )hk2 . ≥ (1 + cos απ ) · kϕ (T )hk2 = cos2 2 2

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The second of the inequalities (iv) is a consequence of (iii), by the Schwarz inequality. The first inequality (iv) follows from the first inequality (ii): 1 k[Reϕ (T )]hk ≤ [kϕ (T )hk + kϕ (T)∗ hk] 2 1  cos2 (απ /2) 1 1+ · kϕ (T )hk = · kϕ (T )hk. ≤ 2 cos απ cos απ

Finally, (v) is derived from (i) as follows. By (i) we have

2 Re(ϕ (T )h, ϕ (T )∗ h) ≥ cos απ · [kϕ (T )hk2 + kϕ (T )∗ hk2 ]. Because cos απ =

1 − tan2 (απ /2) , 1 + tan2 (απ /2)

we obtain kϕ (T )hk2 − 2 Re(ϕ (T )h, ϕ (T )∗ h) + kϕ (T )∗ hk2  απ  · kϕ (T )hk2 + 2 Re(ϕ (T )h, ϕ (T )∗ h) + kϕ (T )∗ hk2 , ≤ tan2 2

and this yields (v).

Theorem 3.2. Let ϕ = u/v be as in Theorem 2.1(ii); that is, u and v satisfy condition (Q), v belongs to E reg , and does not vanish on C except at the points of a set of ET measure O. Let us also suppose that, for λ ∈ D, | arg ϕ (λ )| ≤

απ 2

(0 ≤ α ≤ 1).

Inequality (3.2) is then valid for every h in the domain of ϕ (T ). If, moreover, 0 ≤ α < 21 then the operators ϕ (T ), ϕ (T )∗ , and hence also 1 1 Re ϕ (T ) = [ϕ (T ) + ϕ (T )∗ ] and Im ϕ (T ) = [ϕ (T ) − ϕ (T )∗ ] 2 2i have the same domain D, and the inequalities (i)–(v) of Proposition 3.1 hold for every h ∈ D. Also, Re ϕ (T ) is then a positive self-adjoint operator. Proof. Because ϕr ∈ A for 0 ≤ r < 1, we can apply Proposition 3.1 to ϕr . By virtue of Theorem 2.1 we have ϕ (T ) = lim ϕr (T ). r→1

and hence inequality (3.2) is valid for ϕ (T ) and for every h ∈ D[ϕ (T )]. By the same Theorem 2.1, D[ϕ (T )] consists of the elements h for which sup kϕr (T )hk < ∞. r

(3.3)

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Similarly, ϕ (T )∗ = ϕ ˜(T ∗ ) and ϕr (T )∗ = (ϕ ˜)r (T ∗ ), therefore D[ϕ (T )∗ ] consists of the elements h for which sup kϕr (T )∗ hk < ∞. (3.4) r

Now if 0 ≤ α < equalities

1 2,

conditions (3.3) and (3.4) are equivalent on account of the in-

kϕr (T )hk ≥ cos απ · kϕr (T )∗ hk and kϕr (T )∗ hk ≥ cos απ · kϕr (T )hk. Thus D[ϕ (T )] = D[ϕ (T )∗ ], and because for an element h of this common domain D we have ϕr (T )h → ϕ (T )h and

ϕr (T )∗ h = ϕ r˜(T ∗ )h → ϕ ˜(T ∗ )h = ϕ (T )∗ h as r → 1 − 0, inequalities (i)–(v) of Proposition 3.1 are valid in the limit case r = 1 also. From (3.2) it follows that ([Re ϕ (T )]h, h) = Re(ϕ (T )h, h) ≥ 0 for h ∈ D, and thus Re ϕ (T ) is a positive symmetric operator. It is actually self-adjoint. To this end it suffices to show that (I + Re ϕ (T ))D = H.4 From inequality (v) and from the positivity of Re ϕ (T ) it follows that kIm ϕ (T )hk ≤ ck(I + Re ϕ (T ))hk

with c = tan

απ < 1; 2

as a consequence we deduce that there exists an operator B, which can even be assumed to be defined everywhere on H, such that Im ϕ (T ) = B[I + Re ϕ (T )] and kBk ≤ c. Hence I + ϕ (T ) = I + Re ϕ (T ) + iIm ϕ (T ) = (I + iB)(I + Re ϕ (T )). Now [I + ϕ (T )]D = H because −1 does not belong to the spectrum of ϕ (T ) (cf. Proposition 1.2), and I + iB is boundedly invertible because kiBk < 1. As a result we have (I + Re ϕ (T ))D = (I + iB)−1(I + ϕ (T ))D = (I + iB)−1 H = H, and this concludes the proof. 4

See Sec. 125 in [Func. Anal].

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171

4 Accretive and dissipative operators 1. Let A0 be an operator, not necessarily bounded, but with domain D(A0 ) dense in H. A0 is said to be accretive if Re(A0 f , f ) ≥ 0

for every

f ∈ D(A0 ),

Im(A0 f , f ) ≥ 0

for every

f ∈ D(A0 ).

and dissipative if

As dissipative operators can be derived from accretive ones by multiplication by i, all we say about accretive operators carries over immediately to dissipative operators. For an accretive A0 we have kA0 f + f k2 ≥ kA0 f k2 + k f k2 ≥ kA0 f − f k2 ,

(4.1)

and hence we see that (A0 + I) f = 0 implies f = 0 Thus A0 + I is invertible, and furthermore it follows that the operator T0 defined by T0 (A0 + I) f = (A0 − I) f

( f ∈ D(A0 ))

(4.2)

is a contraction of (A0 + I)D(A0 ) onto (A0 − I)D(A0 ); we have T0 = (A0 − I)(A0 + I)−1.

(4.3)

Hence T0 = I − 2(A0 + I)−1 and I − T0 = 2(A0 + I)−1 ;

(4.4)

A0 = (I + T0)(I − T0 )−1 .

(4.5)

thus I − T0 is invertible and

We call T0 the Cayley transform of the accretive operator A0 . From the reciprocal relations (4.3), (4.5) it follows that an accretive operator A1 is a proper extension of A0 if and only if its Cayley transform T1 is a proper extension of T0 . An accretive operator that has no proper accretive extension is said to be maximal accretive. An obvious sufficient condition for the accretive operator A0 to be maximal is that its Cayley transform T0 be defined on the whole space H: that is D(T0 ) = (A0 + I)D(A0 ) = D. This condition turns out also to be necessary. Moreover, we prove that every accretive operator has a maximal accretive extension. To this end let us suppose that D(T0 ) 6= H. Let T be an extension of T0 to a contraction defined everywhere on H. (Such a T exists because T0 extends to the subspace D(T0 ) by continuity and then to H by linearity, setting, e.g., T h = 0 for h ⊥ D(T0 ).) There is

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no nonzero invariant vector for T . In fact, if h is invariant for T then it is invariant for T ∗ too (cf. Proposition I.3.1), and we have for every f ∈ D(A0 ): (h, (A0 + I) f ) = (T ∗ h, (A0 + I) f ) = (h, T (A0 + I) f ) = (h, T0 (A0 + I) f ) = (h, (A0 − I) f ); hence (h, f ) = 0, and as D(A0 ) is dense in H this implies that h = 0. Now every contraction T on H that has no nonzero invariant vector is the Cayley transform of an accretive operator A, indeed of A = (I + T )(I − T )−1 .

(4.6)

As a matter of fact, because D(A) consists of the vectors of the form f = (I − T )g, we have Re(A f , f ) = Re((I + T )g, (I − T )g) = kgk2 − kT gk2 ≥ 0, and these vectors f are dense in H because T ∗ has no nonzero invariant vector. Thus A is accretive, and it is obvious that its Cayley transform equals T . As D(T ) = H, we have that A is maximal accretive. Let us also observe that whenever T is a contraction on H not having the eigenvalue 1, then so is T ∗ (cf. Proposition I.3.1). Hence I − T ∗ is invertible, (I − T ∗ )−1 = [(I − T )−1 ]∗ , and (I + T ∗ )(I − T ∗ )−1 = −I + 2(I − T ∗ )−1

= [−I + 2(I − T )−1 ]∗ = [(I + T )(I − T )−1 ]∗ .

Thus the maximal accretive operators corresponding to T and T ∗ are adjoints of one another. Taking account of Theorem III.8.1 and of the relations (III.8.3) and (III.8.4) between generators and cogenerators of semigroups of contractions, we can state the following result. Theorem 4.1. Every accretive operator in H has a maximal accretive extension. For an operator A in H the following conditions are equivalent. (a) (b) (c) (d)

A is maximal accretive. A is accretive and (A + I)D(A) = H. A = (I + T )(I − T )−1 with a contraction T not having the eigenvalue 1. −A is the generator of a continuous semigroup of contractions {Ts }s≥0 .

If A is maximal accretive then so is A∗ , and the corresponding Cayley transforms are adjoints of one another. Let us note that if A is maximal accretive, then the semigroup of contractions generated by −A has its cogenerator equal to the Cayley transform of A (compare the definitions (III.8.3) and (4.5)). Let us also observe that a normal operator A (bounded or not) is maximal accretive if and only if its spectrum is contained in the half-plane Re s ≥ 0 of the plane of the complex numbers s.

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173

2. The following result is not used in the remainder of this book. Proposition 4.2. Let A0 , B0 be two accretive operators in H, such that (A0 f , g) = ( f , B0 g) for

f ∈ D(A0 ),

g ∈ D(B0 ).

(4.7)

Then we can extend A0 and B0 to maximal accretive operators A and B so that we still have (A f , g) = ( f , Bg) for f ∈ D(A), g ∈ D(B). (4.8) Proof. Equation (4.7) implies for f ∈ D(A0 ) and g ∈ D(B0 ) that ((A0 + I) f , (B0 − I)g) = (A0 f , B0 g) − (A0 f , g) + ( f , B0 g) − ( f , g) = (A0 f , B0 g) − ( f , B0 g) + (A0 f , g) − ( f , g) = ((A0 − I) f , (B0 + I)g).

Hence it follows that for the Cayley transforms T0 of A0 and S0 of B0 we have (T0 ϕ , ψ ) = (ϕ , S0 ψ )

(ϕ ∈ D(T0 ), ψ ∈ D(S0 )).

(4.9)

If we construct extensions of T0 and S0 to contractions T and S defined on the whole space H, such that (T h, k) = (h, Sk) for h, k ∈ H, (4.10)

then the corresponding maximal accretive operators A and B will satisfy (4.8). This is in fact a consequence of the following relation which follows from (4.10): ((I + T )h, (I − S)k) = (h, k) + (T h, k) − (h, Sk) − (Th, Sk) = (h, k) + (h, Sk) − (Th, k) − (T h, Sk) = ((I − T )h, (I + S)k) (h, k ∈ H).

To obtain T and S we first extend T0 and S0 by continuity to the closures of their domains; these extensions are also denoted by T0 and S0 ; they satisfy (4.9). For the sake of brevity, we write L for D(S0 ) (this is now a subspace of H), and we set M = H ⊖ L. For any fixed h in H, (h, S0 ψ ) defines a conjugate-linear form on L such that |(h, S0 ψ )| ≤ khk · kψ k. Thus there exists a (unique) h∗ ∈ L such that (h, S0 ψ ) = (h∗ , ψ ) and kh∗ k ≤ khk. Setting h∗ = L0 h we have defined a contraction L0 of H into L such that (L0 h, ψ ) = (h, S0 ψ ) (h ∈ H, ψ ∈ L). (4.11)

In particular, if h = ϕ ∈ D(T0 ) then (L0 ϕ , ψ ) = (ϕ , S0 ψ ) = (T0 ϕ , ψ ); hence it follows that L0 ϕ is the orthogonal projection of T0 ϕ into L: L0 ϕ = PL T0 ϕ .

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Thus kPM T0 ϕ k2 = kT0 ϕ k2 − kL0ϕ k2 ≤ kϕ k2 − kL0 ϕ k2

for ϕ ∈ D(T0 ).

We conclude, by an argument due to M. G. Kre˘ın, that there exists an extension of PM T0 to an operator L1 defined on the whole space H, with values in M, and such that kL1 hk2 ≤ khk2 − kL0 hk2 for every h ∈ H; see [Func. Anal.] Sec. 125. The operator T defined by T h = L0 h + L1h

for h ∈ H

is thus a contraction and T ϕ = L0 ϕ + L1 ϕ = PL T0 ϕ + PM T0 ϕ = T0 ϕ

for ϕ ∈ D(T0 );

that is, T ⊃ T0 . Moreover, (T h, ψ ) = (L0 h + L1h, ψ ) = (L0 h, ψ ) = (h, S0 ψ ) for h ∈ H, ψ ∈ D(S0 ). This shows that T ∗ ⊃ S0 ; setting S = T ∗ the pair {T, S} satisfies(4.10), T ⊃ T0 , and S ⊃ S0 , so the proof is complete. Corollary. For every accretive A0 with D(A0 ) ⊂ D(A∗0 ), there exists a maximal accretive A so that A0 ⊂ A ⊂ B∗0

with

B0 = A∗0 |D(A0 ).

In fact, we have (A0 f , g) = ( f , A∗0 g) = ( f , B0 g) for f , g ∈ D(A0 ) = D(B0 ), and in particular Re( f , B0 f ) = Re(A0 f , f ) ≥ 0. Thus B0 is accretive and there exists a maximal accretive extension A or A0 such that (A f , g) = ( f , B0 g) for all f ∈ D(A) and g ∈ D(B0 ); hence A ⊂ B∗0 . 3. Let A be maximal accretive in H and let T be its Cayley transform; T is a contraction defined on the whole space H, and T = (A − I)(A + I)−1,

A = (I + T )(I − T )−1 .

(4.12)

By virtue of these relations, a subspace K of H reduces5 A if and only if it reduces T ; it also follows that T is unitary if and only if A has the form iH with self-adjoint H. The operators A = iH are characterized by the property A∗ = −A, 5

That is, PA ⊂ AP, P being the orthogonal projection of H onto K.

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175

so we call them antiadjoint operators. An operator is completely nonantiadjoint if no nonzero subspace reduces it to an antiadjoint operator. A maximal accretive, completely nonantiadjoint operator is said to be purely maximal accretive. The canonical decomposition of the Cayley transform T of a maximal accretive A generates a decomposition of A. Proposition 4.3. For every maximal accretive operator A on H there exists a decomposition H = H0 ⊕ H1, reducing A, and such that the part of A in H0 is antiadjoint and the part of A in H1 is purely maximal accretive. This decomposition is unique; H0 or H1 may equal {0}. It follows from Proposition I.3.1 that if T is a contraction on H, then every subspace Ha = {h : h ∈ H, T h = ah} with |a| = 1 reduces T . This implies that if A is maximal accretive in H, then every subspace { f : f ∈ H, AF = b f }, with purely imaginary number b, reduces A. In particular, the subspace NA = { f : A f = 0}

(4.13)

reduces A, or what amounts to the same, A f = 0 implies A∗ f = 0, and conversely. If A is purely maximal accretive, then necessarily NA = {0}. We conclude that in this case A−1 exists and its domain is dense in H. 4. The relations (4.12) between a maximal accretive operator A and its Cayley transform T yield a method for constructing a functional calculus for A based on the functional calculus for T . To this end let us consider the homography

λ →δ =

1+λ ≡ ω (λ ) 1−λ

(4.14)

that maps the unit disc D onto the right half-plane

∆ = {δ : Re δ > 0}; the inverse map is

δ −1 . δ +1 We then define the functions of A by the formula δ →λ =

f (A) = ( f ◦ ω )(T )

(4.15)

(4.16)

whenever f ◦ ω ∈ NT . In this way our functional calculus for contractions gives rise to a functional calculus for maximal accretive operators. As 1 is not an eigenvalue of T , the one-point set {1} is of measure O with respect to the spectral measure ET corresponding to the unitary part of T . Therefore the class HT∞ contains the functions v ∈ H ∞ that are continuous on D\{1}. Because D\{1} is mapped by the homography (4.14) onto the closed half-plane ∆¯ = {δ : Re δ ≥ 0}

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the point δ = ∞ not included, we see that the class of functions f (δ ) admitted contains in particular the functions that are continuous and bounded on ∆ , and holomorphic on ∆ . One such function is e−t δ = exp(t(λ + 1)/(λ − 1)) = et (λ ) for t ≥ 0. We conclude that e−tA makes sense for t ≥ 0 andRequals et (T ). For fixed z (Re z < 0) and M (0 < M < ∞) the integral fM (z; δ ) = 0M etz e−t δ dt is the limit, by bounded convergence on ∆ , of the corresponding Riemann sums; hence it folR lows that fM (z; A) = 0M etz e−tA dt. As M → ∞, fM (z; δ ) tends to (δ − z)−1 uniformly on ∆ ; thus on account of Theorem III.2.3(c′) and Theorem 1.1(iv) we have fM (z; A) → (A− zI)−1 . So we arrive within the framework of our functional calculus at the relation Z ∞ (A − zI)−1 = etz e−tA dt, (4.17) 0

valid for any maximal accretive operator A and for any complex number z with Re z < 0.

5. The results of this section also apply in an obvious way to dissipative operators A′ : one just has to consider the accretive operator A = −iA′ . Thus the Cayley transform of the dissipative operator A′ will be, by definition, equal to the Cayley transform of the accretive operator A = −iA′ ; A′ and T are therefore connected by the relations T = (A′ − iI)(A′ + iI)−1 ,

A′ = i(I + T )(I − T )−1 .

(4.18)

In the case of a maximal dissipative A′ , the canonical decomposition of T generates a decomposition of A′ into the orthogonal sum of a self-adjoint operator and of a purely maximal dissipative operator.

5 Fractional powers 1. Let A be a maximal accretive operator in the space H, and let T be its Cayley transform. In this section we define and study the powers Aα of A, where 0 < α < 1. For this purpose let us consider the functions fα (δ ) = δ α

(α ≥ 0)

on ∆ (for z = reiϕ (r > 0, |ϕ | ≤ π /2) we define zα by rα eiαϕ ). Then we have ( fα ◦ ω )(λ ) = ω α (λ ) = with

uα (λ ) = (1 + λ )α ,

uα (λ ) vα (λ )

vα (λ ) = (1 − λ )α

(λ ∈ D).

The functions uα and vα are continuous on D, holomorphic on D, and belong to the class E ∞ (cf. Proposition III.1.3). Moreover, uα has only a zero at −1 and vα has

5. F RACTIONAL POWERS

177

only a zero at +1. Because ET ({1}) = O, it follows that ω α ∈ NT and that fα (A) = ω α (T ) = vα (T )−1 uα (T ).

(5.1)

Furthermore, by virtue of Proposition 1.3 we have for any α , β ≥ 0, vβ (T )−1 uα (T ) = uα (T )vβ (T )−1 .

(5.2)

We use the shorter notation Aα for fα (A). This is justified if we show that the following relations hold. A1 = A, and Aα Aβ = Aα +β

A0 = I,

for α , β ≥ 0.

Now A0 = I, because ω 0 (λ ) = 1. Then, on account of (5.2), A1 = v1 (T )−1 u1 (T ) = u1 (T )v1 (T )−1 = (I + T )(I − T )−1 = A and

Aα +β = ω α +β (T ) = (ω α · ω β )(T ) = ω α (T )ω β (T ) = Aα Aβ

(cf. Proposition 1.3). Because (ω α )˜ = ω α , the same proposition also implies (Aα )∗ = [ω α (T )]∗ = ω α (T ∗ ) = (A∗ )α , because T ∗ is the Cayley transform of A∗ (cf. Theorem 4.1). The inequality απ | arg ω α (λ )| ≤ on D, 2 for 0 ≤ α ≤ 1, allows us to apply Theorem 3.2 to Aα = ω α (T ) and to (Aα )∗ . We consider, then, the functions uβ (λ ) = (1 + λ )β ,

vβ (λ ) = (1 − λ )β ,

wα (λ ) =

uα (λ ) − vα (λ ) , uα (λ ) + vα (λ )

(5.3)

where 0 ≤ β and 0 ≤ α ≤ 1, and show that they satisfy the hypotheses of Theorem 1.1(viii). In fact, uα (λ ) and vα (λ ) have no common zero and their values lie in the sector n απ o , z : | arg z| ≤ 2

thus it follows that uα (λ ) + vα (λ ) has no zero in D for 0 ≤ α < 1. Therefore the function wα (λ ) is continuous on D and holomorphic on D; moreover, it is easy to see that |wα (λ )| < 1

on D with the exception of the points λ = ±1, where we have wα (1) = 1 and wα (−1) = −1; otherwise if α = 1 then w1 (λ ) ≡ λ . So we have for 0 ≤ α ≤ 1

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and for β ≥ 0 Cwα = ∅,

w−1 α (Cuβ ) = ∅

and wα−1 (C0vβ ) = w−1 α ({1}) = {1},

and all these sets are of measure O with respect to ET . Applying Theorem 1.1(viii) it follows that (5.4)

Tα = wα (T ) is a contraction, ω β (Tα ) and (ω β ◦ wα )(T ) exist, and

ω β (Tα ) = (ω β ◦ wα )(T ).

(5.5)

Now, because (ω β ◦ wα )(λ ) =



1 + wα (λ ) 1 − wα (λ )

relation (5.5) means that

β

=



uα (λ ) vα (λ )

β

= ω αβ (λ ),

ω β (Tα ) = Aαβ .

(5.6)

Aα = ω 1 (Tα ) = v1 (Tα )−1 · u1 (Tα ) = (I − Tα )−1 (I + Tα );

(5.7)

For β = 1 this relation becomes

this shows that Aα is maximal accretive, with the Cayley transform Tα . Hence we have f (Aα ) = f ◦ ω (Tα ) whenever f ◦ ω (Tα ) makes sense, so in particular (Aα )β = ω β (Tα ). Thus (5.6) becomes (Aα )β = Aαβ

(0 ≤ α ≤ 1, β ≥ 0).6

2. Because Aα is maximal accretive if 0 ≤ α ≤ 1, there exists a continuous semigroup {Tα (s)}s≥0 of contractions whose generator is −Aα . The cogenerator of this semigroup is then equal to the Cayley transform of Aα , that is, to Tα . By virtue of Theorem III.8.1 we have therefore Tα (s) = es (Tα ) = es (wα (T ))

(5.8)

with es (λ ) = exp(s(λ + 1)(λ − 1)). This function is holomorphic and bounded on D, and continuous on D\{1}, therefore we can apply Theorem 1.1 (viii) to obtain that es (wα (T )) = (es ◦ wα )(T ). (5.9) 6 The restriction α ≤ 1 is motivated by the fact that, for α > 1, Aα is not necessarily accretive (this is the case, e.g., if A is antiadjoint and A 6= O).

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179

Now we have   wα (λ ) + 1 (es ◦ wα )(λ ) = exp s = exp(−sω α (λ )); wα (λ ) − 1

hence we see that if 0 ≤ α < 1 this function belongs to the class HT∞ and is bounded in absolute value by 1 on D, for all the real or complex values of the parameter s lying in the sector n πo ∆ 1−α = s : | arg s| ≤ (1 − α ) . 2 The identities for s1 , s2 ∈ ∆ 1−α ,

(es1 ◦ wα )(es2 ◦ wα ) = es1 +s2 ◦ wα imply that the operators

Tα (s) = (es ◦ wα )(T ) (s ∈ ∆ 1−α ) form a semigroup: Tα (0) = I,

Tα (s1 )Tα (s2 ) = Tα (s1 + s2 )

(s1 , s2 ∈ ∆ 1−α ).

Moreover, this semigroup is holomorphic in the interior ∆1−α of ∆ 1−α . To prove this, let us fix a point s0 > 0 and a point s lying in the largest open disc contained in ∆1−α , with center s0 , that is, such that |s − s0 | < ρ0 = s0 cos

απ . 2

Let us consider then the expansion α

α

α

∞

e−sδ = e−(s−s0 )δ · e−s0 δ = ∑

 s − s n

n=0

where bn (δ ) =

ρ0

0

bn (δ ) (δ ∈ ∆ ),

(5.10)

(−1)n α (ρ0 δ α )n e−s0 δ ; n!

then we have ∞

1 |ρ0 δ α |m · exp(−s0 Re δ α ) = exp[ρ0 |δ α | − s0 Re δ α ] ≤ 1, m=0 m!

|bn (δ )| ≤ ∑ because

Re δ α ≥ |δ α |cos

απ 2

for δ ∈ ∆ .

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The expansion (5.10) is therefore uniformly convergent with respect to δ on ∆ . Hence it follows for the corresponding operators:   ∞ s − s0 n Tα (s) = ∑ Bn , ρ0 n=0 where Bn = bn (A) = (bn ◦ ω )(T ), kBn k ≤ 1. This concludes the proof of the fact that Tα (s) is a holomorphic function of s on ∆1−α . Finally we note that Theorem 3.2 can be applied to the functions ω α (λ ), 0 ≤ α ≤ 1. Summing up, we have proved the following result. Theorem 5.1. Let A be a maximal accretive operator in H and let Aα (α ≥ 0) be the operator that corresponds to the function δ α in the sense of the functional calculus f → f (A) defined in Sec. 4.4. Aα is then a closed operator with domain dense in H, and we have A0 = I;

A1 = A;

Aα +β = Aα Aβ (α , β ≥ 0);

(Aα )∗ = (A∗ )α

(α ≥ 0).

For 0 ≤ α ≤ 1, Aα is maximal accretive and we have (Aα )β = Aαβ for β ≥ 0; moreover, απ | arg(Aα h, h)| ≤ for h ∈ D(Aα ). 2 Let {Tα (s)}s≥0 be the semigroup of contractions whose generator is equal to −Aα . If 0 ≤ α < 1 then this semigroup can be extended to the complex values of the parameter s lying in the sector | arg s| ≤ (1 − α ) π /2 so that the semigroup property is preserved and Tα (s) is a contraction valued function of s, holomorphic in this sector. If 0 ≤ α < 12 , the operators Aα , Aα ∗ , and hence also Re Aα and Im Aα have the same domain; moreover, Re Aα is positive self-adjoint and the following inequalities hold. (i) Re(Aα h, Aα ∗ h) ≥ cos απ · max{kAα hk2 , kAα ∗ hk2 }.

(ii) kAα hk ≥ cos απ · kAα ∗hk,

kAα ∗ hk ≥ cos απ · kAα hk. απ · kAα hk2 . (iii) Re(Aα h, [Re Aα ]h) ≥ cos2 2 . h i απ απ cos απ · kAα hk ≥ k[Re Aα ]hk ≥ cos2 · kAα hk. (iv) cos2 2 2 απ · k[Re Aα ]hk. (v) k[Im Aα ]hk ≤ tan 2 3. This theorem implies in particular that every maximal accretive operator A possesses, for n ≥ 2, an nth root B = A1/n that is also maximal accretive and such that π | arg(Bh, h)| ≤ . (5.11) 2n

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181

We prove that there is only one nth root with these properties. To this end we need some supplementary lemmas. Lemma 5.2. (a) Let P ≥ O and Q = Q∗ satisfy |(Qh, h)| ≤ kPhk2

for all h ∈ H.

Then the operator |Q| = (Q2 )1/2 satisfies k|Q|1/2 hk2 = (|Q|h, h) ≤ 21/2 khkkPhkkPk for all h ∈ H. (b) Let T be a contraction and let R = (T + T ∗ )/2, Q = (T − T ∗ )/2i. If Q satisfies for some 0 ≤ θ < ∞, |(Qh, h)| ≤ θ kDT hk2

for all h ∈ H

then kDR hk2 ≤ (1 + (21/2θ )1/2 )2 khk3/2kDT hk1/2

for all h ∈ H.

(c) With the notation above, if DR h0 = 0 for some h0 ∈ H, then h = h+1 + h−1,

where

T h±1 = Rh±1 = ±h±1.

R kQk

Proof. Part (a). Let Q = −kQk λ dEλ be the spectral representation of Q and let E+ = EkQk − E0− , E− = E0− . For h± = E± h± and any complex scalars a± we have |a+ |2 (Qh+ , h+ ) + |a−|2 (Qh− , h− ) ≤ kP(a+ h+ + a−h− )k2 .

Note that q± = ±(Qh± , h± ) ≥ 0, q± ≤ kPh± k2 , and |a+ |2 q+ − |a−|2 q− ≤ |a+ |2 kPh+k2 + 2 Re[a+ a¯− (Ph+ Ph− )] + |a−|2 kPhk2. Because a+ , a− are arbitrary, we infer that

|λ 2 q+ − q− | ≤ λ 2 kPh+k2 − 2λ |(Ph+, Ph− )| + kPh−k2 for all real values of λ . In other words,

λ 2 (kPh+ k2 ∓ q+ ) − 2λ |(Ph+, Ph− )| + (kPh−k2 ± q− ) ≥ 0 for all λ , and taking discriminants we obtain |(Ph+ , Ph− )|2 ≤ (kPh+k2 ∓ q+ )(kPh− k2 ± q−), and consequently |(Ph+ , Ph− )| ≤ (kPh+ k4 − q2+)1/4 (kPh−k4 − q2− )1/4 .

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Setting ρ± = q± /kPh±k2 , we apply this relation to h± = E± h (h ∈ H) to obtain kPhk2 = kPh+ + Ph−k2 = kPh+ k2 + 2 Re(Ph+ , Ph− ) + kPh−k2

≥ kPh+ k2 − 2kPh+kkPh−k(1 − ρ+2 )1/4 (1 − ρ−2 )1/4 + kPh−k2 ≥ kPh+ k2 (1 − (1 − ρ+2 )1/2 ) + kPh−k2 (1 − (1 − ρ−2 )1/4 ) ≥ (kPh+ k2 ρ+2 + kPh−k2 ρ−2 )/2

= (q2+ /kPh+k2 + q2−/kPh−k2 )/2

≥ (q2+ /kh+k2 + q2−/kh− k2 )/(2kPk2) ≥ (q+ + q− )2 /(2khk2kPk2 ) = (|Q|h, h)2 /(2khk2kPk2 ),

where in the last inequality we used the fact that for a, b, c > 0, 0 < x < c, (a2 /x) + (b2 /(c − x)) ≥ (a + b)2/c. Part (b). Let h ∈ H. Then

kDT hk2 = khk2 − kT hk2 = kDR hk2 − i(Rh, Qh) + i(Qh, Rh) − kQhk2, hence, by applying (a) and taking into account that Q, R, and DT are contractions, we have kDR hk2 ≤ kDT hkkhk + 2k|Q|1/2hkkhk + k|Q|1/2hk2

≤ kDT hkkhk + 2(21/2θ )1/2 khk3/2kDT hk1/2 + 21/2θ khkkDT hk ≤ (1 + 2(21/2θ )1/2 + 21/2θ )khk3/2kDT hk1/2.

Part (c). If kDR h0 k2 = 0, then (because R is a self-adjoint contraction) it is obvious that h0 = h+1 + h−1 where Rh±1 = ±h±1 . But in kh±1 k2 = ±(Rh±1, h±1 ) = ±(T h±1 , h±1 ) = |(T h±1 , h±1 )| ≤ ≤ kT h±1 kkh±1k ≤ kh±1 k2

all inequalities turn out to be equalities, thus we must have T h±1 = ±h±1 = Rh±1 and this concludes the proof of the lemma. Lemma 5.3. Let A be a maximal accretive operator in H satisfying the condition | arg(A f , f )| ≤ απ /2 for all

f ∈ D(A)

(5.12)

with some α ∈ [0, 1). Then (A f0 , f0 ) = 0 implies A f0 = 0. Proof. Let T be the Cayley transform of A, that is, the contraction T on H defined by T h = (A − I) f for h = (A + I) f , f ∈ D(A).

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183

Then (with the notation of Lemma 5.2(b) 4(A f , f ) = ((I + T )h, (I − T )h) = kDT hk2 + 2i(Qh, h), hence T satisfies the conditions of Lemma 5.2(b) with θ = tan(πα /2)/2. If (A f0 , f0 ) = 0, then DT h0 = 0 and by virtue of Lemma 5.2(b) and (c) we have h0 = h+1 +h−1 where T h±1 = ±h±1. But 1 is not an eigenvalue of T , hence h0 = h−1 and T h0 = −h0 . It follows that 2A f0 = (T + I)h0 = 0, which completes the proof. We also use the following commutativity property. Lemma 5.4. Let A, A′ be two maximal accretive operators such that D(A) ⊂ D(A′ ) and A′ A ⊂ AA′ .

(5.13)

Then their Cayley transforms T, T ′ commute. Proof. The operators A + I and A′ + I are boundedly invertible, thus for every f ∈ H there exists g ∈ H such that f = (A′ + I)(A + I)g.

(5.14)

Because g ∈ D(A), from (5.13) we have also g ∈ D(A′ ); hence Ag = (A + I)g − g ∈ D(A′ ). So we can write (5.14) in the form f = (A′ A + A′ + A + I)g; from the second relation (5.13) we have then f = (AA′ + A′ + A + I)g = (A + I)(A′ + I)g.

(5.15)

The relations (5.14) and (5.15) imply (A + I)−1(A′ + I)−1 f = (A′ + I)−1 (A + I)−1 f

(for all

f ∈ H).

The relations T = I − 2(A + I)−1 and T ′ = I − 2(A′ + I)−1 show now that T and T ′ commute. After these preliminaries we can turn to the proof of the following result. Proposition 5.5. For a maximal accretive A and for every integer n ≥ 2 there exists a maximal accretive B, and only one, satisfying the conditions: Bn = A, π | arg(B f , f )| ≤ 2n

(5.16) (for all f ∈ D(B)).

(5.17)

Proof. Let {Aα } be the system of the fractional powers of A constructed in the proof of Theorem 5.1; B = A1/n is then a solution of our problem. We just have to show

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C HAPTER IV. E XTENDED F UNCTIONAL C ALCULUS

that no other solution exists. Let B′ be any solution, and let S and S′ be the Cayley transforms of B and B′ , respectively; by virtue of (5.4) we have S = w1/n (T ), where T denotes the Cayley transform of A and w1/n is the function defined by (5.3). Let us observe that B′ A = B′ B′n = B′n+1 = B′n B′ = AB′ and that D(A) = D(B′n ) ⊂ D(B′ ); applying Lemma 5.4 we obtain then that S′ commutes with T and consequently with every function of T , in particular S′ commutes with S. Let B′ = B′0 ⊕ B′1 be the canonical decomposition of the maximal accretive operator B′ into the orthogonal sum of its antiadjoint part B′0 and its purely maximal accretive part B′1 . Let f be such that B′n f = 0. If f = f0 + f1 is the corresponding ′n ′ decomposition of f , then we have B′n 0 f 0 = 0 and B1 f 1 = 0. Now B1 is invertible (cf. the end of Sec. 4.3), and hence f1 = 0. On the other hand, because iB′0 = H with a self-adjoint H, it follows from the spectral theory for H that B′n 0 f 0 = 0 implies B′0 f0 = 0. Thus B′n f = 0 implies B′ f = 0. As B′n = A, we conclude that the null spaces of A and B′ are the same, and as all this applies to B too (instead of B′ ), it follows that the maximal accretive operators A, B, B′ all have the same null space N. By virtue of the remark following Proposition 4.3, N reduces A, B, B′ , and hence their Cayley transforms too, and for f ∈ N we have B f = B′ f (= 0). Thus to prove B = B′ it suffices to consider in the sequel the parts of these operators in M = H⊖ N. Let us set V = (I + S)(I − S′) and W = (I + S′)(I − S). As V and W commute we have

n−1

V n − W n = ∏ (V − ε kW ),

ε = e2π i/n .

(5.18)

k=0

Because I + S = B(I − S), I + S′ = B′ (I − S′), and Bn = A = B′n , we have V n = Bn (I − S)n(I − S′)n = B′n (I − S′)n (I − S)n = W n , and thus

n−1

∏ (V − ε kW ) = O.

(5.19)

k=0

Let g ∈ M be such that for some k (V − ε kW )g = 0.

(5.20)

6. N OTES

185

Setting f = (I − S)(I − S′ )g we have then B f = (I + S)(I − S′ )g = V g = ε kW g = ε k (I + S′)(I − S)g = ε k B′ f , k

(B f , f ) = ε (B f , f ). ′

(5.21) (5.22)

If 1 ≤ k ≤ n − 1 this implies (B f , f ) = (B′ f , f ) = 0, for otherwise the relations | arg(B f , f )| ≤

π , 2n

| arg(B′ f , f )| ≤

π , 2n

arg ε k =

2π k n

(1 ≤ k ≤ n − 1)

would contradict (5.22). In view of Lemma 5.2, relation (5.20) implies therefore in this case that B f = 0; because B is invertible on M it follows that f = 0, and hence g = 0. Thus (5.19) implies V − W = O; that is, (I − S)(I + S′ ) = (I + S)(I − S′), hence S = S′ and therefore B = B′ . (All this for the parts of the operators on M.) This concludes the proof of the proposition.

6 Notes The functional calculus of contractions, for not necessarily bounded functions, was developed in S Z .-N.–F. [VI]. There one studied the corresponding calculus for maximal accretive operators also; a slightly more restricted calculus was proposed independently (for maximal dissipative operators) by L ANGER [2]. In earlier work 7 one considered only holomorphic functions; meromorphic functions are admitted in the present work. This natural and almost immediate extension is motivated by some results of the authors on the commutant and bicommutant of contractions of class C0 with finite defect indices; see S Z .-N.–F. [14], [15]. The method of extending an accretive (or dissipative) operator to a maximal one via Cayley transforms, modeled on von Neumann’s theory of symmetric operators, is due to P HILLIPS [2]. The characterization of the infinitesimal generators of continuous semigroups of contractions by their Cayley transforms (i.e. the equivalence of conditions (c) and (d) in Theorem 4.1) was given in S Z .-NAGY [II]; also see F OIAS¸ [2]. Proposition 4.2 is new. In a slightly more restricted form, Proposition 4.3 (on the decomposition of maximal accretive or dissipative operators) was given by L ANGER [1]. Fractional powers of operators A in Hilbert space, or even in Banach spaces, such that −A is the infinitesimal generator of a continuous one-parameter semigroup of contractions, have been constructed by different authors using different methods. 7

Including the original French edition of this book.

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C HAPTER IV. E XTENDED F UNCTIONAL C ALCULUS

The definition proposed by B OCHNER [1] and P HILLIPS [1] introduces Aα (0 < α < 1) by means of the continuous semigroup Tα ,t = e−tA That is, one defines Tα ,t =

Z ∞ 0

α

(t ≥ 0).8

Ts dmα ,t (s),

where Ts = e−sA and the measure is determined by the Laplace integral e−t ρ

α

Z ∞ 0

e−sρ dmα ,t (s)

(t ≥ 0; Re ρ ≥ 0; 0 < α < 1).

Further formulas, due to B ALAKRISHNAN [1], are the following: Aα h =

sin απ π

Z ∞ 0

Aα h = Γ (−α )−1

λ α −1 (λ I + A)−1Ah d λ ,

Z ∞ 0

λ −α −1 (e−λ A − I)h d λ

(h ∈ D(A)); see also YOSIDA [1], [2] p. 259. In the Hilbert space situation, these formulas may be obtained easily by our functional calculus, in a similar way as (4.17). The uniqueness theorem (our Proposition 5.5) is due (in its form on dissipative operators) to M ACAEV AND PALANT [1] (in the case of bounded operators) and to L ANGER [2] (in the general case); our proof is a (slightly simplified) variant of that of Langer. Later, N OLLAU [1] extended the uniqueness theorem to operators in arbitrary Banach spaces. As to Theorem 5.1, the fact that the semigroup is holomorphic in a sector of the complex plane, was first proved by Y OSIDA [1]; the fact that for 0 ≤ α < 1/2 the operators Aα and Aα ∗ have the same domain of definition and satisfy the inequalities (i)–(v), was first established (partially with other constant factors) by K ATO [1]. His methods are different from ours. Lemma 5.2, which corrects and completes the argument presented in the earlier editions of this book, may be useful in other applications. In connection with this chapter see also D OLPH [1]; D OLPH AND P ENZLIN [1]; L ANGER AND N OLLAU [1]; and N OLLAU [2].

7 Further results If T is a unilateral shift of multiplicity one, then Lemma V.3.2 implies that every operator X commuting with T is of the form u(T ) for some u ∈ H ∞ . S ARASON [5],[6] proved that this statement extends to unbounded operators. Thus, if X is a closed, densely defined operator satisfying T X ⊂ X T , then there exists a function 8

This notation only indicates that Tα ,t is the semigroup whose generator is −Aα .

7. F URTHER RESULTS

187

u ∈ NT such that X = u(T ). Both of these statements remain true if T is replaced by an operator of class C0 with dT = 1. This is proved in S ARASON [3] and [6] for the bounded and unbounded case, respectively. Similar results can be proved for operators in the double commutant of an arbitrary operator of class C0 ; see Theorem X.4.2. In connection with this material, see also M ARTIN [1]; B ERCOVICI , D OUGLAS , F OIAS , AND P EARCY [1]; and B ERCOVICI [6].

Chapter V

Operator-Valued Analytic Functions 1 The spaces L2 (A) and H 2 (A) 1. For any separable Hilbert space A we denote by L2 (A) the class of functions v(t) (0 ≤ t ≤ 2π ) with values in A, measurable1 (strongly or weakly, which are equivalent due to the separability of A) and such that 1 kvk = 2π 2

Z2π 0

kv(t)k2A dt < ∞.

(1.1)

With this definition of the norm kvk, L2 (A) becomes a (separable) Hilbert space; it is understood that two functions in L2 (A) are considered identical if they coincide almost everywhere (with respect to Lebesgue measure). If dim A = 1 (i.e., if L2 (A) consists of scalar-valued functions), we write L2 instead of L2 (A). Let {vn (t)} (n = 1, 2, . . .) be a sequence converging to v(t) in L2 (A), that is, in the mean: kvn − vk2 =

1 2π

Z2π 0

kvn (t) − v(t)k2A dt → 0 (n → ∞).

Then we can choose a subsequence {vnk (t)} (k = 1, 2, . . .) such that ∑ k

Z2π 0

kvnk (t) − v(t)k2A dt < ∞.

By virtue of the theorem of Beppo Levi we have then a.e. ∑ kvnk (t) − v(t)k2A < ∞ n

1

For the theory of integration of vector-valued functions as well as for that of analytic vectorvalued function see H ILLE [1], Chap. III.

B.Sz.-Nagy et al., Harmonic Analysis of Operators on Hilbert Space, Universitext, DOI 10.1007/978-1-4419-6094-8_5, © Springer Science + Business Media, LLC 2010

189

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C HAPTER V. O PERATOR -VALUED A NALYTIC F UNCTIONS

and consequently a.e. kvnk (t) − v(t)kA → 0

(k → ∞).

Thus from every sequence converging in L2 (A) we can choose a subsequence that converges pointwise a.e. For any integer k let us denote by Ek the subspace of L2 (A) consisting of the functions of the form eikt a (a ∈ A). Clearly, Ek ⊥ E j for k 6= j and, moreover, we have ∞ L L2 (A) = Ek . (1.2) −∞

2

Indeed, let v ∈ L (A) be orthogonal to all Ek : 1 2π

Z 2π 0

e−ikt (v(t), a)A dt = 0

(a ∈ A; k = 0, ±1, ±2, . . .).

Then we have (v(t), a)A = 0 everywhere, with the possible exception of the points t of a set Ea depending on a and of zero measure. Letting a run over a countable dense subset of A, and taking the union of the corresponding sets Ea one obtains a set E of zero measure; then v(t) = 0 for every point t not belonging to E, and consequently v = 0 as an element of L2 (A). This proves (1.2). Let us also observe that keikt akL2 (A) = kakA

(a ∈ A).

(1.3)

From (1.2) and (1.3) it follows that there exists a one-to-one correspondence be2 tween the elements v of L2 (A) and the sequences {ak }∞ −∞ (ak ∈ A) with ∑ kak kA < ∞, such that for corresponding v and {ak } we have ∞

v(t) = ∑ eikt ak −∞

and

∞

kvk2 = ∑ kak k2A , −∞

where (1.4) is understood in the sense of convergence in the mean: Z 2π

0

2 n

v(t) − ∑ eikt ak dt → 0 (m, n → ∞);

−m A

by virtue of (1.4) we have ak =

1 2π

Z 2π 0

e−ikt v(t) dt

that is, (1.4) is the Fourier series of v.

(k = 0, ±1, . . .);

(1.4)

(1.5)

1. T HE SPACES L2 (A) AND H 2 (A)

191

An important subspace of L2 (A), which we denote by L2+ (A), consists of those functions for which ak = 0 (k < 0). With any function ∞

v(t) = ∑ eikt ak ∈ L2+ (A) 0

we associate the function

∞

u(λ ) = ∑ λ k ak 0

of the complex variable λ ; u(λ ) is defined and holomorphic on the unit disc D, because

 1/2 n n

n k

∑ λ ak ≤ ∑ |λ |k kak kA ≤ (1 − |λ |2)−1/2 ∑ kak k2 →0 A

m

A

m

m

for n > m → ∞ and for |λ | < 1, uniformly for |λ | ≤ r0 < 1. One can recover v(t) from u(λ ) as a radial limit in the mean, in fact, 1 2π

Z 2π 0

kv(t) − u(re

it

)k2A

1 dt = 2π ∞

Z 2π

∞ 0

2

∑(1 − rk )eikt ak dt

0

A

= ∑(1 − rk )2 kak k2A → 0 0

as r → 1 − 0; moreover, we have 1 2π

Z 2π 0

∞

∞

0

0

ku(reit )k2A dt = ∑ r2k kak k2A ≤ ∑ kak k2A < ∞

(0 ≤ r < 1).

Let us denote by H 2 (A) the class of functions ∞

u(λ ) = ∑ λ k ak 0

with values in A, holomorphic on D, and such that 1 2π

Z 2π 0

ku(reit )k2A dt

(0 ≤ r < 1)

2k 2 has a bound independent of r. This integral being equal to ∑∞ 0 r kak kA , the last 2 < ∞. Hence we see that every condition is equivalent to the condition ∑∞ ka k k A 0 function u(λ ) ∈ H 2 (A) arises from a function v(t) ∈ L2+ (A), indeed from v(t) = ikt ∑∞ 0 e ak . Because u(λ ) and v(t) determine each other, we can identify the classes H 2 (A) and L2+ (A), thus providing H 2 (A) with the Hilbert space structure of L2+ (A) and embedding it in L2 (A) as a subspace.

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C HAPTER V. O PERATOR -VALUED A NALYTIC F UNCTIONS

2. The functions u(λ ) and v(t) thus associated are also connected by Poisson’s formula Z 1 2π u(reit ) = Pr (t − s)v(s) ds (0 ≤ r < 1), (1.6) 2π 0 where 1 − r2 Pr (t) = . (1.7) 1 − 2r cost + r2 This is a simple consequence of the elementary relation (reit )k =

1 2π

Z 2π 0

Pr (t − s)eiks ds

(k = 0, 1, . . .)

and of the fact that v(t) is the limit in the mean of its Fourier series. Making use of the formula (1.6) we can prove that v(t) is the radial limit of u(reit ) not only in the mean but also pointwise, almost everywhere. More precisely, u(λ ) tends to v(t) strongly (in A) as λ tends to eit nontangentially with respect to the unit circle at every point t such that 1 2s

Z t+s t−s

v(τ ) d τ → v(t) strongly (s → 0),

(1.8)

thus a.e. (generalized Fatou theorem). The proof is the same as in the scalar-valued case; see H OFFMAN [1]. By virtue of this theorem we may write u(eit ) instead of v(t) when considering functions in L2+ (A).

2 Inner and outer functions 1. Consider a function Θ (λ ) whose values are bounded operators from a Hilbert space A to a Hilbert space A∗ , both separable, and which has a power series expansion ∞ (2.1) Θ (λ ) = ∑ λ kΘk 0

whose coefficients are bounded operators from A to A∗ ; the series is supposed to be convergent in the open unit disc D (weakly, strongly, or in the norm, which amounts to the same for power series). Let us suppose, moreover, that kΘ (λ )k ≤ M

(bounded independent of λ in D).

Such a function {A, A∗ , Θ (λ )} is called a bounded analytic function (on D). Condition (2.2) implies 1 2π

Z 2π 0

kΘ (reit )ak2A∗ dt ≤ M 2 kak2A

(0 ≤ r < 1)

(2.2)

2. I NNER AND OUTER FUNCTIONS

193

and consequently (by Sec. 1) ∞

∑ kΘk ak2A∗ ≤ M 2 kak2A 0

(2.3)

for all a ∈ A. It follows by virtue of Sec. 1 that the limit lim Θ (λ )a

(λ → eit nontangentially)

(2.4)

exists in the strong sense in A∗ , everywhere, with the possible exception of the points t of a set Ea of zero measure. Letting a run over a countable dense subset of A and taking the union of the corresponding sets Ea we obtain a set E of zero measure, such that, on account of (2.2), the limit (2.4) will exist for every t not belonging to E and for every a ∈ A. Hence Θ (eit ) = lim Θ (λ ) (λ → eit nontangentially) (2.5) exists a.e. as a strong limit of operators. In particular, we have a.e.

Θ (eit ) = lim Θ (reit ) (strongly). r→1−0

(2.6)

Moreover, also by virtue of Sec. 1, Θ (reit )a converges in the mean (i.e., in L2 (A∗ )) to Θ (eit )a as r → 1 − 0, and this limit has the Fourier expansion ∞

Θ (eit )a = ∑ eikt Θk a, 0

(2.7)

which converges in L2 (A∗ ). With the bounded analytic function {A, A∗, Θ (λ )} we associate its “adjoint” {A∗ , A, Θ ˜(λ )} defined by

Θ ˜(λ ) = Θ (λ¯ )∗

(λ ∈ D)

(2.8)

which is also analytic and bounded by the same bound; in fact we have ∞

Θ ˜(λ ) = ∑ λ kΘk∗ 0

and kΘ ˜(λ )k = kΘ (λ¯ )∗ k = kΘ (λ¯ )k ≤ M for λ ∈ D. Consequently

Θ ˜(eit ) = lim Θ ˜(λ ) (λ → eit nontangentially) exists a.e. as a strong limit of operators. This implies that Θ ˜(e−it ) is a.e. the strong limit of Θ (λ )∗ as λ → eit nontangentially. On the other hand, Θ (λ ) tends to Θ (eit ) strongly, hence Θ (λ )∗ tends to Θ (eit )∗ weakly, as λ → eit nontangentially, for almost every t. Thus for almost every t, Θ ˜(e−it ) = Θ (eit )∗ and

Θ (eit )∗ = lim Θ (λ )∗

(as λ → eit nontangentially)

(2.9)

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C HAPTER V. O PERATOR -VALUED A NALYTIC F UNCTIONS

in the sense of the strong convergence of operators. In particular, we have

Θ (eit )∗ = lim Θ (reit )∗ r→1−0

strongly, a.e.

(2.10)

2. In the sequel we deal mostly with analytic functions such that kΘ (λ )k ≤ 1 on D: we call them contractive analytic functions. Such a function {A, A∗ , Θ (λ )} is said to be purely contractive if kΘ (0)ak < kak for all

a ∈ A, a 6= 0.

At the other extreme, if Θ (0) is a unitary operator from A onto A∗ , then the maximum principle implies (see the proof of the following result) that Θ (λ ) = Θ (0) for every λ ∈ D. Such a function is called a unitary constant.

Proposition 2.1. For every contractive analytic function {A, A∗, Θ (λ )} there exist uniquely determined decompositions A = A0 ⊕ A′ , A∗ = A0∗ ⊕ A′∗ so that, for every fixed λ , Θ 0 (λ ) = Θ (λ )|A0 has its range in A0∗ and Θ ′ (λ ) = Θ (λ )|A′ has its range in A′∗ , and that {A0 , A0∗ , Θ 0 (λ )} is a purely contractive analytic function, and {A′ , A′∗ , Θ ′ (λ )} is a unitary constant. The function Θ 0 (λ ) is called the “purely contractive part” of Θ (λ ). Proof. Let us set A′ = {a : a ∈ A, a = Θ (0)∗Θ (0)a},

A∗ = {a∗ : a∗ ∈ A∗ , a∗ = Θ (0)Θ (0)∗ a∗ }.

For a ∈ A′ we have Θ (0)a = Θ (0)Θ (0)∗Θ (0)a, and hence Θ (0)a ∈ A′∗ , that is, Θ (0)A′ ⊂ A′∗ . By similar reasoning, Θ (0)∗ A′∗ ⊂ A′ ; by virtue of the relation a∗ = Θ (0)Θ (0)∗ a∗ (a∗ ∈ A′∗ ) this implies A′∗ ⊂ Θ (0)A′ . Thus Θ (0) maps A′ onto A′∗ . This is a unitary operator, because for any a ∈ A′ we have kΘ (0)ak2 = (Θ (0)∗Θ (0)a, a) = (a, a) = kak2. For a ∈ A we have fa (λ ) = Θ (λ )a ∈ H 2 (A∗ ). If in particular a ∈ A′ , then Z

1 2π (Θ (eit )a, Θ (0)a) dt 2π 0 = (Θ (0)a, Θ (0)a) = kak2 = kakkΘ (0)ak  Z 2π 1/2 1 it 2 ≥ kΘ (e )ak dt kΘ (0)ak 2π 0

( fa , Θ (0)a)H 2 (A∗ ) =

= k fa kH 2 (A∗ ) kΘ (0)akH 2 (A∗ ) .

By virtue of the Schwarz inequality we have therefore that fa (eit ) = αΘ (0)a for almost every t, and hence fa (λ ) = αΘ (0)a for every λ ∈ D and some numerical constant α . Obviously α = 1 if a 6= 0 and hence Θ (λ ) = Θ (0)a for every a ∈ A′ . Thus Θ ′ (λ ) = Θ (λ )|A′ is a constant unitary operator from A′ onto A′∗ for λ ∈ D.

2. I NNER AND OUTER FUNCTIONS

195

If we replace in this reasoning Θ (λ ) by Θ ˜(λ ), the spaces A′ and A′∗ interchange their roles and we obtain that the function Θ ˜(λ )|A′∗ is on D a constant unitary operator from A′∗ onto A′ . Consequently, we have for all a ∈ A0 = A ⊖ A′ and for all a∗ ∈ A′∗ : (Θ (λ )a, a∗ ) = (a, Θ (λ )∗ a∗ ) = (a, Θ ˜(λ¯ )a∗ ) = 0

(λ ∈ D);

hence Θ (λ )a ∈ A0∗ = A∗ ⊖ A′∗; that is, Θ 0 (λ ) = Θ (λ )|A0 maps A0 into A0∗ . The analytic function {A0 , A0∗ , Θ 0 (λ )} thus obtained is purely contractive. In fact, if kΘ 0 (0)ak = kak for an a ∈ A0 , then ((I − Θ (0)∗Θ (0))a, a) = kak2 − kΘ (0)ak2 = kak2 − kΘ 0(0)ak2 = 0, and hence (I − Θ (0)∗Θ (0))a = 0, that is, a ∈ A′ . As A′ ⊥ A0 , this implies a = 0. The decompositions A = A0 ⊕ A′ , A∗ = A0∗ ⊕ A′∗ constructed above satisfy therefore the condition stated in the Proposition. It remains to prove uniqueness. To this effect, consider any decompositions A = B0 ⊕ B′ , A∗ = B0∗ ⊕ B′∗ satisfying the same conditions. Because Θ (0) maps B′ unitarily onto B′∗ , we have kak = kΘ (0)ak for a ∈ B′ , and hence (I − Θ (0)∗Θ (0))a = 0 (i.e.,a ∈ A′ ); thus B′ ⊂ A′ . If there existed in A′ an element a 6= 0 orthogonal to B′ , we should have kΘ (0)ak = kak because a ∈ A′ , and kΘ (0)ak < kak because a ∈ B0 ; a contradiction. Thus B′ = A′ , and hence B′∗ = Θ (λ )B′ = Θ (λ )A′ = A′∗ , B0 = A ⊖ B′ = A ⊖ A′ = A0 , B0∗ = A∗ ⊖ B′∗ = A∗ ⊖ A′∗ = A0∗ . This completes the proof. 3. With every bounded analytic function {A, A∗ , Θ (λ )} we associate the operator Θ from L2 (A) into L2 (A∗ ) defined by (Θ v)(t) = Θ (eit )v(t)

for v ∈ L2 (A),

and the operator Θ+ from H 2 (A) into H 2 (A∗ ) defined by (Θ+ u)(λ ) = Θ (λ )u(λ ) for u ∈ H 2 (A). By virtue of the possible identification of H 2 (A) with the subspace L2+ (A) of L2 (A) (and the same for A∗ ) we can consider Θ+ as the restriction of Θ to the subspace H 2 (A) of L2 (A). Obviously, we have kΘ k = ess supkΘ (eit )k = sup kΘ (λ )k 0≤t≤2π

|λ <1

so that if the function Θ (λ ) is contractive then the operators Θ and Θ+ are contractions. Let us also observe that for the adjoint of the operator Θ we have (Θ ∗ v)(t) = Θ (eit )∗ v(t). Proposition 2.2. In order that the operator Θ+ from H 2 (A) into H 2 (A∗ ) be an isometry it is necessary and sufficient that Θ (eit ) be an isometry from A into A∗ for almost every value of t.

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C HAPTER V. O PERATOR -VALUED A NALYTIC F UNCTIONS

Proof. The sufficiency of the condition is obvious. To prove its necessity suppose that Θ+ is an isometry, that is, kΘ+ uk = kuk for all u ∈ H 2 (A). From this we infer that Θ also is an isometry. This follows from the obvious relations kΘ e−int uk = ke−int Θ uk = kΘ+ uk = kuk = ke−int uk for u ∈ H 2 (A) and n ≥ 1, and from the fact that the functions e−int u(eit ) (u ∈ H 2 (A); n ≥ 1) are dense in L2 (A). Thus the relation Z 2π 0

kΘ (eit )v(t)k2A∗ dt =

Z 2π 0

kv(t)k2A dt

holds true for every function v ∈ L2 (A). Let us choose in particular v(t) = ε (τ , δ ;t)a, where a ∈ A and ε (τ , δ ;t) is the characteristic function of the interval (τ , τ + δ ). Thus we obtain, dividing by δ , 1 δ and this implies

Z τ +δ τ

kΘ (eit )ak2A∗ dt = kak2A ,

kΘ (eit )akA∗ = kakA

(2.11)

almost everywhere, the set Ea of the exceptional points t depending on a. Letting a run over a countable set {an } dense in A, and taking the union of the corresponding sets Ea , we obtain a set E of zero measure; if t ∈ / E, equation (2.11) holds simultaneously for every an , and hence for every a ∈ A; thus Θ (eit ) is an isometry. Now we make the following definition. Definitions. The contractive analytic function {A, A∗ , Θ (λ )} is said to be

(i) Inner if Θ (eit ) is an isometry from A into A∗ for almost every t, or equivalently (see Proposition 2.2), if Θ+ is an isometry from H 2 (A) into H 2 (A∗ ). (ii) Outer if Θ+ H 2 (A) = H 2 (A∗ ), the closure being taken in L2 (A∗ ). (iii) ∗-inner if the function {A∗ , A, Θ ˜(λ )} is inner. (iv) ∗-outer if the function {A∗ , A, Θ ˜(λ )} is outer. (v) Inner from both sides if it is both inner and ∗-inner, that is, if Θ (eit ) is unitary a.e. (vi) Outer from both sides if it is both outer and ∗-outer. In the scalar case (i.e., if both A and A∗ are of dimension 1) definitions (i) and (ii) reduce to those given in Sec. III.1.1. Indeed, for a scalar-valued analytic function u(λ ) such that |u(λ )| ≤ 1 on D, (i) means that |u(eit )| = 1 a.e., and (ii) means that 2 the functions {λ k u(λ )}∞ k=0 span the space H , and this property characterizes outer functions by Beurling’s theorem; see Sec. III.1.2. Moreover, it is obvious that in the scalar case every inner function is also ∗-inner, and every outer function is also ∗-outer.

2. I NNER AND OUTER FUNCTIONS

197

Proposition 2.3. The only contractive analytic functions {A, A∗ , Θ (λ )} that are simultaneously inner and outer, are the constant unitary functions, that is, for which Θ (λ ) ≡ Θ0 , where Θ0 is a unitary operator from A to A∗ . Proof. The fact that {A, A∗, Θ (λ )} is simultaneously inner and outer means that Θ+ is a unitary operator from H 2 (A) to H 2 (A∗ ). As Θ+ commutes with multiplication by the variable λ , we have

Θ+ [H 2 (A) ⊖ λ · H 2 (A)] = H 2 (A∗ ) ⊖ λ · H 2 (A∗ ). Now H 2 (A) ⊖ λ · H 2 (A) and H 2 (A∗ ) ⊖ λ · H 2 (A∗ ) consist of the constant functions with values in A and in A∗ , respectively. For every a ∈ A we have therefore Θ (λ )a ≡ a∗ with some a∗ ∈ A∗ , and a∗ runs over A∗ when a runs over A. Thus Θ (λ ) is constant, its value being an operator from A onto A∗ . The limit values of Θ (λ ) on the unit circle have to be isometries, therefore we conclude that the constant value of Θ (λ ) is a unitary operator. This proves one implication in the proposition. The other one is obvious. Proposition 2.4. For every outer function {A, A∗, Θ (λ )} we have (a) Θ (λ )A = A∗ for all λ ∈ D (b) Θ (eit )A = A∗ for almost all t ∈ (0, 2π )

the closure being taken in A∗ .

Proof. Part (a): By virtue of the Cauchy integral formula we have Z

1 2π (Θ (eit )u(eit ), a∗ )A∗ dt 2π 0 1 − λ0e−it = (Θ (λ0 )u(λ0 ), a∗ )A∗

(Θ u, (1 − λ¯ 0λ )−1 a∗ )L2 (A∗ ) =

for u ∈ H 2 (A), a∗ ∈ A∗ , and |λ0 | < 1. Hence, if a∗ is orthogonal to Θ (λ0 )A and a∗ 6= 0, then (1 − λ¯ 0λ )−1 a∗ is a nonzero element of H 2 (A∗ ), orthogonal to Θ H 2 (A). Part (b): Because Θ H 2 (A) is dense in H 2 (A∗ ), it follows in particular that for every constant function a∗ (λ ) ≡ a∗ ∈ A∗ there exists a sequence of elements un ∈ H 2 (A) such that Θ un converges to a∗ in the mean, and also pointwise a.e. So we have Θ (eit )un (eit ) → a∗ (n → ∞)

at every point t with the possible exception of the points t of a set Ea∗ of zero measure. Consequently, for t not in Ea∗ , the closure of Θ (eit )A contains the vector a∗ . Let a∗ run over a countable set {a∗n } dense in A∗ , and let E denote the union of the sets Ea∗n ; E is also of zero measure. For t not in E the closure of Θ (eit )A will contain all the vectors a∗n , hence it will coincide with the whole space A∗ .

As an immediate corollary of Propositions 2.2 and 2.4 we state that if the function {A, A∗ , Θ (λ )} is inner, then dim A ≤ dim A∗ , whereas if it is outer, then dim A ≥ dim A∗ .

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4. The following definition extends a notion already used for inner functions in H ∞. Definition. We say that the functions {A, A∗ , Θ (λ )} and {A′ , A′∗ , Θ ′ (λ )} (λ ∈ D) coincide if there exist a unitary operator τ from A to A′ and a unitary operator τ∗ from A∗ to A′∗ such that Θ ′ (λ ) = τ∗Θ (λ )τ −1 (|λ | < 1).

3 Lemmas on Fourier representations 1. Let U be a bilateral shift on a (complex, separable) Hilbert space R and let A be a generating subspace for U (i.e., such that R = M(A)), where M(A) =

∞ L

−∞

U k A.

We denote by Φ A the unitary transformation from M(A) to L2 (A) defined by     ∞ ∞ ∞ A 2 k ikt ak ∈ A; ∑ kak k < ∞ , Φ ∑ U ak (t) = ∑ e ak (3.1) −∞

−∞

−∞

the series on the right-hand side being convergent in the mean. If we denote by U × the operator of multiplication by eit on the space L2 (A), then we have

Φ AU = U × Φ A ;

(3.2)

that is, Φ A transforms U to U × . We call Φ A the Fourier representation of M(A). Similarly, if U+ is a unilateral shift on a (complex, separable) Hilbert space R+ with the generating subspace A, that is, R+ = M+ (A), where M+ (A) =

∞ L U n A, 0

+

then the Fourier representation of M+ (A), denoted by Φ+A , is the unitary transformation from M+ (A) to H 2 (A) defined by     ∞ ∞ ∞ A k k 2 ak ∈ A; ∑ kak k < ∞; |λ | < 1 . Φ+ ∑ U+ ak (λ ) = ∑ λ ak (3.3) 0

0

0

If U+× denotes multiplication by λ on H 2 (A), we have in analogy to (3.2):

Φ+AU+ = U+× Φ+A .

(3.4)

2. The following lemmas play an important role in the sequel. Lemma 3.1. Let U and U ′ be bilateral shifts on the (complex, separable) Hilbert spaces R and R′ , with the generating subspaces A and A′ , respectively. Let Q be a

3. L EMMAS ON F OURIER REPRESENTATIONS

199

contraction of R into R′ such that

and

QU = U ′ Q

(3.5)

QM+ (A) ⊂ M+ (A′ ).

(3.6)

Then there exists a unique contractive analytic function {A, A′ , Θ (λ )} such that ′

Φ A Q = Θ Φ A.

(3.7)

Proof. By virtue of (3.6) we have in particular for a ∈ A: ∞

Qa = ∑ U ′k a′k

with a′k ∈ A′ ,

0

∞

∑ ka′k k2 = kQak2 ≤ kak2. 0

Setting a′k = Θk a we define a sequence of bounded operators Θk (k = 0, 1, . . .), indeed contractions of A into A′ . So we have ′

Φ A Qa = va ,

where

∞

va (t) = ∑ eikt (Θk a) (convergence in L2 (A′ )). 0

On account of (3.5) and (3.2) this implies ′

′

′

Φ A Qϕ (U)a = Φ A ϕ (U ′ )Qa = ϕ (U × )Φ A Qa = ϕ va for every trigonometric polynomial ϕ (eit ) with scalar coefficients, and hence ′

kϕ (U)ak2R ≥ kQϕ (U)ak2R′ = kΦ A Qϕ (U)ak2L2 (A′ ) =

1 2π

Z 2π 0

|ϕ (eit )|2 kva (t)k2A′ dt.

On the other hand we have kϕ (U)ak2R = kΦ A ϕ (U)ak2L2 (A) = kϕ (U × )Φ A ak2L2 (A) = kϕ ak2L2 (A) 1 = 2π

Z 2π 0

|ϕ (eit )|2 dt · kak2A.

These two results together imply Z 2π 0

|ϕ (eit )|2 kva (t)k2A′ dt ≤

Z 2π 0

|ϕ (eit )|2 dt · kak2A.

This inequality extends by virtue of the Weierstrass approximation theorem to all continuous functions ρ (t) ≥ 0 of period 2π , thus Z 2π 0

ρ (t)kva (t)k2A′ dt ≤

Z 2π 0

ρ (t) dt · kak2A .

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Next, using Lebesgue’s dominated convergence theorem, we extend this inequality to all functions ρ (t) ≥ 0, measurable and bounded on (0, 2π ). Choosing for ρ (t) in particular the characteristic function, divided by δ , of the interval (τ , τ + δ ), and letting δ → 0 we obtain that a.e.

kva (t)kA′ ≤ kakA

(3.8)

By virtue of the Poisson formula (1.6) connecting the function va (t) with the associated function ∞ ua (λ ) = ∑ λ kΘk a (λ ∈ D) (3.9) 0

of class implies

H 2 (A′ ),

and by well-known properties of the kernel Pr (t), inequality (3.8) kua (λ )kA′ ≤ kakA

(λ ∈ D).

(3.10)

As the series (3.9) converges strongly in A′ for each a ∈ A, the operator series ∞

Θ (λ ) = ∑ λ kΘk 0

(λ ∈ D)

also converges and we have Θ (λ )a = ua (λ ); on account of (3.10), {A, A′ , Θ (λ )} is thus a contractive analytic function. Therefore the nontangential strong operator limit Θ (eit ) exists a.e., and we have a.e.:

Θ (eit )a = lim Θ (λ )a = lim ua (λ ) = va (t) (λ → eit nontangentially); thus

′

Φ A Qa = Θ (eit )a (a ∈ A).

(3.11)

From (3.5), (3.2) and (3.11) we deduce ′

′

′

Φ A Q ∑ U k ak = ∑ Φ A U ′k Qak = ∑ eikt Φ A Qak k

=∑ k

Hence it follows

k

k

eikt Θ (eit )a

k

= Θ (eit ) ∑ eikt ak = Θ (eit )Φ A ∑ U k ak . k

k

′

Φ A Qh = Θ Φ A h

first for finite sums h = ∑k U k ak (ak ∈ A), and then by continuity for all h ∈ M(A). This completes the proof of Lemma 3.1. We deduce now an analogous lemma for unilateral shifts along with some results on possible properties of the function Θ (λ ). Lemma 3.2. Let U+ and U+′ be unilateral shifts on the (complex, separable) Hilbert spaces R+ and R′+ , and let A and A′ be the corresponding generating subspaces. Let Q be a contraction of R+ into R′+ such that QU+ = U+′ Q.

(3.12)

3. L EMMAS ON F OURIER REPRESENTATIONS

201

Then there exists a unique contractive analytic function {A, A′ , Θ (λ )} such that ′

Φ+A Q = Θ+ Φ+A .

(3.13)

In order that this function be (a) (b) (c) (d) (e)

Purely contractive Inner Outer A unitary constant Boundedly invertible, with a uniform bound k, that is, with kΘ (λ )−1 k ≤ k

for λ ∈ D

(f) A constant operator it is necessary and sufficient that the following conditions hold, respectively. (a) kPA′ Qak < kak for all nonzero a ∈ A, PA′ denoting the orthogonal projection of R′+ into A′ . (b) Q is an isometry from R+ into R′+ . (c) QR+ = R′+ . (d) Q is unitary from R+ to R′+ . (e) Q is boundedly invertible with kQ−1 k ≤ k. (f) Q∗U+′ = U+ Q∗ . Proof. By virtue of Proposition I.2.2 the unilateral shifts U+ , U+′ can be extended to bilateral shifts U,U ′ on the spaces R, R′ , respectively, with multiplicities preserved: we have ∞ ∞ L L R = U n A, R′ = U ′n A′ . −∞

−∞

If −∞ < p < q < ∞ and an ∈ A, then q

q

q

p

p

p

n−p ∑ U n an = U p ∑ U n−pan = U p ∑ U+ an

and q

q

q

q

p

p

p

p

n−p n−p ∑ U ′n Qan = U ′ p ∑ U ′n−p Qan = U ′ p ∑ U ′ + Qan = U ′ p Q ∑ U+ an ;

hence







q ′n

q

q

q

∑ U Qan = Q ∑ U+n−p an ≤ ∑ U+n−p an = ∑ U n an .

p

p

p

p

This shows that setting

q

q

p

p

b ∑ U n an = ∑ U ′n Qan Q

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b of the linear manifold of all vectors of the form ∑qp U n an , we obtain a contraction Q ′ b extends by continuity to a contraction of R into R′ , which we into R . Now Q b We have denote by the same letter Q. q

q

q

0

0

0

b ∑ U+n an = ∑ U ′n+ Qan = Q ∑ U+n an Q

(q ≥ 0),

b ⊃ Q. Moreover, we have and hence it follows that Q q

q

q

q

p

p

p

p

b ∑ U n+1 an = ∑ U ′n+1 Qan = U ′ Q b ∑ U n an = Q b ∑ U n an , QU

b = U ′ Q. b Because we also have and thus QU

b + (A) = QR+ ⊂ R′+ = M+ (A′ ), QM

b Thus there exists a contractive we can apply Lemma 3.1 to R, R′ , U, U ′ , and Q. ′ analytic function {A, A , Θ (λ )} such that and, taking the restriction to R+ ,

′ b = Θ ΦA ΦA Q ′

Φ+A Q = Θ+ Φ+A . Let us now consider the conditions for the properties (a)–(e). For a ∈ A we have ∞

′

Θ (λ )a = Θ (λ )(Φ+A a)(λ ) = (Φ+A Qa)(λ ) = ∑ λ n a′n 0

(|λ | < 1),

(3.14)

where the coefficients a′n are determined by the expansion ∞

Qa = ∑ U ′n a′n 0

in particular we have

(a′n ∈ A′ );

a′0 = PA′ Qa.

(3.15)

(3.16)

From (3.14), Θ (0)a = a′0 . Thus, in order that Θ (λ ) be purely contractive, it is necessary and sufficient that the inequality ka′0 k < kak hold for every nonzero a ∈ A. By virtue of (3.16) this establishes the case (a). The cases (b) and (c) are immediate consequences of the fact that the operator Θ+ is unitarily equivalent to the operator Q; see (3.13). By virtue of Proposition 2.3, the case (d) follows from (b) and (c). Case (e): Suppose Q is boundedly invertible, with kQ−1 k ≤ k. Then Q1 = (1/k)Q−1 is a contraction of R′+ into R+ ; and (3.12) implies that Q1U+′ = U+ Q1 . Thus there exists, by the first assertion (already proved) of our lemma, a contractive

3. L EMMAS ON F OURIER REPRESENTATIONS

203

analytic function {A′ , A, Ω1 (λ )} such that ′

Φ+A Q1 = Ω1+ Φ+A . Setting Ω (λ ) = kΩ1 (λ ) we obtain a bounded analytic function {A′ , A, Ω (λ )}, kΩ (λ )k ≤ k, such that ′ Φ+A Q−1 = Ω+ Φ+A . (3.17) From (3.17) and (3.13) we deduce that the operators Θ+ and Ω+ are the inverses of each other: Ω+Θ+ = IH 2 (A) , Θ+ Ω+ = IH 2 (A′ ) . This implies

Ω (λ )Θ (λ ) = IA ,

Θ (λ )Ω (λ ) = IA′

(λ ∈ D),

and hence Ω (λ ) = Θ (λ )−1 and kΘ (λ )−1 k ≤ k for λ ∈ D. Conversely, if Θ (λ ) has a uniformly bounded inverse Ω (λ ) = Θ (λ )−1 on D, say kΩ (λ )k ≤ k, then this is a holomorphic function on D, because its derivative exists (in the operator norm topology): d d Ω (λ ) = −Θ (λ )−1 · Θ (λ ) · Θ (λ )−1 . dλ dλ The bounded analytic function {A′ , A, Ω (λ )} induces a bounded operator Ω+ from H 2 (A′ ) into H 2 (A), kΩ+ k ≤ k. Because (Ω+Θ+ v)(λ ) = Ω (λ )(Θ+ v)(λ ) = Ω (λ )Θ (λ )v(λ ) = v(λ ) (v ∈ H 2 (A)) and (Θ+ Ω+ u)(λ ) = Θ (λ )(Ω+ u)(λ ) = Θ (λ )Ω (λ )u(λ ) = u(λ ) (u ∈ H 2 (A′ )), ′

Ω+ is the inverse of Θ+ . The operator (Φ+A )−1 Ω+ Φ+A , which is also bounded by k, is then the inverse of the operator ′

(Φ+A )−1Θ+ Φ+A = Q. Finally, concerning the case (f), if Θ (λ ) = Θ0 is constant, the adjoint of the operator Θ+ is the multiplication by Θ0∗ from H 2 (A′ ) into H 2 (A) and hence it obviously satisfies Θ+∗ U+′× = U+×Θ+∗ , where U+× and U+′× are the multiplications by λ on H 2 (A) and H 2 (A′ ), respectively. Due to the relation (3.2) for U+ and the similar relation for U+′ , we also have Q∗U+′ = U+ Q∗ . Conversely the latter equation implies that Q takes the kernel of U+∗ into that of U+′∗ . Consequently, Θ+ takes the kernel A(⊂ H 2 (A)) of U+×∗ into the kernel A′ (⊂ H 2 (A′ )) of U+′×∗ . This means that Θ (λ ) is a constant function (in λ ). This concludes the proof of Lemma 3.2. 3. As a first application of Lemma 3.2 we determine the invariant subspaces for unilateral shifts of countable multiplicity, on a complex Hilbert space. By virtue of the Fourier representation of such operators (cf. Subsect. 1), our problem reduces to

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finding the invariant subspaces for the operator U+× on the space H 2 (E), E being a complex, separable Hilbert space. Theorem 3.3. For the operator U+× of multiplication by λ on the space H 2 (E), the invariant subspaces are precisely those of the form H = Θ+ H 2 (F)

(3.18)

where {F, E, Θ (λ )}, is an arbitrary inner function.

Proof. If {F, E, Θ (λ )} is an inner function, the corresponding operator Θ+ is isometric, and hence Θ+ H 2 (F) is closed, that is, a subspace of H 2 (E). Its invariance for U+× is obvious. We shall show that every invariant subspace H for U+× has the form (3.18). To this end we first embed E in H 2 (E) as a subspace by identifying the element e ∈ E with the constant function e(λ ) ≡ e; E is then wandering for U+× and H 2 (E) =

∞ L U+×n E 0

(= M+ (E));

see (1.2).

(3.19)

Let V denote the restriction of U+× to the invariant subspace H; this is an isometry on H. We have ∞ ∞ T T V nH ⊂ U+×n H 2 (E) = {0}, n=0

n=0

and thus V has no unitary part so that the corresponding Wold decomposition is of the form ∞ L H = V n F, where F = H ⊖ V H. (3.20) 0

Let us now apply Lemma 3.2 to R+ = H,

and

U+ = V,

A = F;

R′+ = H 2 (E),

U+′ = U+× ,

A′ = E

Q = the identity transformation of H into H 2 (E);

(3.12) is satisfied obviously. Thus there exists an inner function {F, E, Θ (λ )} such that (3.21) Φ+E Q = Θ+ Φ+F (on H). Because E consists of the constant functions in H 2 (E), the Fourier representation of H 2 (E) with respect to U+× is the identity transformation. On the other hand, we have Qh = h for h ∈ H. Thus (3.21) reduces to the relation h = Θ+ Φ+F h and hence we have which completes the proof.

(h ∈ H)

H = Θ+ Φ+F H = Θ+ H 2 (F),

4. FACTORIZATIONS

205

4. Let us state—without proof—that the subspaces H 6= {0} of H 2 (E), which are hyperinvariant for the unilateral shift U+× , are precisely those that can be represented in the form H = u · H 2(E) with a scalar inner function u (i.e., in this case, F = E and Θ (λ ) = u(λ )IE ).

4 Factorizations 1. We begin with the following result. Proposition 4.1. (a) Let {E, F, Θ (λ )} and {E, F1 , Θ1 (λ )} be two contractive analytic functions, the second one being outer, and suppose that

Θ (eit )∗Θ (eit ) ≤ Θ1 (eit )∗Θ1 (eit ) a.e.

(4.1)

Then there exists a contractive analytic function {F1 , F, Θ2 (λ )} such that

Θ (λ ) = Θ2 (λ )Θ1 (λ )

(λ ∈ D).

(4.2)

(b) If in (4.1) the equality sign holds a.e., then Θ2 (λ ) is an inner function. If, moreover, Θ (λ ) is outer, then Θ2 (λ ) is a unitary constant. Proof. Part (a): Inequality (4.1) implies that the operator X defined by X (Θ1 u) = Θ u

(u ∈ H 2 (E))2

(4.3)

is a contraction of Θ1 H 2 (E) into H 2 (F); this extends by continuity to a contraction (also denoted by X) of H 2 (F1 ) = Θ1 H 2 (E) into H 2 (F). We obviously have Θ · λ u = λ · Θ u and Θ1 · λ u = λ · Θ1 u, therefore it follows from the definition (4.3) that × XU1+ = U+× X, × denote the operators of multiplication by the variable λ on where U+× and U1+ 2 2 H (F) and H (F1 ), respectively. Let us apply Lemma 3.2 to the case

R+ = H 2 (F1 ),

× U+ = U1+ ;

R′+ = H 2 (F),

U+′ = U+× ;

Q=X

(F and F1 are supposed to be embedded in H 2 (F) and H 2 (F1 ) in the usual way, as the subspaces of the constant functions). It follows that there exists a contractive analytic function {F1 , F, Θ2 (λ )} such that Xv = Θ2 v

for v ∈ H 2 (F1 )

(4.4)

As the operator Θ+ can be considered as a restriction of the operator Θ , we are allowed to write Θ u instead of Θ+ u if u ∈ H 2 (E), and we do so often in the sequel. But of course we have to distinguish between Θ+∗ u(= (Θ+ )∗ u) and Θ ∗ u even for u ∈ H 2 (F). 2

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(observe that the Fourier representations Φ+F and Φ+F1 of the spaces H 2 (F) and H 2 (F1 ) are the identity transformations). Equations (4.3) and (4.4) imply for v = Θ1 u (u ∈ H 2 (E)): Θ u = Θ2Θ1 u; choosing u(λ ) ≡ h (constant function, h ∈ H), we obtain the relation (4.2). Part (b): When equality holds in (4.1) a.e., then X is an isometry. By virtue of Lemma 3.2, Θ2 (λ ) is then an inner function. If, moreover, Θ (λ ) is an outer function, then H 2 (F) ⊃ Θ2 H 2 (F1 ) ⊃ Θ2Θ1 H 2 (E) = Θ H 2 (E) = H 2 (F),

so that Θ2 (λ ) is also an outer function. By virtue of Proposition 2.3, Θ2 (λ ) is then a unitary constant, and this concludes the proof.

2. Consider now a function N(t) (0 ≤ t ≤ 2π ), whose values are self-adjoint operators on a (separable) Hilbert space E, and which is measurable (strongly or weakly, which amounts to the same thing because E is separable); moreover, assume that O ≤ N(t) ≤ I. (4.5)

The formula (Nv)(t) = N(t)v(t) defines a self-adjoint operator N on L2 (E), bounded by O and I. Proposition 4.2. There exists a contractive outer function {E, F1 , Θ1 (λ )} with the following properties: (i) N(t)2 ≥ Θ1 (eit )∗Θ1 (eit ) a.e. (ii) For every other contractive analytic function {E, F, Θ (λ )} such that

we also have

N(t)2 ≥ Θ (eit )∗Θ (eit ) a.e.

(4.6)

Θ1 (eit )∗Θ1 (eit ) ≥ Θ (eit )∗Θ (eit ) a.e.

(4.7)

Moreover, these properties determine the outer function Θ1 (λ ) up to a constant unitary factor from the left. In order that equality hold in (i) a.e., it is necessary and sufficient that the condition T int e NH 2 (E) = {0}

n≥0

(4.8)

be satisfied. Proof. Let U × denote multiplication by eit on L2 (E). As N commutes with U × , and as H 2 (E) is invariant for U × , the subspace N = NH 2 (E)

(4.9)

of L2 (E) is also invariant for U × . Thus U × induces an isometry on N. Let N = M+ (F1 ) ⊕ N0

(4.10)

4. FACTORIZATIONS

207

be the Wold decomposition of N corresponding to this isometry: F1 = N ⊖ U ×N,

M+ (F1 ) =

L

n≥0

U ×n F1 ,

N0 =

T

n≥0

U ×n N.

(4.11)

Let P denote the orthogonal projection of N onto M+ (F1 ) and N+ denote the multiplication operator from H 2 (E) into L2 (E); because M+ (F1 ) reduces U × |N we have U × P = P(U × |N). Because N commutes with U × we obtain: U × PN+ = P(U × |N)N+ = PU × N+ = PN+U+× . Hence × U1+ X = XU+× ,

where X = PN+ ,

× U1+ = U × |M+ (F1 ), and U+× = U × |H 2 (E).

We can therefore apply Lemma 3.2 to the case R+ = H 2 (E), R′+ Because

= M+ (F1 ),

U+ = U+× , U+′

A = E;

× = U1+ ,

A′ = F1 , and Q = X.

QR+ = XH 2 (E) = PNH 2 (E) = PN = M+ (F1 ) = R′+ ,

we obtain that there exists a contractive outer function {E, F1 , Θ1 (λ )} such that

Φ+F1 Xu = Θ1 u for u ∈ H 2 (E);

(4.12)

here we have also used the fact that Φ+E u = u, because E is the subspace of H 2 (E) consisting of the constant functions. The transformation Φ+F1 is unitary, thus (4.12) implies kNuk ≥ kPNuk = kX uk = kΘ1uk for u ∈ H 2 (E).

(4.13)

Now the elements of the form Nu (u ∈ H 2 (E)) are dense in N (cf. (4.9)), so there is equality in (4.13) for all u if, and only if, P = IN . This condition is equivalent to the condition N0 = {0}, that is, to (4.8). Let us set in (4.13) u(λ ) = p(λ )h, where h ∈ E, and p(λ ) is a (scalar) polynomial of λ . We get: 1 2π

Z 2π 0

|p(eit )|2 kN(t)hk2 dt = kNuk2 ≥ kΘ1 uk2 =

1 2π

Z 2π 0

|p(eit )|2 kΘ1 (eit )hk2 dt.

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Every trigonometric polynomial ϕ (eit ) is of the form e−int p(eit ) for an (algebraic) polynomial p(λ ), therefore we have Z 2π 0

|ϕ (eit )|2 kN(t)hk2 dt ≥

Z 2π 0

|ϕ (eit )|2 kΘ1(eit )hk2 dt

(4.14)

for all trigonometric polynomials ϕ (eit ). Hence we deduce, just as in the proof of Lemma 3.1, that kN(t)hk2 ≥ kΘ1 (eit )hk2 a.e. (4.15)

Because E is separable, the exceptional set of zero measure can be chosen to be independent of h. Thus the inequality (i) holds a.e. Let us observe, moreover, that equality in (4.13) for all u ∈ H 2 (E) implies equality in (4.14) for all ϕ , and hence equality in (4.15) and in (i) a.e. By virtue of what we have already stated concerning equality in (4.13) it follows that the outer function Θ1 (λ ) constructed above satisfies (i) with the equality sign a.e. if, and only if, condition (4.8) holds. Now let {E, F, Θ (λ )} be an arbitrary contractive analytic function, satisfying (4.6). Then there exists a contraction Y of N into H 2 (F) such that Y (Nu) = Θ u

(u ∈ H 2 (E)).

(4.16)

We obviously have Y · eit Nu = Y · Neit u = Θ · eit u = eit · Θ u = eit ·Y Nu (u ∈ H 2 (E)); hence

Y · eit v = eit ·Y v

This implies Y N0 ⊂

T

n≥0

Y · eint N =

(v ∈ N).

T int T int 2 e YN ⊂ e H (F) = {0},

n≥0

n≥0

and hence Y N0 = {0}, Y (IN − P) = O. It follows that Y Nu = Y PNu = Y X u

(u ∈ H 2 (E));

(4.17)

using (4.12), (4.16), and the fact that Φ+F1 is unitary and Y is a contraction, we conclude: kΘ1 uk = kXuk ≥ kY Xuk = kY Nuk = kΘ uk

(u ∈ H 2 (E)).

(4.18)

The method that has led us from (4.13) to (4.15) and to property (i) of Θ1 (λ ), can be applied to (4.18) and leads to (4.7). This argument completes the proof of property (ii) for Θ1 (λ ).

4. FACTORIZATIONS

209

It remains to prove uniqueness. Let {E, F′1 , Θ1′ (λ )} be any contractive outer function with the properties (i) and (ii). Then we have

Θ1 (eit )∗Θ1 (eit ) = Θ1′ (eit )∗Θ1′ (eit ) a.e. By virtue of Proposition 4.1(b) this implies that Θ1′ (λ ) = Z · Θ1 (λ ), with a unitary operator Z from F1 to F′1 . This completes the proof of Proposition 4.2. Remark. From Proposition 4.2 it follows at once that if the function N(t)2 is factorable in the form Θ (eit )∗Θ (eit ) with a contractive analytic function Θ (λ ), outer or not, then condition (4.8) is satisfied. 3. From Propositions 4.1 and 4.2 we can deduce that every contractive analytic function {E, E∗ , Θ (λ )} can be factored into the product of inner and outer factors. In fact, setting N(t) = [Θ (eit )∗Θ (eit )]1/2 , we can apply Proposition 4.2, and obtain an outer function {E, F, Θe (λ )} with properties (i) and (ii). As Θ (eit )∗Θ (eit ) = N(t)2 , these properties imply that N(t)2 ≥ Θe (eit )∗Θe (eit ) ≥ Θ (eit )∗Θ (eit ) = N(t)2

a.e.,

and hence Θe (eit )∗Θe (eit ) = Θ (eit )∗Θ (eit ) a.e. Thus, by virtue of Proposition 4.1, there exists an inner function {F, E∗ , Θi (λ )} such that

Θ (λ ) = Θi (λ )Θe (λ )

(λ ∈ D).

(4.19)

This is called the canonical factorization of Θ (λ ) into the product of its outer factor Θe (λ ) and inner factor Θi (λ ). This factorization is unique in the sense that if Θ (λ ) = Θi′ (λ )Θe′ (λ ) is any factorization with some outer Θe′ (λ ) and inner Θi′ (λ ), and with some intermediary space F′ , then there exists a unitary operator Z from F to F′ such that

Θe′ (λ ) = Z · Θe (λ ) and Θi′ (λ ) = Θi (λ ) · Z −1

(λ ∈ D).

This follows readily from Proposition 4.1(b). In particular, Θe′ (λ ) coincides with Θe (λ ), and Θi′ (λ ) coincides with Θi (λ ), in the sense of Sec. 2.4. If we take the canonical factorization of the adjoint function Θ ˜(λ ) and then return to Θ (λ ), we arrive at a factorization

Θ (λ ) = Θ∗e (λ )Θ∗i (λ ) (λ ∈ D)

(4.20)

with a ∗-outer factor Θ∗e (λ ) and a ∗-inner factor Θ∗i (λ ). This is called the ∗canonical factorization of Θ (λ ); uniqueness in the same sense follows as for the canonical factorization. 4. Let u(λ ) (λ ∈ D) be a scalar-valued contractive analytic function, that is, u ∈ H ∞ , |u(λ )| ≤ 1 on D. If u(λ ) 6≡ 0, then to every Borel subset α of the unit circle

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C there corresponds a factorization u(λ ) = u2α (λ )u1α (λ ) into the product of two scalar-valued contractive analytic functions, such that u1α (λ ) is outer, and a.e.  |u(eit )| for eit ∈ α , |u1α (eit )| = (4.21) 1 for eit ∈ α ′ = C\α , and u2α (eit ) =



1 for eit ∈ α , it |u(e )| for eit ∈ α ′ .

(4.22)

We can simply choose  Z  eit + λ 1 it u1α (λ ) = exp log|u(e )| dt , 2π (α ) eit − λ

(4.23)

where (α ) = {t : 0 ≤ t < 2π , eit ∈ α }. In fact, if u1α (λ ) is the function corresponding to α ′ in the same manner, then u1α ′ (λ )u1α (λ ) is equal to the outer factor of u(λ ), and hence u2α (λ ) = u(λ )/u1α (λ ) (4.24) is equal to the inner factor of u(λ ) multiplied by u1α ′ (λ ); thus u2α (λ ) is a contractive analytic function. Equation (4.21) follows immediately from the definition (4.23), and (4.22) follows from (4.21) and (4.24), because u(eit ) 6= 0 a.e. Applying our preceding results we can generalize these facts to operator-valued analytic functions as follows. Proposition 4.3. Let {E, E∗ , Θ (λ )} be a contractive analytic function such that

Θ (eit )−1

exists a.e. (not necessarily boundedly).

(4.25)

(a) To every Borel subset α of C there corresponds a factorization

Θ (λ ) = Θ2α (λ )Θ1α (λ ) (λ ∈ D)

(4.26)

of Θ (λ ) into the product of an outer function {E, F, Θ1α (λ )} and a contractive analytic function {F, E∗ , Θ2α (λ )}, such that

Θ1α (eit ) is unitary for almost every t ∈ (α ′ ), it

Θ2α (e ) is isometric for almost every t ∈ (α ).

(4.27) (4.28)

4. FACTORIZATIONS

211

(b) The factorization (4.26) is unique in the sense that if {E, F′ , Θ1′ α (λ )} and {F′ , E∗ , Θ2′ α (λ )} satisfy the same conditions, then we have

Θ1′ α (λ ) = Z · Θ1α (λ ) and Θ2′ α (λ ) = Θ2α (λ ) · Z −1 with a constant unitary operator Z from F to F′ . (c) If Θ (eit ) is not a.e. isometric on (α ) as well as on (α ′ ), then the factorization (4.26) is nontrivial (i.e., neither factor is a unitary constant). Remark. Equations (4.26) and (4.28) imply

Θ1α (eit )∗Θ1α (eit ) = Θ (eit )∗Θ (eit ) a.e. on (α ).

(4.27′)

Proof. Part (a): Let us set 1

Nα (t) = [Θ (eit )∗Θ (eit )] 2 for t ∈ (α ) then we have

and Nα (t) = IE for t ∈ (α ′ );

Nα (t)2 ≥ Θ (eit )∗Θ (eit ) a.e.,

and hence

kNα ukL2 (E) ≥ kΘ u||H 2 (E∗ )

(4.29) (4.30)

(u ∈ H 2 (E)).

Thus there exists a contraction Y of Nα H 2 (E) into H 2 (E∗ ) such that Y (Nα u) = Θ u

(u ∈ H 2 (E)).

As the operators Nα and Θ commute with multiplication by the function eit , so does Y , and hence Y

T int T int 2 T int e Θ H 2 (E) ⊂ e H (E∗ ) = {0}. e Nα H 2 (E) ⊂ n≥0

n≥0

n≥0

Thus, in order to prove the validity of condition (4.8) for the function Nα (t), it suffices to show that the only element w of Nα H 2 (E) for which Y w = 0, is w = 0. In other words, we have to show that if a sequence of functions vn (t) ∈ H 2 (E) satisfies the conditions Nα vn → w,

Θ vn → 0 (convergence in the mean),

then w = 0. Replacing the sequence, if necessary, by a subsequence (so that ∑n kNα vn − wk2 and ∑n kΘ vn k2 converge), we have Nα (t)vn (t) → w(t) and Θ (eit )vn (t) → 0 a.e. in the norm of E and E∗ , respectively. Thus we have a.e. in (α ′ )

Θ (eit )w(t) = Θ (eit ) lim Nα (t)vn (t) = Θ (eit ) lim vn (t) = lim Θ (eit )vn (t) = 0 n

n

n

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and a.e. in (α ) kΘ (eit )w(t)k ≤ kw(t)k = lim kNα (t)vn (t)k = lim kΘ (eit )vn (t)k = 0, n

and hence

n

Θ (eit )w(t) = 0 a.e. in (0, 2π ).

By virtue of the hypothesis (4.25) this implies w(t) = 0 a.e., that is, w = 0. Thus Nα (t) satisfies condition (4.8). Hence, by virtue of Proposition 4.2, there exists a contractive outer function {E, F, Θ1α (λ )} such that

Θ1α (eit )∗Θ1α (eit ) = Nα (t)2

a.e.

(4.31)

From (4.30) and (4.31) it follows by Proposition 4.1 that there exists a contractive analytic function {F, E∗ , Θ2α (λ )} such that equation (4.26) holds. The factor Θ1α (λ ) satisfies (4.27) by virtue of (4.29) and (4.31) and because, being outer, it verifies the relation Θ1α (eit )E = F a.e. in (0, 2π ); (4.32) see Proposition 2.4. In order to prove (4.28), we deduce from (4.26), (4.29), and (4.31) that for almost every t ∈ (α ) and every h ∈ E, kΘ2α (eit )Θ1α (eit )hk = kΘ (eit )hk = kNα (t)hk = kΘ1α (eit )hk. Hence it follows by (4.32) that kΘ2α (eit )h∗ k = kh∗ k for every h∗ ∈ E∗ , a.e. in (α ). Part (b): Recalling Remark (4.27′) we see that in the present situation

Θ1′ α (eit )∗Θ1′ α (eit ) = Θ1α (eit )∗Θ1α (eit ) a.e.; thus there exists by virtue of Proposition 4.1(b) a unitary operator Z from F to F′ such that Θ1′ α (λ ) = Z · Θ1α (λ ). So we have

Θ2α (λ )Θ1α (λ ) = Θ (λ ) = Θ2′ α (λ )Θ1′ α (λ ) = Θ2′ α (λ )Z · Θ1α (λ ). The space Θ1α (λ )E is dense in F (cf. Proposition 2.4), therefore

Θ2α (λ ) = Θ2′ α (λ )Z,

Θ2′ α (λ ) = Θ2α (λ )Z −1 .

Part (c): This is obvious. In fact, if Θ1α (λ ) is a unitary constant, (4.26) and (4.28) imply that Θ (eit ) is isometric a.e. in (α ). On the other hand, if Θ2α (λ ) is a unitary constant, (4.26) and (4.27) imply that Θ (eit ) is isometric a.e. in (α ′ ). This concludes the proof of Proposition 4.3.

4. FACTORIZATIONS

213

Remarks. (1) Condition (4.25) is equivalent to the condition

Θ (eit )∗ E∗ = E a.e.

(4.33)

Now by virtue of Proposition 2.4, condition (4.33) holds for every ∗-outer function Θ (λ ). (2) If Θ (λ ) is not an inner function, then Θ (eit )∗Θ (eit ) is different from IE on a set ρ of points eit , of positive measure. Then we can choose α so that both α ∩ ρ and α ′ ∩ ρ are of positive measure. By virtue of Proposition 4.3(c) we can state a corollary. Corollary 4.4. For every contractive analytic function, which is ∗-outer and not inner, there exist nontrivial factorizations of type (4.26). 5. The following example is interesting for several reasons. Let A be a self-adjoint operator on the Hilbert space E, satisfying the inequalities O ≤ A ≤ I and such that 0 and 1 are not eigenvalues of A. Define the function {E, E, Θ (λ )} by setting Θ (λ ) ≡ A. This function is obviously analytic and purely contractive. (Here we use the assumption that 1 is not an eigenvalue of A.) If u is an element of H 2 (E) orthogonal to AH 2 (E), then we have Au(λ ) = 0 for λ ∈ D; because 0 is not an eigenvalue of A this implies that u(λ ) = 0 (i.e., u = 0). Hence AH 2 (E) is dense in H 2 (E): the function Θ (λ ) ≡ A is outer. Moreover, Θ ˜(λ ) ≡ A∗ = A ≡ Θ (λ ), so that the function Θ (λ ) ≡ A is outer from both sides. We now determine the factorization (4.26) of this function for any Borel set α on C. To this end we introduce the function 1 ωα (λ ) = 2π

Z

eis + λ ds is (α ) e − λ

which is holomorphic on D. For λ = reit (0 ≤ r < 1) we can also write 1 ωα (λ ) = 2π

Z

i Pr (t − s) ds + 2π (α )

Z

(α )

Qr (t − s) ds,

(4.34)

where P and Q denote the Poisson kernel and the conjugate Poisson kernel. By well-known properties of integrals with these kernels (cf. Z YGMUND [1] Secs. 3.4 and 7.1), we infer that the nontangential limit of ωα (λ ) exists at a.e. eit ∈ C and equals χα (t) + iχ˜ α (t), where χα (t) denotes the characteristic function of the set (α ), and χ˜ α (t) the (trigonometric) conjugate function. From (4.34) it follows that Re ωα (λ ) ≥ 0 (and even Re ωα (λ ) > 0 unless α is of zero measure). Hence |aωα (λ ) | ≤ 1 for 0 ≤ a ≤ 1 and λ ∈ D. Now we define Ωα (λ ) = Aωα (λ ) (λ ∈ D), (4.35) the operator on the right hand side being understood as the integral of the function aωα (λ ) on the interval 0 ≤ a ≤ 1 with respect to the spectral measure of A. (Note that

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C HAPTER V. O PERATOR -VALUED A NALYTIC F UNCTIONS

the points 0 and 1 have zero spectral measure.) We obtain readily that Ωα (λ ) is a contractive analytic function. From the relations 1 ωα (λ ) + ωα ′ (λ ) = 2π and

Z 2π is e +λ 0

eis − λ

ds = 1

(λ ∈ D)

Re ωα (eit ) = χα (t) (a.e.)

we infer by well-known properties of spectral integrals that

Ωα (λ )Ωα ′ (λ ) = Ωα ′ (λ )Ωα (λ ) = A (λ ∈ D)

(4.36)

and, a.e.,

Ωα (eit )∗ Ωα (eit ) = Ωα (eit )Ωα (eit )∗ = A2χα (t) =



A2 if t ∈ (α ), I if t ∈ (α ′ ).

(4.37)

From (4.36) we deduce that AH 2 (E) = Ωα Ωα ′ H 2 (E) ⊂ Ωα H 2 (E), and this shows that Ωα H 2 (E) is dense in H 2 (E). Because the relation (4.36) implies Ω α˜ (λ )Ω α˜′ (λ ) = A we obtain similarly that Ω α˜ H 2 (E) is also dense in H 2 (E). Thus the function Ωα (λ ) is outer from both sides. Relations (4.36) and (4.37) (when applied for α ′ as well as for α ) prove that for Θ (λ ) ≡ A the factors in (4.26) are given by Θ1α (λ ) = Ωα (λ ) and Θ2α (λ ) = Ωα ′ (λ ). Some further remarks are of interest. From (4.37) we see that Ωα (eit ) is unitary for almost every t ∈ (α ′ ). But if A is not boundedly invertible and α is not of zero measure then at every point z = eit , where Ωα (z) is unitary, Ωα (λ ) does not tend to Ωα (z) in the operator norm as λ → z. Indeed, otherwise Ωα (λ ) would be boundedly invertible for λ close enough to z, and this is impossible because Ωα (λ )∗ Ωα (λ ) = A2 Re ωα (λ ) and Re ωα (λ ) > 0. Hence it follows in particular that if A is not boundedly invertible and α is an arc of positive length of C, then Ωα (λ ) has unitary (nontangential) limit almost everywhere on α ′ , but cannot be extended analytically through α ′ . This remark is useful later in illuminating the role of Proposition 6.7, and we return to it in the Notes to Chap. VII.

5 Nontrivial factorizations As we show in Chap. VII, the problem of finding the nontrivial invariant subspaces for operators on Hilbert space is equivalent to the problem of finding the factorizations of contractive analytic functions into the product of two functions of the same kind, neither of which is a unitary constant (i.e., nontrivial factorizations), and which, moreover, satisfy a certain regularity condition.

5. N ONTRIVIAL FACTORIZATIONS

215

It is therefore natural to ask if every (nonconstant unitary) contractive analytic function can be factored nontrivially, without requiring first any additional property of this factorization. However, if {E, E∗ , Θ (λ )} is an arbitrary contractive analytic function that is not a constant unitary and {E, F, Z} and {E∗ , F∗ , Z∗ } are constant nonunitary isometries then the factorizations Θ (λ ) = (Θ (λ )Z ∗ )Z = Z∗∗ (Z∗Θ (λ )) are obviously nontrivial. Moreover when N(Θ ) = {e ∈ E : Θ (λ )e = 0 for |λ | < 1} is not {0}, then Θ (λ ) has the obviously nontrivial factorization Θ (λ ) = Θ (λ )(I − P + λ P) where P denotes the orthogonal projection of E onto N(Θ ). Also if N(Θ ˜) 6= {0} and P∗ denotes the orthogonal projection of E∗ onto N(Θ ˜), then Θ (λ ) = (I − P∗ + λ P∗ )Θ (λ ) is again a nontrivial factorization. In view of these examples it is reasonable to call strictly nontrivial a factorization Θ (λ ) = Θ2(λ )Θ1 (λ ), where {E, F, Θ1 (λ )} and {F, E∗ , Θ (λ )} are contractive analytic functions, such that N(Θ1 ) = N(Θ ), N(Θ ˜) = N(Θ 2˜ ) and neither Θ1 (λ ) nor Θ2 (λ ) is constant as a function of λ . Note that if Θ (λ ) = Θ2 (λ )Θ1 (λ ) is strictly nontrivial then so is Θ ˜(λ ) = Θ 2˜ (λ )Θ 1˜ (λ ). If {E, E∗ , Θ (λ )} is a constant partial isometry, say W , then Θ (λ ) has no strictly nontrivial factorization. Indeed if W = Θ2 (λ )Θ1 (λ ), with {E, F, Θ1 (λ )} and {F, E∗ , Θ2 (λ )} contractive analytic functions, is a strictly nontrivial factorization then applying Proposition 2.1 to Θ1 (λ ) and using the property that N(Θ1 ) = N(Θ ) equals the kernel of W we obtain that Θ1 (λ ) is a constant partial isometry W1 with kernel equal to that of W . Applying the same argument to Θ ˜(λ ) we infer that Θ2 (λ ) is a partial isometry W2 such that the kernel of W2∗ coincides with that of W ∗ . Thus both Θ1 (λ ) and Θ2 (λ ) are constant (in λ ) and this contradicts our initial assumption. The aim of the present section is to prove that strictly nontrivial factorizations do exist for any contractive analytic function which is not a constant partial isometry. 1. We begin with some geometrical considerations. Let us introduce the following definition. If V is a unilateral shift on the Hilbert space H, let us denote by π (V ) the class of those bounded operators Q on H, with which it is possible to associate a Hilbert space HQ , a unilateral shift VQ on HQ , and a bounded operator A from HQ to H, in such a way that the following conditions hold, VA = AVQ , AA∗ = Q.

(5.1) (5.2)

Proposition 5.1. Let Q be a bounded self-adjoint operator on H, Q ≥ O. In order that Q belong to π (V ), it is necessary and sufficient that Q − V QV ∗ ≥ O.

(5.3)

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Proof. If Q ∈ π (V ) then (5.1) and (5.2) imply Q − V QV ∗ = AA∗ − VAA∗V ∗ = AA∗ − AVQVQ∗ A∗ = A(I − VQVQ∗ )A∗ ≥ O, because I − VQVQ∗ ≥ O. Let us suppose, conversely, that Q satisfies (5.3). Let us denote by R the positive square root of the operator on the left hand side of (5.3). Then Q = R2 + V QV ∗ ; hence it follows by iteration Q = R2 + V R2V ∗ + · · · + V n R2V ∗n + V n+1 QV ∗n+1

(n = 1, 2, . . .).

Because V is a unilateral shift, V ∗n converges strongly to O as n → ∞. Thus we obtain ∞ Q = ∑ V n R2V ∗n n=0

and consequently ∞

kQ1/2 hk2 = ∑ kRV ∗n hk2 n=0

for all

h ∈ H.

(5.4)

Consider now the Hilbert space HQ of the sequences x = {xn }∞ 0 such that xn ∈ RH 2 < ∞. Let V denote the unilateral shift on H (n = 0, 1, . . .) and kxk2 = ∑∞ kx k n Q Q 0 defined by VQ {x0 , x1 , . . .} = {0, x0 , x1 , . . .}.

Let us set, for h ∈ H,

Bh = {Rh, RV ∗ h, . . . , RV ∗n h, . . .}. From (5.4) it follows that B is an operator from H to HQ such that kBhkHQ = kQ1/2 hkH . Moreover, implies

(5.5)

VQ∗ {x0 , x1 , . . .} = {x1 , x2 , . . .}, BV ∗ h = {RV ∗ h, RV ∗2 h, . . .} = VQ∗ Bh,

and thus BV ∗ = VQ∗ B, V B∗ = B∗VQ . The operator A = B∗ therefore satisfies (5.1) and, on account of (5.5), it also satisfies (5.2). Proposition 5.2. Let Q be an operator on H, of class π (V ) and such that O ≤ Q ≤ I. The operator Qα = α Q + (1 − α )I, where 0 < α < 1, is then also of class π (V ). If V is of infinite multiplicity, one can choose HQ = H and VQα = V , thus in this case there exists an operator Aα on H, commuting with V and such that Qα = Aα Aα∗ . Proof. The first assertion follows immediately from Proposition 5.1. As to the second assertion, let us observe first that if A is the generating subspace for V and if PA

5. N ONTRIVIAL FACTORIZATIONS

217

is the orthogonal projection onto A, then Rα2 = Qα − V Qα V ∗ = α (Q − V QV ∗ ) + (1 − α )(I − VV ∗ ) ≥ (1 − α )PA. Hence it follows that Rα h = 0 implies PA h = 0 and consequently Rα H ⊃ PA H = A; thus dim H ≥ dim Rα H ≥ dim A. (5.6) On the other hand, we have

dim H = ℵ0 · dim A = dim A,

(5.7)

because dim A is infinite. Thus (5.6) and (5.7) imply that dim Rα H = dim A. Let ϕ be a unitary operator from Rα H to A. This induces by ∞

Φ x = ∑ V n (ϕ xn ) 0

a unitary operator Φ from the space HQα of the sequences x = {xn }∞ 0 (xn ∈ Rα H) to the space H. We have ∞

∞

1

1

Φ (VQα x) = ∑ V n (ϕ xn−1 ) = V ∑ V n−1 (ϕ xn−1 ) = V Φ x. Thus, if A′α is the operator from HQα to H, associated with Qα according to the proof of Proposition 5.1, then the operator Aα = A′α Φ ∗ commutes with V and Aα A∗α = A′α Φ ∗ Φ A′∗α = A′α A′∗α = Qα . Proposition 5.3. Let A and B be bounded operators on the space H, commuting with an isometry V on H. Let us suppose, moreover, that A is an isometry, B is a contraction, and BB∗ ≥ AA∗ . (5.8)

Then the operator C = B∗ A is an isometry on H, commuting with V and such that A = BC. Furthermore, if B has dense range and A∗ does not commute with V then C∗ does not commute with V . Proof. Relation (5.8) and the fact that B is a contraction imply I − AA∗ ≥ I − BB∗ ≥ O; hence (I − AA∗ )h = 0 (for an h ∈ H) implies (I − BB∗ )h = 0. Because A is an isometry, we have (I − AA∗ )Ag = Ag − AA∗Ag = Ag − Ag = 0 for every g ∈ H, and hence (I − BB∗ )Ag = 0 or Ag = BB∗ Ag. Thus, setting C = B∗ A, we have A = BC. Because A is an isometry and B,C are contractions, C is necessarily an isometry, too. As A and B commute with V , we have V ∗CV = V ∗ B∗ AV = (BV )∗ (AV ) = (V B)∗ (VA) = B∗V ∗VA = B∗ A = C, and hence

(VV ∗ )CV = VC.

(5.9)

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Now C and V being isometries, so are CV and VC too. On the other hand, VV ∗ is an orthogonal projection (namely to V H), thus we conclude from (5.9) that CV = VC. If C∗ commutes with V , then A∗V B = A∗ BV = C∗V = VC∗ = VA∗B, whence A∗V = VA∗ follows under the assumption that B has dense range. 2. Let A be an isometry on H. Suppose that A commutes with a unilateral shift V on H, of infinite multiplicity, and suppose that A∗ does not commute with V , in particular A is not unitary. Set Q = AA∗ and Qα = α Q + (1 − α )I (0 < α < 1). By virtue of Proposition 5.2 there exists an operator Bα on H, commuting with V and such that Bα B∗α = Qα ; since O ≤ Qα ≤ I, Bα is a contraction on H. Because Qα = α Q + (1 − α )I ≥ Q, we have Bα B∗α ≥ AA∗ . Then, by virtue of Proposition 5.3, the operator Cα = Bα∗ A is an isometry, commuting with V and such that A = Bα Cα . We show that neither Bα nor Cα is a coisometry. In fact, equation Bα B∗α = I is impossible since it would imply Qα = I, and thus Q = I, AA∗ = I, i.e., the isometry A would be unitary, which contradicts the hypothesis. Similarly, Cα Cα∗ = I would imply Bα B∗α = Bα Cα Cα∗ B∗ = AA∗ , Qα = Q, and thus Q = I, which is impossible. Let us show that Bα H is dense in H. In the contrary case there would exist an h 6= 0 such that Bα∗ h = 0, Qα h = Bα Bα∗ h = 0. Since Qα ≥ (1 − α )I, we should have (1 − α )h = 0, which implies h = 0, a contradiction. Since A∗ does not commute with V , we infer from Proposition 5.3 that neither does Cα∗ . Assuming that Bα∗ commutes with V , it follows that Qα and so Q also commute with V . Then AA∗V = QV = V Q = VAA∗ = AVA∗ , and since A is an isometry, we obtain that A∗V = VA∗ , which contradicts our assumption. Consequently, B∗α does not commute with V . So we have proved: Proposition 5.4. Every isometry A on H, commuting with a unilateral shift V on H of infinite multiplicity, but not commuting with V ∗ , is the product A = BC of two noncoisometric operators on H, both commuting with V , but neither commuting with V ∗ , C is an isometry and B is a contraction with dense range in H. It should be remarked that the Hilbert spaces considered in this paragraph are not necessarily separable. 3. Let us now consider a separable Hilbert space E of infinite dimension, and a nonconstant inner function {E, E, Θ (λ )}. The corresponding operator Θ+ on H 2 (E) is then isometric, Θ+ commutes with the operator U × of multiplication by λ in H 2 (E), which is a unilateral shift of infinite multiplicity (because dim E is infinite). Moreover, by Lemma 3.2(f) Θ+∗ does not commute with U × . Let us apply Proposition 5.4 to the case A = Θ+ , V = U × . So we obtain that Θ+ equals the product B2 B1 of two noncoisometric operators on H 2 (E), commuting with U × , and such that B1 is isometric and B2 is a contraction with range dense in H 2 (E), and neither of them constant. Applying then Lemma 3.2 we obtain that the operators Bk are generated by nonconstant contractive analytic functions {E, E, Θk (λ )} so that Bk = Θk+ (k = 1, 2), Θ1 (λ ) being inner and Θ2 (λ ) outer. We have proved the following result.

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219

Theorem 5.5. If E is a Hilbert space of dimension ℵ0 , then every nonconstant inner function {E, E, Θ (λ )}, can be factored into the product

Θ (λ ) = Θ2 (λ )Θ1 (λ ) (λ ∈ D)

(5.10)

of two nonconstant contractive analytic functions {E, E, Θk (λ )} (k = 1, 2), Θ1 (λ ) being inner, Θ2 (λ ) outer, and neither one is ∗-inner.

Remark 1. If E is of finite dimension, such a factorization is impossible. In fact, because Θ (λ ) and Θ1 (λ ) are inner functions, Θ (eit ) and Θ1 (eit ) are isometries on E a.e. (cf. Sec. 2.3); now every isometry on E is unitary. It follows that Θ2 (eit ) = Θ (eit ) · Θ1(eit )−1 is also unitary on E, a.e., and hence the operator Θ2+ generated by the function Θ2 (λ ) on H 2 (E) is isometric. Thus Θ2 (λ ) is an inner function, and it cannot be simultaneously a nonconstant unitary, outer function. Thus Theorem 5.5 illustrates the essential difference one encounters in the study of factoring inner functions, when one passes from the case of a finite-dimensional space E to the case of an infinite-dimensional one. Remark 2. We note that Theorem 5.5 is valid also for any nonconstant inner function {E, E∗ , Θ (λ )} such that both E and E∗ are of dimension ℵ0 . In this case, the factorization (5.10) is strictly nontrivial if and only if N(Θ ˜) = {0}. If N(Θ ˜) 6= {0}, then E′∗ = E∗ ⊖ N(Θ ˜) is still of dimension ℵ0 because Θ (eit ) is isometric a.e. Moreover we can consider instead of {E, E∗ , Θ (λ )} the function {E, E′∗ , Θ (λ )}. For this function, Theorem 5.5 provides the strictly nontrivial factorization Θ (λ ) = Θ2 (λ )Θ1 (λ ) where {E, F, Θ1 (λ )} is inner and {F, E′∗ , Θ2 (λ )} is outer. By replacing this last function with {F, E∗ , Θ2 (λ )} we obviously obtain a strictly nontrivial factorization of the initial function {E, E∗ , Θ (λ )}. 4. We are now able to prove the following result. Theorem 5.6. Every contractive analytic function {E, E∗ , Θ (λ )} (with separable E, E∗ ), that is not a constant partial isometry, can be factored into the product of two contractive analytic functions, say {E, F, Θ1 (λ )} and {F, E∗ , Θ2 (λ )} (with separable F), such that the factorization Θ (λ ) = Θ2 (λ )Θ1 (λ ) is strictly nontrivial. Proof. Let Θ (λ ) = Θi (λ )Θe (λ ) = Θ∗e (λ )Θ∗i (λ ) be the canonical and the ∗-canonical factorizations of Θ (λ ); see Sec. 4.3. There are two possibilities: (a) Θ (λ ) has a nonconstant inner or ∗-inner factor, and (b) Θ (λ ) is outer and ∗-outer. In case (b) it follows from Proposition 2.3 and from Corollary 4.4 that Θ (λ ) has a strictly nontrivial factorization. In case (a) it suffices to get a factorization of the inner or of the ∗-inner factor, respectively. Strictly nontrivial factorizations Θ ′ (λ ) = Θ ′′ (λ )Θ ′′′ (λ ) give rise to strictly nontrivial factorizations Θ ′˜(λ ) = Θ ′′′˜(λ )Θ ′′˜(λ ), thus our problem reduces to finding strictly nontrivial factorizations for a nonconstant inner function {E, E∗ , Θ (λ )}. In this case dim E ≤ dim E∗ ≤ ℵ0 (cf. Sec. 2.3). Thus we can suppose that E is a subspace of E∗ , and we can embed E∗ in a space F so that F ⊖ E∗ is also of dimension ℵ0 . Then both F ⊖ E and F ⊖ E∗ have dimension ℵ0 , so there exists a

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partially isometric operator Z on F with initial domain F ⊖ E and range F ⊖ E∗ . Let us set b (λ ) = Θ (λ )PE + Z, Θ (5.11)

where PE = I − Z ∗ Z denotes the orthogonal projection of F onto E. Then we have for f ∈ F b (eit ) f k2 = kΘ (eit )PE f + Z f k2 = kΘ (eit )PE f k2 + kZ f k2 kΘ

= kPE f k2 + kZ f k2 = ((I − Z ∗ Z) f , f ) + (Z ∗ Z f , f ) = k f k2

at every point t where Θ (eit ) is isometric, and hence almost everywhere. Thus b (λ )} is a nonconstant inner function, so we can apply to it Remark 2 fol{F, F, Θ lowing Theorem 5.5 and obtain a strictly nontrivial factorization b (λ ) = Θ b2 (λ ) · Θ b1 (λ ) (λ ∈ D) Θ

(5.12)

Θ (λ ) = Θ2 (λ ) · Θ1(λ ) (λ ∈ D),

(5.13)

bk (λ )} (k = 1, 2). with nonconstant contractive analytic factors {F, F, Θ b1 (λ )|F ⊖ E is a constant isometry W1 In view of Proposition 2.1, the restriction Θ b2 (λ )|F1 is a from F ⊖ E onto a subspace F1 of F. It is clear that the restriction Θ constant isometry W2 from F1 onto the subspace F ⊖ E∗ , and Z = W2W1 . Taking into b1 (λ ) (λ ∈ D) is a contraction, we obtain that Θ b1 (λ ) transforms E into account that Θ b F0 = F ⊖ F1 , and in a similar way Θ2 (λ ) transforms F0 into E∗ Now it is easy to verify that the factorization where the factors are defined by b1 (λ )|E} {E, F0 , Θ1 (λ ) = Θ

b2 (λ )|F0 }, and {F0 , E∗ , Θ2 (λ ) = Θ

is strictly nontrivial. This concludes the proof of Theorem 5.6.

Remark. In a factorization of type (5.13) it is essential to allow intermediary spaces F different from the spaces E, E∗ . For example, the scalar function λ has no nontrivial scalar factorizations, but we have the factorization λ = Θ2 (λ )Θ1 (λ ) with    √  1 λ λ /√ 2 and Θ1 (λ ) = Θ2 (λ ) = √ √ 1/ 2 2 2 (in this case E = E∗ = E 1 and F = E 2 ).

6 Scalar multiples 1. An important case, when the study of the operator-valued function Θ (λ ) can be reduced to the study of a scalar function, is indicated by the following

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Definition. The contractive analytic function {E, E∗ , Θ (λ )} is said to have the scalar multiple δ (λ ), if δ (λ ) is a scalar-valued analytic function, δ (λ ) 6≡ 0, and there exists a contractive analytic function {E∗ , E, Ω (λ )} such that

Ω (λ )Θ (λ ) = δ (λ )IE ,

Θ (λ )Ω (λ ) = δ (λ )IE∗

(λ ∈ D).

(6.1)

A simple class of such functions Θ (λ ) is exhibited by the following Proposition 6.1. If dim E = dim E∗ = n < ∞, then every contractive analytic function {E, E∗ , Θ (λ )} such that Θ (λ ) is invertible for at least one λ in D, has a scalar multiple. In particular, the determinant d(λ ) of the matrix of Θ (λ ) with respect to two orthonormal bases, in E and in E∗ , is a scalar multiple of Θ (λ ), and the matrix of the corresponding function Ω (λ ) is the algebraic adjoint of the matrix of Θ (λ ). Proof. Let {ei } and {ei∗ } (i = 1, . . . , n) be orthonormal bases in E and in E∗ . The corresponding matrix ϑ (λ ) = [ϑi j (λ )] (i, j = 1, . . . , n) of Θ (λ ) is defined by n

Θ (λ )e j = ∑ ϑi j (λ )e∗i i=1

( j = 1, . . . , n)

(i.e., ϑi j (λ ) = (Θ (λ )e j , e∗i ).). Let ω (λ ) = [ωi j (λ )] be the algebraic adjoint of ϑ (λ ), that is, ωi j (λ ) is the determinant, multiplied by (−1)i+ j , of the matrix obtained from the matrix ϑ (λ ) by deleting its ith column and its jth row. By their construction, the scalar-valued functions ϑi j (λ ), ωi j (λ ), and d(λ ) = det ϑ (λ ) are holomorphic and bounded on D; moreover, the matrix ϑ (λ ) is regular for at least one value of λ in D, thus d(λ ) 6≡ 0. The operator-valued function {E∗ , E, Ω (λ )} defined by n

Ω (λ )e∗ j = ∑ ωi j (λ )ei i=1

( j = 1, . . . , n)

is also holomorphic on D, and it follows from the matrix relations ω (λ )ϑ (λ ) = ϑ (λ )ω (λ ) = d(λ )In (In is the identity matrix of order n) that the operator relations (6.1) are satisfied with δ = d. It remains to show that the function Ω (λ ) is contractive. The self-adjoint operator Θ (λ )∗Θ (λ ) has an orthonormal set of eigenvectors f1 (λ ), . . . , fn (λ ) corresponding to the eigenvalues ρ1 (λ ), . . . , ρn (λ ); as Θ (λ ) and therefore Θ (λ )∗Θ (λ ) are contractive functions, we have 1 ≥ ρi ≥ 0 for i = 1, . . . , n. Let us observe next that |d(λ )|2 = d(λ ) · d(λ ) = det ϑ (λ )∗ · det ϑ (λ ) = det ϑ (λ )∗ ϑ (λ ).

Now ϑ (λ )∗ ϑ (λ ) is the matrix of Θ (λ )∗Θ (λ ) with respect to the orthonormal basis {ei }. Passing to the orthonormal basis { fi (λ )} by a unitary transformation U(λ ), the determinant of the matrix is not changed, and hence |d(λ )|2 = ρ1 (λ ) · · · ρn (λ ) ≤ ρ j (λ )

( j = 1, 2, . . . , n).

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Consequently, for every e ∈ E n

n

j=1

j=1

|d(λ )|2 kek2 = |d(λ )|2 ∑ |(e, f j (λ ))|2 ≤ ∑ ρ j (λ )|(e, f j (λ ))|2 = (Θ (λ ) Θ (λ )e, e) = kΘ (λ )ek2 . ∗

On the other hand, (6.1) gives (for δ = d) kΩ (λ )Θ (λ )ek = kd(λ )ek = |d(λ )| kek, and thus we have kΩ (λ )Θ (λ )ek ≤ kΘ (λ )ek. (6.2) For λ such that d(λ ) 6= 0 the range of Θ (λ ) equals E∗ , and then (6.2) yields kΩ (λ )e∗ k ≤ ke∗ k for every

e∗ ∈ E∗ .

(6.3)

The zeros of d(λ ) in D form a discrete set, thus the validity of (6.3) extends by continuity to all λ in D. This concludes the proof of Proposition 6.1. 2. Now we consider a contractive analytic function {E, E∗ , Θ (λ )} with E and E∗ not necessarily finite-dimensional. Theorem 6.2. If Θ (λ ) has a scalar multiple δ (λ ) and if

Θ (λ ) = Θi (λ )Θe (λ ),

δ (λ ) = δi (λ )δe (λ )

are the corresponding canonical factorizations, then δi (λ ) is a scalar multiple of Θi (λ ), and δe (λ ) is a scalar multiple of Θe (λ ). Particular consequences are: (a) If Θ (λ ) is inner or outer, then δi (λ ) or δe (λ ) is also a scalar multiple of Θ (λ ), respectively. (b) If δ (λ ) is inner or outer, then Θ (λ ) is also inner or outer, respectively. (c) If Θ (λ ) is inner or outer, then it is so from both sides. Proof. We begin with the particular case (a), then we prove the general assertion, and finally we deduce the particular cases (b) and (c). Case (a): (1) If Θ (λ ) is inner, then (6.1) implies for almost every t kΩ (eit )e∗ k = kΘ (eit )Ω (eit )e∗ k = kδ (eit )e∗ k = |δe (eit )| ke∗ k (e∗ ∈ E∗ ). (6.4) Let {E∗ , E∗ , Ω1 (λ )} be the contractive analytic function defined by Ω1 (λ ) = δe (λ )IE∗ . By virtue of (6.4) we have

Ω (eit )∗ Ω (eit ) = Ω1 (eit )∗ Ω1 (eit ) a.e. Because Ω1 (λ ) is obviously outer, it follows from Proposition 4.1(b) that there exists an inner function {E∗ , E, Ω2 (λ )} such that

Ω (λ ) = Ω2 (λ )Ω1 (λ ) = δe (λ )Ω2 (λ ) (λ ∈ D).

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Introducing this expression of Ω (λ ) into (6.1) we obtain

δe (λ )[Ω2 (λ )Θ (λ ) − δi (λ )IE ] = O,

δe (λ )[Θ (λ )Ω2 (λ ) − δi (λ )IE∗ ] = O.

As δe (λ ) 6= 0 on D, this implies

Ω2 (λ )Θ (λ ) = δi (λ )IE ,

Θ (λ )Ω2 (λ ) = δi (λ )IE∗ ;

that is, δi (λ ) is a scalar multiple of Θ (λ ). (2) If Θ (λ ) is outer, then (6.1) implies

Ω H 2 (E∗ ) = ΩΘ H 2 (E) = ΩΘ H 2 (E) = δ H 2 (E) = δi H 2 (E) = δi H 2 (E).

(6.5)

Let Ω (λ ) = Ωi (λ )Ωe (λ ) be the canonical factorization of Ω (λ ) with the inner factor {G, E, Ωi (λ )} and the outer factor {E∗ , G, Ωe (λ )}. Then we have

Ω H 2 (E∗ ) = Ωi Ωe H 2 (E∗ ) = Ωi Ωe H 2 (E∗ ) = Ωi H 2 (G) = Ωi H 2 (G). Compared with (6.5) this yields:

Ωi H 2 (G) = δi H 2 (E). Thus to every u ∈ H 2 (G) there corresponds a v ∈ H 2 (E), and conversely, so that

Ωi u = δi v, and we have

kuk = kΩi uk = kδi vk = kvk.

Hence we see that u → v = Qu is a unitary transformation from H 2 (G) to H 2 (E), and Ωi u = δi Qu for u ∈ H 2 (G). (6.6) It is obvious that Q commutes with multiplication by the variable λ : Q · λ u = λ · Qu (u ∈ H 2 (G)). We apply Lemma 3.2(d) to Q, and deduce the existence of a unitary transformation Z from G to E such that (Qu)(λ ) = Z · u(λ )

(λ ∈ D, u ∈ H 2 (G)).

We deduce, using (6.6), that Ωi (λ ) = δi (λ )Z (λ ∈ D). Substituting this value in (6.1) we obtain

δi (λ )(Z Ωe (λ ) · Θ (λ ) − δe(λ )IE ) = O,

δi (λ )(Θ (λ ) · Z Ωe (λ ) − δe (λ )IE∗ ) = O.

The functions appearing in these relations are holomorphic on D, and δi (λ ) has only a discrete set of zeros in D, therefore we conclude that Z Ωe (λ ) · Θ (λ ) = δe (λ )IE ,

Θ (λ ) · Z Ωe (λ ) = δe (λ )IE∗

(λ ∈ D).

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This result shows that δe (λ ) is a scalar multiple of Θ (λ ). The general case: Let F be the intermediate space for the canonical factorization of Θ (λ ) (cf. Sec. 4.3). By virtue of the characteristic property of the outer factors Θe (λ ) and δe (λ ) we have

Θe ΩΘi H 2 (F) = Θe ΩΘiΘe H 2 (E) = Θe ΩΘ H 2 (E) = Θe δ H 2 (E) = δΘe H 2 (E) = δΘe H 2 (E) = δ H 2 (F) = δi δe H 2 (F) = δi H 2 (F) = δi H 2 (F); the last equation follows from the fact that δi (λ ) is an inner function. Hence for every u ∈ H 2 (F) there exists a v ∈ H 2 (F) such that

Θe ΩΘi u = δi v.

(6.7)

Applying Θi to both sides of (6.7) and taking account of (6.1) we obtain

δΘi u = Θi δi v,

Θi (δ u − δi v) = 0.

Because Θi is an isometry we deduce δ u − δiv = 0, and thus (6.7) implies

Θe ΩΘi u = δ u; as u is arbitrary, it follows that

Θe (λ )Ω (λ )Θi (λ ) = δ (λ )IF .

(6.8)

On the other hand, relations (6.1) imply

Θi (λ ) · Θe (λ )Ω (λ ) = Θ (λ )Ω (λ ) = δ (λ )IE∗ and

Ω (λ )Θi (λ ) · Θe (λ ) = Ω (λ )Θ (λ ) = δ (λ )IE . These relations, combined with (6.8), show that δ (λ ) is a scalar multiple of both Θi (λ ) and Θe (λ ). The proof concludes by applying case (a) to Θi (λ ) and to Θe (λ ). Case (b): This follows readily from the general case and from the obvious fact that a contractive analytic function cannot have the scalar multiple 1 unless it is a unitary constant. Case (c): This is an immediate consequence of cases (a) and (b), and of the fact that if Θ (λ ) has the scalar multiple δ (λ ) then Θ ˜(λ ) has the scalar multiple δ ˜(λ ), which is inner or outer when δ (λ ) is inner or outer, respectively. This concludes the proof of the theorem.

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Corollary 6.3. Let {E n , E n , ϑ (λ )} be a matrix-valued contractive analytic function, such that d(λ ) = det ϑ (λ ) 6≡ 0. In order that ϑ (λ ) be inner or outer it is necessary and sufficient that d(λ ) be inner or outer, respectively. Proof. By virtue of Proposition 6.1, the sufficiency of the condition is contained in Theorem 6.2(b). Let us now suppose that ϑ (λ ) is inner or outer. From Theorem 6.2(a) it follows that there exists a contractive analytic (matrix) function {E n , E n , ω (λ )} such that

ω (λ )ϑ (λ ) = ϑ (λ )ω (λ ) = δ (λ )In

(λ ∈ D)

(6.9)

with a scalar function δ (λ ), which is inner or outer, respectively. Denoting the determinant of ω (λ ) by d∗ (λ ) we obtain from (6.9), by taking determinants, d∗ (λ )d(λ ) = δ (λ )n

(λ ∈ D).

Hence d(λ ) is a divisor (in H ∞ ) of δ (λ )n , and this is inner or outer when δ (λ ) is inner or outer, respectively. Thus d(λ ) is of the same type (inner or outer) as δ (λ ). 3. Suppose now that the contractive analytic function {E, E∗ , Θ (λ )} is the product of the contractive analytic functions {E, F, Θ1 (λ )} and {F, E∗ , Θ2 (λ )},

Θ (λ ) = Θ2 (λ )Θ1 (λ )

(λ ∈ D).

(6.10)

It is obvious that if Θ1 (λ ) and Θ2 (λ ) have the scalar multiples δ1 (λ ) and δ2 (λ ), then Θ (λ ) has the scalar multiple δ (λ ) = δ2 (λ )δ1 (λ ). The converse is in general not true: the existence of a scalar multiple of Θ (λ ) does not imply the existence of scalar multiples of Θ1 (λ ) and Θ2 (λ ). For example, the function Θ (λ ) ≡ λ has the trivial scalar multiple δ (λ ) ≡ λ , but the factors in its factorizations considered at the end of Sec. 5 have no scalar multiples at all. However, we can prove a partial converse. Proposition 6.4. If, in (6.10), Θ (λ ) and one of the factors Θ j (λ ) ( j = 1, 2) have scalar multiples then so has the other factor, and in this case every scalar multiple of Θ (λ ) is also a scalar multiple of Θ1 (λ ) and Θ2 (λ ). Proof. Suppose Θ (λ ) and Θ1 (λ ) have the scalar multiple δ (λ ) and δ1 (λ ). Thus (6.1) holds, and

Ω1 (λ )Θ1 (λ ) = δ1 (λ )IE ,

Θ1 (λ )Ω1 (λ ) = δ1 (λ )IF

(6.11)

hold. Multiplying (6.1) from the left by Θ1 (λ ) and from the right by Ω1 (λ ), and taking (6.11) into account, we obtain

δ1 (λ )Θ1 (λ )Ω (λ )Θ2 (λ ) = δ1 (λ )δ (λ )IF

(λ ∈ D).

Because δ1 (λ ) 6≡ 0, the analyticity of the functions under consideration implies that

Θ1 (λ )Ω (λ )Θ2 (λ ) = δ (λ )IF

(λ ∈ D).

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This relation, together with the second relation (6.1), shows that δ (λ ) is a scalar multiple of Θ2 (λ ) too. If it is the factor Θ2 (λ ) which is supposed to have a scalar multiple δ2 (λ ), we obtain by an analogous reasoning that Θ1 (λ ) has the scalar multiple δ (λ ). We conclude that both factors Θ j (λ ) ( j = 1, 2) have the scalar multiple δ (λ ), as asserted. 4. For a matrix-valued contractive analytic function {E n , E n , ϑ (λ )}, the function d(λ ) = det ϑ (λ ) belongs to H ∞ . Moreover, it follows from the relation d(λ )d(λ ) = det ϑ (λ )∗ · det ϑ (λ ) = det[ϑ (λ )∗ ϑ (λ )] that |d(eit )| = 1 at any point eit where ϑ (eit ) is isometric (hence unitary, because the space is finite-dimensional). This fact extends to every contractive analytic function {E, E∗ , Θ (λ )} with a scalar multiple δ (λ ) in the following manner.

Proposition 6.5. Let α be the set of points ζ = eit at which Θ (ζ ) exists and is an isometry (from E to E∗ ). Then (i) Θ (ζ ) is even unitary a.e. in α . (ii) Θ (λ ) has a scalar multiple δ2α (λ ) for which |δ2α (ζ )| = 1 a.e. in α ; such is in particular the function δ2α (λ ) = δ (λ )/δ1α (λ ), where  Z  eit + λ 1 it δ1α (λ ) = exp log|δ (e )| dt (λ ∈ D). 2π (α ) eit − λ (See the remark at the beginning of Sec. 4.4.) Proof. Let {E, F, Θe (λ )} and {F, E∗ , Θi (λ )} be the outer and inner factors of {E, E∗ , Θ (λ )}: Θ (λ ) = Θi (λ )Θe (λ ). By virtue of Theorem 6.2 these factors also have scalar multiples and Θi (ζ ) is unitary a.e. in C. Hence Θe (ζ ) = Θi (ζ )−1Θ (ζ ) is an isometry a.e. in α . Now as Θe (λ ) is an outer function we have

Θe (ζ )E = F a.e. in C. We conclude that Θe (ζ ) is unitary a.e. on α , and therefore so is the product Θi (ζ )Θe (ζ ) = Θ (ζ ). This proves (i). As to (ii), let us observe first that if Ω (λ ) is the function associated with Θ (λ ) and δ (λ ) in the sense of (6.1), then for any fixed e ∈ E, e∗ ∈ E∗ we have |(Ω (ζ )e∗ , e)| ≤ ke∗ kkek a.e. in C, and |(Ω (ζ )e∗ , e)| = |(Θ (ζ )Ω (ζ )e∗ , Θ (ζ )e)| = |δ (ζ )||(e∗ , Θ (ζ )e)| ≤ |δ (ζ )| · ke∗ kkek a.e. in α . Because |δ1α (ζ )| = |δ (ζ )| a.e. in α and |δ1α (ζ )| = 1 a.e. in α ′ = C\α , it follows that |(Ω (ζ )e∗ , e)| ≤ |δ1α (ζ )| · ke∗ kkek a.e. in C. (6.12)

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The functions (Ω (λ )e∗ , e) and δ1α (λ ) are holomorphic on D and δ1α (λ ) is an outer function, therefore (6.12) implies |(Ω (λ )e∗ , e)| ≤ |δ1α (λ )| · ke∗ kkek (λ ∈ D); see H OFFMAN [1] p. 62. Hence

Ω ′ (λ ) = δ1α (λ )−1 · Ω (λ ) is a contractive analytic function on D. From (6.1) it follows, dividing by δ1α (λ ), that δ2α (λ ) = δ (λ )/δ1α (λ ) is a scalar multiple of Θ (λ ). This concludes the proof. 5. We need the following lemma. Lemma 6.6. Let S be a domain in the plane of complex numbers, given by {λ = ρ eit : r0 < ρ < R0 ,t1 < t < t2 } with 0 ≤ r0 < 1 < R0 ≤ ∞, 0 < t2 −t1 ≤ 2π . Let ϕ− (λ ) and ϕ+ (λ ) be functions with values in a Banach space X, defined and holomorphic on S− and S+ , respectively, where S− denotes the part of S interior to the unit circle C, and S+ the part exterior to C. Let us assume that the limits

ψ− (t) = lim ϕ− (reit ), r→1−0

ψ+ (t) = lim ϕ+ (Reit ) R→1+0

(6.13)

exist a.e. in the interval (t1 ,t2 ), and also in the mean L1 , that is, Z t2

t1 Z t2 t1

kϕ− (reit ) − ψ− (t)kX dt → 0

(r0 < r → 1 − 0),

(6.14)

kϕ+ (Reit ) − ψ+ (t)kX dt → 0 (R0 > R → 1 + 0).

Furthermore, let us suppose that

ψ− (t) = ψ+ (t)

(6.15)

a.e. in the interval (t1 ,t2 ). Then the functions ϕ− (λ ) and ϕ+ (λ ) are analytic continuations of each other through the arc α = {eit : t1 < t < t2 } of C.

Proof. Let β = {eit : τ1 ≤ t ≤ τ2 } be a closed sub-arc of α such that the limits (6.13) exist for t = τ1 and for t = τ2 , and are equal: on account of our hypotheses we can choose β as close to α as we wish. Choose r1 and R1 such that r0 < r1 < 1 < R1 < R0 and denote by Γ the contour a1 A1 A2 a2 indicated by the figure, where ak = r1 eiτk and Ak = R1 eiτk (k = 1, 2), and let Σ be the union of Γ and its interior. Let us denote by Σ− and Σ+ the parts of Σ situated in the interior and in the exterior of C, respectively. Let us define the function F(λ ) on Σ \β by setting  ϕ− (λ ) for λ ∈ Σ− , F(λ ) = ϕ+ (λ ) for λ ∈ Σ+ ;

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due to the manner in which we have chosen β , the definition of F(λ ) extends to the points of intersection of Γ and C so that F(λ ) will be a continuous function on the whole contour Γ . Consequently, the integral G(λ ) =

1 2π i

Z

Γ

F(ζ ) dζ ζ −λ

(6.16)

exists and is a holomorphic function of λ on the whole interior of Σ . We show that G(λ ) = ϕ− (λ ) in the interior of Σ− , and G(λ ) = ϕ+ (λ ) in the exterior of Σ+ ; and this concludes the proof. To this end, let us fix a point λ0 in the interior of Σ− , and choose r and R such that |λ0 | < r < 1 < R < R1 . The arcs βr and βR , with radii r and R, cut Σ into three parts whose contours we denote by Γ1 , Γ2 , Γ3 ; see the figure below. The integral (6.16) equals the sum of the integrals along these contours; the first of them is equal to ϕ− (λ0 ), and the third equal to 0, because λ0 is interior to Γ1 and exterior to Γ3 and because ϕ− and ϕ+ are holomorphic on S− and on S+ , respectively. As regards the integral along Γ2 , it tends to 0 as R − r → 0. This is obvious for the integrals along the two segments joining the extremities of βr and βR . On the other hand, it follows from (6.14) and (6.15) that if r → 1 − 0 and R → 1 + 0 then A2

βR C

Γ3 A1

β

Γ2

βr λ0

a2

C

Γ1

•

a1

1 2π

Z

βR

−

Z  βr

F(ζ ) 1 dζ = ζ − λ0 2π →

Z t2  t1

1 2π

 Reit reit it it ϕ+ (Re ) − it ϕ− (re ) dt Reit − λ0 re − λ0

Z t2 t1

eit [ψ+ (t) − ψ−(t)] dt = 0 eit − λ0

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229

(here we used the fact that Reit /(Reit − λ0) and reit /(reit − λ0 ) tend to eit /(eit − λ0 ) uniformly on (τ1 , τ2 )). We obtain that G(λ0 ) = ϕ− (λ0 ). Similar reasoning proves that for λ0 in the interior of Σ+ we have G(λ0 ) = ϕ+ (λ0 ), thus concluding the proof. We apply this lemma to prove the following result. Proposition 6.7. Let {E, E∗ , Θ (λ )} be a contractive outer function with a scalar multiple. Suppose Θ (ζ ) is an isometry (from E into E∗ ) at almost every point ζ = eit of an open arc α of the unit circle C. Then the function Θ (λ ) has an analytic continuation through α to the exterior of C. Proof. By virtue of Theorem 6.2(a), Θ (λ ) also has an outer scalar multiple δ (λ ). On the other hand, by Proposition 6.5, Θ (ζ ) is unitary at almost every point ζ ∈ α , and Θ (λ ) has as a scalar multiple the function δ2α (λ ) = δ (λ )/δ1α (λ ). Note that  Z  eit + λ 1 it δ2α (λ ) = exp log|δ (e )| dt , (6.17) 2π (α ′ ) eit − λ where α ′ = C\α . If Ω (λ ) is the function associated with Θ (λ ) and δ2α (λ ) in the sense of (6.1), we have

Θ (λ )−1 =

1 Ω (λ ), δ2α (λ )

kΘ (λ )−1 k ≤

1 . |δ2α (λ )|

Using (6.17) we conclude that Θ (λ )−1 is defined and holomorphic on the interior D of C, and that it has bounded norm on every subset of D which is at a positive distance from α ′ . This implies that the function

Φ+ (λ ) = [Θ (1/λ¯ )−1 ]∗ is defined and holomorphic on the exterior of C, and is bounded on every subset of this domain which is at positive distance from α ′ . Let α1 be a closed arc in the interior of the arc α and let ∆ be the domain limited by α1 and by an arc β1 having the same endpoints as α1 , but lying otherwise in the exterior of C. Φ+ (λ ) is then holomorphic and bounded on ∆ . Performing a conformal mapping of ∆ onto D and applying Fatou’s theorem in its generalized form (cf. Sec. 2.1) to the transform of the function Φ+ (λ ), and then returning to the initial function, we obtain that Φ+ (λ ) has a nontangential strong limit Φ+ (ζ ) at almost every point ζ ∈ α1 . On the other hand we have as r → 1 − 0,   1 it it ∗ it ∗−1 it ∗ e → Θ (eit )∗ Φ+ (eit ) IE = Θ (re ) Θ (re ) = Θ (re ) Φ+ r strongly (cf. (2.10)) a.e. in α1 ; thus IE = Θ (ζ )∗ Φ+ (ζ )

a.e. in α1 .

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As Θ (ζ ) is unitary a.e. in α , we conclude that

Φ+ (ζ ) = Θ (ζ ) a.e. in α1 . Let us set

Φ− (λ ) = Θ (λ ) for |λ | < 1. Then we can apply Lemma 6.6 to the functions Φ− (λ )e, Φ+ (λ )e and to the arc α1 (for all e ∈ E). In fact, these functions being bounded on some domains adjacent to the arc α1 (and lying in the interior and in the exterior of C, respectively), the pointwise (strong) convergences

Φ− (reit ) → Θ (eit ),

Φ+ (Reit ) → Θ (eit ) (r → 1 − 0, R → 1 + 0),

a.e. on α1 , imply the convergences in the mean of type (6.14). Thus we obtain that, for e ∈ E, the functions Φ− (λ )e = Θ (λ )e (|λ | < 1) and Φ+ (λ )e (|λ | > 1) are analytic continuations of each other along α1 . This implies that the operator-valued function extends analytically across α1 to Φ+ (λ ). As α1 is an arbitrary closed arc in the interior of α , this concludes the proof of Proposition 6.7. Remark. As shown by the example of the functions Ωα (λ ) considered in Sec. 4.5, Proposition 6.7 does not hold in general if we omit the assumption that the function admits a scalar multiple. 6. Here is a further useful remark on functions with scalar multiples. Proposition 6.8. If the contractive analytic function {E, E∗ , Θ (λ )} has a scalar multiple δ (λ ), then its purely contractive component {E0 , E0∗ , Θ 0 (λ )} also has the scalar multiple δ (λ ), and conversely. Proof. Let us set E′ = E ⊖ E0 , E′∗ = E∗ ⊖ E0∗ ; the function Θ (λ )|E′ is then constant, its value being a unitary operator Z (from E′ to E′∗ ). Let us denote by P0 the orthogonal projection of E onto E0 , and by P∗0 the orthogonal projection of E∗ onto E0∗ . Let us assume first that Θ (λ ) has the scalar multiple δ (λ ), thus there exists a contractive analytic function {E∗ , E, Ω (λ )} satisfying

Ω (λ )Θ (λ ) = δ (λ )IE ,

Θ (λ )Ω (λ ) = δ (λ )IE∗ .

(6.18)

We have Θ (λ ) = Θ 0 (λ )P0 + Z(IE − P0 ), therefore the second relation (6.18) yields

Θ 0 (λ )P0 Ω (λ )e0∗ − δ (λ )e0∗ = −Z(IE − P0 )Ω (λ )e0∗

for e0∗ ∈ E0∗ .

For every fixed λ ∈ D, the left-hand side of this relation is an element of E0∗ , and the right-hand side is an element of E′∗ . Therefore both equal 0, and hence we have

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231

Θ 0 (λ )P0 Ω (λ )e0∗ = δ (λ )e0∗ on D. On the other hand, the first relation (6.18) implies P0 Ω (λ )Θ 0 (λ )e0 = P0 Ω (λ )Θ (λ )e0 = P0 δ (λ )e0 = δ (λ )e0

for e0 ∈ E0 .

Setting Ω0 (λ ) = P0 Ω (λ )|E0∗ we obtain therefore a contractive analytic function {E0∗ , E0 , Ω0 (λ )} satisfying the relations

Ω0 (λ )Θ 0 (λ ) = δ (λ )IE0 ,

Θ 0 (λ )Ω0 (λ ) = δ (λ )IE0∗

(λ ∈ D).

(6.19)

We have proved that every scalar multiple δ (λ ) of Θ (λ ) is also a scalar multiple of Θ 0 (λ ). Conversely, if δ (λ ) is a scalar multiple of Θ 0 (λ ) (i.e., if there exists a contractive analytic function {E0∗ , E0 , Ω0 (λ )} satisfying relations (6.19)), then it is easy to verify that the function {E∗ , E, Ω (λ )} defined by

Ω (λ ) = Ω0 (λ )P∗0 + δ (λ )Z ∗ (IE∗ − P∗0) is contractive analytic and satisfies (6.18). This completes the proof.

7 Factorization of functions with scalar multiple 1. For functions with a scalar multiple some of the results of Sec. 4 can be improved. Let N(t) be as in Sec. 4.2, that is, a strongly measurable function whose values are self-adjoint operators on a (separable) Hilbert space E, and such that O ≤ N(t) ≤ I (0 ≤ t ≤ 2π ). Let m(t) denote its lower bound function: m(t) = inf{(N(t)e, e) : e ∈ E, kek = 1}; m(t) is a scalar-valued, nonnegative, measurable function. Proposition 7.1. (a) If there exists a contractive analytic function {E, F, Θ (λ )} with a scalar multiple δ (λ ), satisfying the inequality N(t)2 ≥ Θ (eit )∗Θ (eit ) a.e.,

(7.1)

log m(t) ∈ L1 (0, 2π ).

(7.2)

|δ1 (eit )| = m(t) a.e.

(7.4)

then we have

(b) Conversely, if (7.2) holds, then there exists a contractive analytic function, even an outer one {E, F1 , Θ1 (λ )}, having a scalar multiple δ1 (λ ) and satisfying the equality N(t)2 = Θ1 (eit )∗Θ1 (eit ) a.e.; (7.3) moreover, we can assume that

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(c) If dim E < ∞, condition (7.2) is equivalent to the following one, log det[N(t)] ∈ L1 (0, 2π ),

(7.5)

where [N(t)] denotes the matrix of the operator N(t) with respect to any basis in E. Proof. Part (a): Suppose δ (λ ) is a scalar multiple of Θ (λ ); that is, δ 6≡ 0 and

Ω (λ )Θ (λ ) = δ (λ )IE ,

Θ (λ )Ω (λ ) = δ (λ )IF

(7.6)

for some contractive analytic function {F, E, Ω (λ )}. From (7.1) and (7.6) we deduce for every h ∈ E khk ≥ kN(t)hk ≥ kΘ (eit )hk ≥ kΩ (eit )Θ (eit )hk = |δ (eit )|khk a.e.; hence log|δ (eit )| ∈

1 ≥ m(t) ≥ |δ (eit )| a.e.

(7.7)

L1 (0, 2π )

(cf. Sec. III.1), we have (7.2). Because Part (b): Condition (7.2) implies that there exists a scalar-valued outer function δ1 (λ ) satisfying (7.4) (cf. Sec. III.1); because m(t) ≤ 1, we have |δ1 (λ )| ≤ 1. Observe next that for v ∈ L2 (E), Z

1 2π kN(t)v(t)k2 dt 2π 0 Z Z 1 2π 1 2π m(t)2 kv(t)k2 dt = |δ1 (eit )|2 kv(t)k2 dt = kδ1 vk2 . ≥ 2π 0 2π 0

kNvk2 =

Considering in particular the functions u ∈ H 2 (E) we conclude that there exists a contraction X from N = NH 2 (E) into H 2 (E) for which X(Nu) = δ1 u

(u ∈ H 2 (E)).

Let us show that Xw = 0 (for a w ∈ N) implies w = 0. In fact, for such a w there exists a sequence {un } in H 2 (E) such that Nun → w and δ1 un = X Nun → X w = 0 (convergence in L2 (E)). Hence we obtain

δ1 Nun → δ1 w and N δ1 un → N0 = 0; from δ1 N = N δ1 we infer δ1 w = 0; that is,

δ1 (λ )w(λ ) = 0 (λ ∈ D). Because δ1 (λ ) 6≡ 0, this implies w(λ ) ≡ 0 (λ ∈ D).

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233

The operator X obviously commutes with U × (multiplication by eit in L2 (E)), thus we have X

T

n≥0

U ×n N ⊂

T

n≥0

U ×n XN ⊂

as Xw = 0 implies w = 0 we obtain T

n≥0

T

n≥0

U ×n H 2 (E) = {0};

U ×n N = {0}.

By virtue of Proposition 4.2 there exists therefore a contractive outer function {E, F1 , Θ (λ )} satisfying (7.3). On the other hand, (7.4) implies |δ1 (eit )|IE ≤ N(t). Taking account of (7.3), we obtain |δ1 (eit )|2 IE ≤ Θ1 (eit )∗Θ1 (eit )

a.e.

By Proposition 4.1 (applied to Θ1 (λ ) and to δ1 (λ )IE ) there exists therefore a contractive analytic function {F1 , E, Θ1 (λ )} such that

δ1 (λ )IE = Ω1 (λ )Θ1 (λ ) (λ ∈ D). Multiplying by Θ1 (λ ) from the left yields [δ1 (λ )IF1 − Θ1 (λ )Ω1 (λ )]Θ1 (λ ) = O (λ ∈ D). Because Θ1 (λ ) is outer, we have Θ1 (λ )E = F1 (cf. Proposition 2.4), so we conclude that δ1 (λ )IF1 = Θ1 (λ )Ω1 (λ ) (λ ∈ D). Thus δ1 (λ ) is a scalar multiple of Θ1 (λ ) and the proof of part (b) is complete. Part (c): We simply observe that if ρk (t) (k = 1, . . . , n; n = dim E) are the eigenvalues of N(t) arranged in a nondecreasing order, we have 0 < ρ1 (t) ≤ ρ2 (t) ≤ · · · ≤ ρn (t) ≤ 1 a.e., and therefore  n ≤ ρ1 (t), m(t) = ρ1 (t) and det[N(t)] = ∏ ρk (t) ≥ ρ1 (t)n . k=1 Thus n · logm(t) ≤ log det[N(t)] ≤ log m(t);

hence if one of the functions logm(t) and log det[N(t)] belongs to L1 (0, 2π ) then so does the other. This completes the proof of Proposition 7.1. 2. If {E, E∗ , Θ (λ )} is a contractive analytic function having a scalar multiple δ (λ ), then (6.1) implies

Θ (eit )−1 =

1 Ω (eit ) a.e. δ (eit )

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Thus Proposition 4.3 applies to Θ (λ ): for every Borel subset α of C there exists a factorization Θ (λ ) = Θ2α (λ )Θ1α (λ ) (λ ∈ D)

with the properties (4.27) and (4.28); moreover Θ1α (λ ) is outer. Set  N(t) (t ∈ (α )), it ∗ it 1/2 N(t) = [Θ (e ) Θ (e )] , Nα (t) = IE (t ∈ (α ′ )), and let m(t) and mα (t) be the corresponding lower bounds; clearly  m(t) (t ∈ (α )), mα (t) = 1 (t ∈ (α ′ )).

(7.8)

By Proposition 7.1(a) we have log m(t) ∈ L1 (0, 2π ); on account of (7.8) we have therefore log mα (t) ∈ L1 (0, 2π ) too. Applying Proposition 7.1(b) we obtain that there exists a contractive outer function Θα (λ ) satisfying the equation Nα (t)2 = Θα (eit )∗Θα (eit ) a.e., and admitting as a scalar multiple the outer function δα (λ ) determined by the relation |δα (eit )| = mα (t) a.e., that is,  Z  eit + λ 1 δα (λ ) = exp log m(t) dt . (7.9) 2π (α ) eit − λ Let us remark that as 1 ≥ m(t) ≥ |δ (eit )| a.e. (cf. (7.7)), the outer function  Z  eit + λ 1 it δ1α (λ ) = exp log|δ (e )| dt 2π (α ) eit − λ

(7.10)

is divisible (in H ∞ ) by δα (λ ) and we have |δ1α (λ )/δα (λ )| ≤ 1 on D; hence δ1α (λ ) is also a scalar multiple of Θα (λ ). Now from (4.27) and (4.27′) it follows that Θ1α (eit )∗Θ1α (eit ) = Nα (t)2 also holds a.e.; by Proposition 4.1(b), Θα (λ ) is therefore equal to Θ1α (λ ) up to a constant unitary factor from the left. Consequently, δ1α (λ ) is a scalar multiple of Θ1α (λ ) also. From Proposition 6.4 we get that Θ2α (λ ) also has the scalar multiple δ (λ ). Because Θ2α (eit ) is isometric at almost every point eit of α (cf. (4.28)), it follows from Proposition 6.5 that the function

δ2α (λ ) = δ (λ )/δ1α (λ )

(7.11)

is also a scalar multiple of Θ2α (λ ). Thus, Proposition 4.3 can be completed as follows. Proposition 7.2. Proposition 4.3 applies to every contractive analytic function Θ (λ ) with a scalar multiple δ (λ ). The factors Θ1α (λ ) and Θ2α (λ ) have the functions δ1α (λ ) and δ2α (λ ) defined by (7.10) and (7.11) as scalar multiples, respectively.

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235

8 Analytic kernels Although the results of this section are not used in the sequel, we include them in this book because they are intimately connected with the subject of the preceding sections, in particular with Sec. 5. Given a separable Hilbert space F we call an analytic kernel in F, or in short a kernel, every function K (µ , λ ) defined on D × D, whose values are bounded operators on F and such that, for any fixed µ ∈ D and f ∈ F, K (µ , λ ) f ∈ H 2 (F)

(as a function of λ ).

(8.1)

We say that the kernel K (µ , λ ) is positive definite, and write K (µ , λ ) ≫ O, if n

n

∑ ∑ (K (µ j , µk ) f j , fk ) ≥ 0

j=1 k=1

(8.2)

for every finite system of complex numbers µ j ∈ D and vectors f j ∈ F ( j = 1, . . . , n). For two kernels, the notation K1 ≫ K2 or K2 ≪ K1 indicates that K1 − K2 ≫ O. The particular kernel 1 I ¯ F 1 − µλ is denoted by J (µ , λ ).

Proposition 8.1. (a) For every contractive analytic function {E, F, Θ (λ )} (with separable E, F) HΘ (µ , λ ) = Θ (λ )Θ (µ )∗

¯ )−1Θ (λ )Θ (µ )∗ and KΘ (µ , λ ) = (1 − µλ

are analytic kernels in F, and we have HΘ ≫ O,

J ≫ KΘ ≫ O.

(b) In order that an analytic kernel K (µ , λ ) in F admits a representation of the form KΘ (µ , λ ) with some contractive analytic function Θ (λ ), it is sufficient (and by virtue of (a) also necessary) that it satisfies the conditions ¯ )K (µ , λ ) ≫ O. O ≪ K (µ , λ ) ≪ J (µ , λ ) and (1 − µλ

(8.3)

Proof. Part (a): The assertion concerning HΘ (µ , λ ) is obvious. As to the others, let us observe first that the functions (of λ ) of the form ¯ )−1 f f µ (λ ) = (1 − µλ

(µ ∈ D; f ∈ F)

(8.4)

belong to H 2 (F), and that (u, f µ )H 2 (F) = (u(µ ), f )F

for u ∈ H 2 (F);

(8.5)

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thus we have in particular (gν , f µ )H 2 (F) = (1 − µ ν¯ )−1 (g, f )F .

(8.6)

From (8.5) we also obtain that the elements of the form f µ span H 2 (F). Let us denote by L the set of the finite linear combinations µ

u(λ ) = ∑ fk k (λ ) (with different µk s); k

L is obviously a linear manifold. A function u(λ ) ∈ L extends to a meromorphic µ function in the whole complex plane and is determined by the fk k (i.e., by its poles and residues). This implies that if R is a transformation in H 2 (F) defined for the elements of the form f µ , such that R(c f )µ = c · R f µ

µ

µ

and R( f1 + f2 )µ = R f1 + R f2 ,

then R extends in a unique way to a linear transformation defined on the whole linear manifold L. We conclude in particular that the formula K (µ , λ ) f = (R f µ )(λ )

( f ∈ F; µ , λ ∈ D)

(8.7)

establishes a one-to-one and linear correspondence K ↔ R between the analytic kernels K (µ , λ ) in F and those linear transformations R in H 2 (F) whose domain of definition is L. Relations (8.7) and (8.5) imply (K (µ , ν ) f , g)F = ((R f µ )(ν ), g)F = (R f µ , gν )H 2 (F) ,

(8.8)

and hence it follows readily that if K ↔ R, then the conditions K ≫O

and R ≥ OL

are equivalent.3 If R = IL , (8.7) gives that K = J ; hence J ≫ O. Now let {E, F, Θ (λ )} be a contractive analytic function (with separable E and F). For any u ∈ H 2 (E) and f ∈ F we have Z 2π

1 dt = (Θ (µ )u(µ ), f )F (Θ (eit )u(eit ), f )F 1 − µ e−it 2π Z 2π 1 dt = (u(µ ), Θ (µ )∗ f )F = (u(eit ), Θ (u)∗ f )F −it 1 − µe 2π 0 = (u, (Θ (µ )∗ f )µ ),

(Θ+ u, f µ ) =

0

and hence (Θ+∗ f µ )(λ ) = 3

The subscript indicates restriction to L.

1 Θ (µ )∗ f . ¯ 1 − µλ

9. N OTES

237

From this we deduce (Θ+Θ+∗ f µ )(λ ) =

1 Θ (λ )Θ (µ )∗ f ¯ 1 − µλ

(µ , λ ∈ D),

thus the kernel corresponding to the operator R = Θ+Θ+∗ |L equals KΘ (µ , λ ). As R ≥ OL and IL − R ≥ OL , this kernel satisfies the conclusion in (a). Part (b): Let us observe first that if V denotes multiplication by λ in H 2 (F), then (V ∗ f µ )(λ ) = and consequently

1 µ [ f (λ ) − f µ (0)] = µ¯ f µ (λ ) λ

(V RV ∗ f µ )(λ ) = λ µ¯ (R f µ )(λ )

for every R; hence if K (µ , λ ) ↔ R then ¯ )K (µ , λ ) ↔ R − V RV ∗ . (1 − µλ Thus conditions (8.3) are equivalent to O ≤ R◦ ≤ I and R◦ −V R◦V ∗ ≥ O, where R◦ denotes the closure of R. By virtue of Proposition 5.1 these conditions imply that R◦ = AA∗ , where A is a bounded operator (indeed, a contraction) of a (separable) Hilbert space HR into H 2 (F), such that VA = AVR , and VR is a unilateral shift on HR . Taking the Fourier representation of HR with respect to VR , A is transformed to the operator Θ+ associated with a contractive analytic function {E, F, Θ (λ )} (with E = HR ⊖ VR HR ); see Lemma 3.2. So we have R◦ = Θ+Θ+∗ . Now the kernel corresponding to R = Θ+Θ+∗ |L is KΘ (µ , λ ). We conclude that this kernel equals the given kernel K (µ , λ ). This completes the proof.

9 Notes Operator-valued analytic functions, namely the resolvent of an operator, have long played a fundamental role in the study of operators, in particular concerning spectrum, functional calculus, spectral decompositions of Riesz–Dunford type, and so on. More recently, some other operator-valued analytic functions have gained importance in several domains of functional analysis. Let us mention the theory of characteristic functions inaugurated by L IV Sˇ IC [1], [2], [3], the prediction theory for multivariate stochastic processes (cf. W IENER AND M ASANI [1], [2]; M ASANI [1], [2]; etc.), the description of the invariant subspaces of unilateral shifts of arbitrary multiplicity (cf. L AX [1] and H ALMOS [2]), and finally the harmonic analysis of the unitary dilation of a contraction, which has led the authors of the present book to the functional models of contractions (cf. S Z .-N.–F. [2], [VIII]) and to various other results dealt with in the following chapters. Proposition 2.1, concerning the decomposition of any contractive analytic function into its purely contractive part and its unitary constant part, was first proved by Sˇ TRAUS [1], [2]; S Z .-N.–F. [IX] found it independently and applied it to the study of the invariant subspaces of a contraction.

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The notions of inner and outer functions can be extended in different ways to the operator-valued case (cf., e.g., H ELSON AND L OWDENSLAGER [2]); the definitions in Sec. 2.2 were given by the authors in connection with the asymptotic behavior of the iterates of contractions (cf. S Z .-N.–F. [3], [VIII] and the following chapter). Lemmas 3.1 and 3.2 on Fourier representations were formulated (as a single lemma) in S Z .-N.–F. [IX]; see also [X]. These lemmas determine in particular the general form of the contractions on H 2 (E) that commute with multiplication by the variable λ ; this form has been wellknown for some time. Theorem 3.3 on the invariant subspaces of the unilateral shift is due to B EURL ING [1] (case dim E = 1), L AX [1] (case dim E < ∞), and H ALMOS [2] (general case). The description of the hyperinvariant subspaces for a unilateral shift, stated in Sec. 3.4, appeared first in the original edition of this book (in French). A proof (simpler than the authors have had in mind, and concerning isometries of general type as well) was given later in D OUGLAS [5]. The problem of factorization of the type N(t)2 = Θ (eit )∗Θ (eit ) for an operatorvalued function N(t) ≥ O by means of an operator-valued analytic function Θ (λ ), is a generalization, first considered by S ZEG O˝ [1], of the representation f (t) = |q(eit )|2 of a scalar-valued function f (t) ≥ 0 by means of a scalar-valued analytic function q(λ ). In its turn, Szeg˝o’s result generalizes a lemma of Fej´er and Riesz, which states that if f (t) is a trigonometric polynomial, f (t) ≥ 0, then q(λ ) can be chosen as a polynomial of λ (cf. [Func. Anal.] Sec. 53). It was much later that Z ASUHIN [1] and W IENER [1] observed the importance of this type of factorization of matrix-valued functions for the prediction theory of multivariate stochastic processes. Indeed, one of the problems of this theory can be formulated as follows. Let E be a (separable) Hilbert space and let N(t) be an operator-valued, measurable function on (0, 2π ), O ≤ N(t) ≤ I. Find for every f ∈ E the distance 

1 p( f ) = inf u 2π

Z 2π 0

kN(t)[e

−it

f − u(e

it

)]k2E

dt

1/2

,

where u runs over H 2 (E). If N(t)2 admits of a factorization Θ (eit )∗Θ (eit ) by means of a contractive outer function {E, E∗ , Θ (λ )}, then, using the fact that Θ H 2 (E) = H 2 (E∗ ), we obtain p( f ) = inf v



1 2π

Z 2π 0

kΘ (eit ) f − v(eit )k2E∗ dt

1/2

,

where v runs over the set of functions in H 2 (E∗ ) with v(0) = 0. So we have in this case p( f ) = kΘ (0) f k.

If there is no such factorization for N(t), then one can show that it is the outer function Θ1 (λ ) appearing in Proposition 4.2 which furnishes the solution of the problem.

9. N OTES

239

A general criterion for the existence of a factorization of the form N(t)2 = Θ (eit )∗Θ (eit ) was obtained by L OWDENSLAGER [1] using the Wold decomposition. The method of this author, combined with Lemmas 3.1 and 3.2 on Fourier representation, leads us to our results in Sec. 4; see also S Z .-N.–F. [IX]. This combination of the two methods lends an evident unity to the reasoning in Sec. 4. The criterion of Lowdenslager (restricted to functions N(t) ≤ I) is the one contained in Proposition 4.2, formula (4.8). (See also D OUGLAS [2].) Proposition 4.1 was given in this form in S Z .-N.–F. [IX]. In the case that E and E∗ are of finite dimension, Proposition 4.1(b) can also be found in H ELSON AND L OWDENSLAGER [2] and Sˇ MUL′ JAN [3]. The canonical factorization appeared in S Z .-N.–F. [3] as a consequence of the canonical triangulation of contractions (cf. Sec. II.4); the way it is obtained in Sec. 4.3 was indicated in S Z .-N.–F. [IX]. Proposition 4.3, due to the present authors, establishes the existence of a rich variety of nontrivial factorizations for any contractive analytic function, outer from both sides. It avoids the use of multiplicative integrals. For the use of these integrals see, for example, P OTAPOV [1]; B RODSKI˘I AND L IV Sˇ IC [1]; G INZBURG [1]; M.S. B ORDSKI˘I [6]; V.M. B RODSKI˘I [1]; and V.M. B RODSKI˘I AND M.S. B ROD SKI˘I [1]. Section 5 reproduces the paper S Z .-N.–F. [XII]. The notion of a scalar multiple, which generalizes to some extent that of the determinant, was introduced in S Z .-N.–F. [7]; the detailed exposition in Sec. 6 appeared for the first time in the French edition of this book. The results of this section are used in Chap. VIII. Corollary 6.3 coincides, for finite matrix-valued contractive outer functions, with a characterization given in H ELSON AND L OWDENSLAGER [2] p. 204; see also H ELSON [1] p. 125. Thus, a matrix-valued function which is outer in the sense of these authors, is outer from both sides in the sense adopted in the present book. Proposition 6.7 (on analytic continuation) was given (in the case of finite dimensional E∗ ) in S Z .-N.–F. [IX∗ ]. Part (b) of Proposition 7.1, concerning the sufficiency of the condition log m(t) ∈ L1 (0, 2π ) in order that N 2 (t) admit a factorization Θ (eit )∗Θ (eit ), is due to D EVINATZ [1]. Part (a) shows that this condition is also necessary, if we also require that Θ (λ ) have a scalar multiple. Part (c) was announced first in Z ASUHIN [1], and proved independently by several authors, including W IENER AND M ASANI [1]; W IENER AND A KUTOWICZ [1]; and H ELSON AND L OWDENSLAGER [1]. Proposition 7.2 is new: it is used in Chap. VIII. It also allows us to compare the factorizations obtained in our Proposition 4.3 with those obtained by means of multiplicative integrals. We have restricted our study throughout this chapter to functions N(t), Θ (λ ), and so on, which are bounded (indeed, contractive), the fundamental lemmas of Sec. 3 being established for bounded operators Q only. This restriction is justified by the nature of the problems to which we apply these results in the following chap-

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ters. However, the results obtained can be extended to some unbounded functions, namely to functions N(t) such that N(t) f ∈ L2 (E) for the elements f of a dense linear manifold in E. We do not go into details of these generalizations here; for E = E n see H ELSON [1]. Analytic kernels play an important role in the work announced by DE B RANGES and ROVNYAK [1]. The fact that I − Θ (λ )Θ (µ )∗ ≫O ¯ 1 − µλ (cf. Proposition 8.1(a)) was proved first by R OVNYAK [1]; another proof can be found in S Z .-NAGY [9]. Part (b) of Proposition 8.1 is contained (implicitly) in DE B RANGES AND ROVNYAK [1],[2]. In connection with this chapter see also D OUGLAS AND P EARCY [1]; D OUGLAS [1]; G INZBURG [4],[5]; and the factorization lemmas in S ARASON [3] and S Z .-N– F. [11].

10 Further results 1. Consider a bounded domain Ω in the complex plane, with analytic boundary. For instance, the boundary of Ω could consist of a finite number of circles. As in the case of D, H ∞ (Ω ) consists of the bounded analytic functions defined on Ω , and H p (Ω ) (0 < p < ∞) consists of those analytic functions f in Ω such that | f | p has a harmonic majorant. We refer to F ISHER [1] for the properties of these spaces. These definitions can be extended to spaces of functions with values in a Hilbert space. For applications to operator theory, it is very useful to consider functions f which at a point λ ∈ Ω take values in a Hilbert space Eλ that depends analytically on λ . Formally, E = (Eλ )λ ∈Ω should be a Hermitian analytic vector bundle. When p = 2, one obtains a Hilbert space H 2 (Ω , E) on which one can define the bundle shift SE by setting (SE f )(λ ) = λ f (λ ), f ∈ H 2 (Ω , E), λ ∈ Ω .

Given two analytic vector bundles E, E∗ , one can consider analytic operator-valued functions Θ such that Θ (λ ) ∈ B(Eλ , E∗λ ) for λ ∈ Ω . In this context, one can define the notion of an inner function and prove a complete analogue of the B EURLING , L AX , AND H ALMOS Theorem 3.3, with the operator U+ replaced by SE . We refer to A BRAHAMSE AND D OUGLAS [1] for these developments, and their applications to the theory of subnormal operators. This theory is instrumental in the study of the class C0 associated with Ω ; see Z UCCHI [1]. 2. Given a Hilbert space H, we denote by H⊗n the Hilbert space tensor product of n copies of H (n = 0, 1, . . . ). Thus, if {e j } j∈J is an orthonormal basis for H, then the vectors {e j1 ⊗ e j2 ⊗ · · · ⊗ e jn } j1 , j2 ,..., jn ∈J form an orthonormal basis in H⊗n . The

10. F URTHER RESULTS

241

space H⊗0 is a copy of the complex scalars. The Fock space T (H) =

∞ M

H⊗n

n=0

can be viewed as an analogue of the space H 2 corresponding to k = dim H noncommuting variables. In this context, the analogue of H ∞ is the weakly closed algebra L∞ H generated by the left creation operators on T (H). The left creation Lh operator associated with a vector h ∈ H is defined by Lh v = h ⊗ v,

v ∈ T (H).

This analogy can be carried surprisingly far. See, for instance, P OPESCU [2] for a version of the B EURLING , L AX , AND H ALMOS theorem. In its simplest form it states that the operators commuting with L∞ H are precisely those in the algebra R∞ H generated by the right creation operators. We refer to P OPESCU [8] for further developments related to these ideas. 3. The existence of a bounded analytic left inverse for a bounded analytic function {E, E∗ , Θ (λ )} is closely related to the classical Corona theorem. This appears in S Z .-N.–F. [27] in connection with similarity problems, and in T EODORESCU [1],[2],[4] in connection with the existence of invariant complements to an invariant subspace for a contraction. The Corona theorem in this context was studied by T REIL [1]–[4]. It is shown that, for spaces E of infinite dimension, an extension of the Corona theorem holds only under some restrictions. See also T REIL AND W ICK [1] for related results. 4. TAKAHASHI [1] gives necessary and sufficient conditions for a positive operator A to be of the form A = B∗ B, where B is the operator of multiplication by a bounded analytic function.

Chapter VI

Functional Models 1 Characteristic functions 1. We recall the definition of the defect operators and defect spaces corresponding to a contraction T on the Hilbert space H: DT = (I − T ∗ T )1/2 ,

DT = DT H,

DT ∗ = (I − T T ∗ )1/2 ,

DT ∗ = DT ∗ H.

Denote by ΛT the set of complex numbers λ for which the operator I − λ T ∗ is boundedly invertible.1 For λ ∈ ΛT we define

ΘT (λ ) = [−T + λ DT ∗ (I − λ T ∗ )−1 DT ]|DT .

(1.1)

From well-known properties of invertible operators (cf. [Funct. Anal.] Sec. 147) we infer that ΛT is an open set containing the unit disc D and ΘT (λ ) is an analytic function on ΛT ; as a consequence of the relation T DT = DT ∗ T (cf. (I.3.4)) the values of ΘT (λ ) are (bounded) operators from DT into DT ∗ . Moreover, by virtue of the same relation we obtain

ΘT (λ )DT = DT ∗ [−T + λ (I − λ T ∗ )−1 (I − T ∗ T )]

= DT ∗ (I − λ T ∗ )−1 [−(I − λ T ∗ )T + λ (I − T ∗ T )],

and hence

ΘT (λ )DT = DT ∗ (I − λ T ∗ )−1 (λ I − T )

(λ ∈ ΛT ).

(1.2)

(µ ∈ ΛT ∗ ).

(1.2)∗

Replacing T by T ∗ we have analogously

ΘT ∗ (µ )DT ∗ = DT (I − µ T )−1 (µ I − T ∗ )

For any operator T , the set ΛT consists of the point λ = 0 and the symmetric image of ρ (T )\{0} with respect to the unit circle C, where ρ (T ) denotes the resolvent set for T . The set ΛT ∗ is the symmetric image of ΛT with respect to the real axis. 1

B.Sz.-Nagy et al., Harmonic Analysis of Operators on Hilbert Space, Universitext, DOI 10.1007/978-1-4419-6094-8_6, © Springer Science + Business Media, LLC 2010

243

244

C HAPTER VI. F UNCTIONAL M ODELS

If λ ∈ ΛT and λ −1 ∈ ΛT ∗ (i.e., both λ and λ¯ −1 belong to ΛT ), then equations (1.2) and (1.2)∗ imply

ΘT (λ )ΘT ∗ (λ −1 )DT ∗ = DT ∗

and ΘT ∗ (λ −1 )ΘT (λ )DT = DT ;

that is, we have

ΘT ∗ (λ −1 ) = ΘT (λ )−1

whenever λ , λ¯ −1 ∈ ΛT .

(1.3)

This relation is used later. Now we set, for λ , µ ∈ ΛT , A(λ , µ ) = I − T ∗ T − DT ΘT (µ )∗ΘT (λ )DT and using (1.2) we obtain A(λ , µ ) = I − T ∗ T − (µ¯ I − T ∗ )(I − µ¯ T )−1 (I − T T ∗ )(I − λ T ∗ )−1 (λ I − T ). We observe that the following relations are valid for λ ∈ ΛT : (I − λ T ∗ )−1 (λ I − T ) = −T + λ (I − λ T ∗ )−1 (I − T ∗ T ),

(λ I − T )(I − λ T ∗ )−1 = −T + λ (I − T T ∗ )(I − λ T ∗ )−1 ; to verify them one multiplies by I − λ T ∗ from the left and from the right, respectively. From these relations we deduce, using the identity (I − T T ∗ )T = T (I − T ∗ T ), that (I − T T ∗ )(I − λ T ∗ )−1 (λ I − T ) = (λ I − T )(I − λ T ∗ )−1 (I − T ∗ T ). It follows that A(λ , µ ) = [I − (µ¯ I − T ∗ )(I − µ¯ T )−1 (λ I − T )(I − λ T ∗ )−1 ](I − T ∗ T )

= [(I − λ T ∗ )(I − µ¯ T ) − (µ¯ I − T ∗ )(λ I − T )](I − µ¯ T )−1 (I − λ T ∗ )−1 (I − T ∗ T ) = (1 − λ µ¯ )(I − T ∗ T )(I − µ¯ T )−1 (I − λ T ∗ )−1 (I − T ∗ T ). Thus we have for any h ∈ H: kDT hk2 − (ΘT (λ )DT h, ΘT (µ )DT h) = (A(λ , µ )h, h) = (1 − λ µ¯ )((I − λ T ∗ )−1 D2T h, (I − µ T ∗ )−1 D2T h),

and consequently, for f = DT h, ( f , f ) − (ΘT (λ ) f , ΘT (µ ) f ) = (1 − λ µ¯ )((I − λ T ∗ )−1 DT f , (I − µ T ∗ )−1 DT f );

1. C HARACTERISTIC FUNCTIONS

245

thisrelation extends by continuity to every element f of DT = DT H. When µ = λ we thus obtain k f k2 − kΘT (λ ) f k2 = (1 − |λ |2 )k(I − λ T ∗ )−1 DT f k2

( f ∈ DT , λ ∈ ΛT ). (1.4)

As an immediate consequence of (1.4) we have k f k2 − kΘT (λ ) f k2 ≥ 0 for

and λ ∈ D.

f ∈ DT

For λ = 0, (1.4) takes the form k f k2 − kΘT (0) f k2 = kDT f k2

( f ∈ DT ).

Observe that DT f = 0 implies that f is orthogonal to every element of the form DT h (h ∈ H) and hence orthogonal to DT ; if f ∈ DT this is impossible unless f = 0. Thus we have kΘT (0) f k < k f k

for every

f 6= 0.

f ∈ DT ,

We have thereby proved that, when considered on the unit disc D, ΘT (λ ) is a purely contractive analytic function (cf. Sec. V.2). Definition. The purely contractive analytic function {DT , DT ∗ , ΘT (λ )} on D is called the characteristic function of the contraction T . From (1.1) it follows that i h ∞ ΘT (λ ) = − T + ∑ λ n DT ∗ T ∗n−1 DT DT n=1

for λ ∈ D,

(1.1)′

the expansion being convergent in the norm. Applying (1.1) to T ∗ as well as to T we obtain

ΘT ∗ (λ ) = ΘT (λ¯ )∗ and thus in particular

for λ ∈ ΛT ∗ ,

ΘT ∗ (λ ) = Θ T˜ (λ ) on D.

(1.5) (1.6)′

If λ is a point on the unit circle C belonging to the resolvent set ρ (T ), then λ = λ¯ −1 ∈ ΛT so that we can apply (1.3). Using (1.5) we also obtain

ΘT (λ )−1 = ΘT ∗ (λ −1 ) = ΘT ∗ (λ¯ ) = ΘT (λ )∗ ; that is, ΘT (λ ) is a unitary operator. Hence, if α is an arc of C belonging to the resolvent set of T then the function ΘT (λ ) is analytic on α and its values on α are unitary operators (from DT to DT ∗ ). 2. Let us now consider two contractions, T1 on H1 and T2 on H2 , and suppose they are unitarily equivalent: T2 = σ T1 σ −1 , where σ is a unitary operator from H1

246

C HAPTER VI. F UNCTIONAL M ODELS

to H2 . From the definitions it follows readily that DT2 = τ DT1 ,

DT2∗ = τ∗ DT1∗ ,

ΘT2 (λ ) = τ∗ΘT1 (λ )τ −1 ,

where τ and τ∗ denote the restrictions of σ to DT1 and DT1∗ , respectively. Making use of the notion of coincidence introduced in Sec. V.2.4, we have therefore that the characteristic functions of unitarily equivalent contractions coincide. The converse is not true, in this generality. Indeed, if H = H0 ⊕ H1 is the decomposition of H corresponding to the unitary part T0 and the c.n.u part T1 of T , then we have DT = O ⊕ DT1 ,

DT ∗ = O ⊕ DT1∗ ,

DT = DT1 ,

DT ∗ = DT1∗ ,

and hence ΘT (λ ) = ΘT1 (λ ). Therefore it suffices to consider c.n.u contractions. We show in Sec. 3 that for such contractions the above proposition has a complete converse. 3. If T is a contraction then so is Ta = (T − aI)(I − aT ¯ )−1 for every complex a with |a| < 1; see Sec. I.4.4. For the characteristic functions of T and Ta we have that {DTa , DTa∗ , ΘTa (λ )} coincides with

 λ + a o n . DT , DT ∗ , ΘT 1 + a¯λ

(1.6)

In fact, we obtain by elementary calculations that I − Ta∗ Ta = S∗ (I − T ∗ T )S where and hence

and I − Ta Ta∗ = S(I − T T ∗ )S∗ ,

S = (1 − |a|2)1/2 (I − aT ¯ )−1 , kDTa hk2 = kDT Shk2,

kDTa∗ hk2 = kDT ∗ S∗ hk2

(1.7)

for h ∈ H. Because S and S∗ map H onto itself, relations (1.7) show that there exist a unitary operator Z from DTa to DT , and a unitary operator Z∗ from DTa∗ to DT ∗ , such that ZDTa = DT S and Z∗ DTa∗ = DT ∗ S∗ . (1.8) Using (1.2) and (1.8) we obtain Z∗ΘTa (λ )Z −1 DT = Z∗ΘTa (λ )DTa S−1 = Z∗ DTa∗ (I − λ Ta∗ )−1 (λ I − Ta )S−1 = DT ∗ S∗ (I − λ Ta∗ )−1 (λ I − Ta )S−1

¯ ) = DT ∗ (I − aT ∗ )−1 (I − λ Ta∗ )−1 (λ I − Ta )(I − aT

= DT ∗ (I − µ T ∗ )−1 (µ I − T ),

2. F UNCTIONAL MODELS FOR A GIVEN CONTRACTION

where

µ= Thus

247

λ +a . 1 + a¯λ

Z∗ΘTa (λ )Z −1 DT = ΘT (µ )DT ;

the operators Z −1 and ΘT (µ ) have the same domain DT , hence Z∗ΘTa (λ )Z −1 = ΘT (µ ), and (1.6) follows. L

4. Let H = α Hα and, correspondingly, T = on the space Hα . Clearly, DT = and hence

L α

DTα ,

DT ∗ =

L α

ΘT (λ ) =

DTα∗ ,

L α

DT =

ΘTα (λ )

L

α Tα ,

L α

where Tα is a contraction

DTα ,

DT ∗ =

L α

DTα∗ ,

(λ ∈ D).

As every contraction on a nonseparable Hilbert space is the orthogonal sum of contractions on separable Hilbert spaces, it suffices to restrict our further investigations on characteristic functions to the case of separable spaces. From now on we only consider separable Hilbert spaces. Then the defect spaces DT and DT ∗ are separable too, and as a consequence the results of Chap. V on contractive analytic functions can be applied to the characteristic functions. Therefore the strong operator limit

ΘT (eit ) = lim ΘT (λ ) (λ ∈ D, λ → eit nontangentially to C) exists at almost every point eit of the unit circle C; moreover, for every fixed f ∈ DT and for r → 1 − 0 the function ΘT (reit ) f of t converges, as an element of L2 (DT ∗ ), to the function ∞ ΘT (eit ) f = −T f + ∑ eint DT ∗ T ∗n−1 DT f ; (1.9) n=1

the infinite sum is also to be understood in the sense of convergence in the mean, that is, in the space L2 (DT ∗ ) (cf. Secs. V.1 and 2).

2 Functional models for a given contraction 1. We have defined in Sec. 1 the characteristic function of a contraction T on the space H without giving a motivation for this definition. We now show that this definition arises in a natural way in the context of our theory of unitary dilations. As the space H is separable, so is the space K of the minimal unitary dilation U of T . Let K+ be the space of the minimal isometric dilation U+ of T : we always view K+ as a subspace of K and U+ as the restriction of U to K+ .

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C HAPTER VI. F UNCTIONAL M ODELS

By virtue of Theorem II.2.1, K and K+ admit the following decompositions K = M(L∗ ) ⊕ R,

K+ = M+ (L∗ ) ⊕ R = H ⊕ M+ (L),

(2.1)

where L = (U − T )H,

L∗ = (I − UT ∗ )H

(2.2)

are subspaces of K+ wandering for U+ (and hence for U), and where R reduces U and U+ to their residual part R, which is unitary. The subspace M(L∗ ) reduces U, and consequently PL∗ (the orthogonal projection of K onto M(L∗ )) commutes with U and, invoking the same theorem once more, PL∗ M+ (L) ⊂ M+ (L∗ ). Thus we can apply Lemma V.3.1 to the bilateral shifts induced by U on M(L) and M(L∗ ), and to the contraction Q = PL∗ |M(L) of M(L) into M(L∗ ). We obtain that there exists a contractive analytic function {L, L∗ , ΘL (λ )} such that

Φ L∗ P L∗ f = Θ L Φ L f

for

f ∈ M(L),

(2.3)

where Φ L and Φ L∗ denote the Fourier representations of M(L) and M(L∗ ) on the functional spaces L2 (L) and L2 (L∗ ); see Sec. V.3.1. Because dimL = dT and dim L∗ = dT ∗ , L and L∗ do not both equal {0} unless T is unitary, a case which we exclude in the sequel. We also make the additional assumption that T is completely nonunitary on H (6= {0}). Then M(L) ∨ M(L∗ ) = K,

(2.4)

and consequently, (I − PL∗ )M(L) = R Set

(cf. Theorem II.2.1).

∆L (t) = [IL − ΘL(eit )∗ΘL (eit )]1/2

(2.5)

for those t at which ΘL (eit ) exists, thus a.e. For t fixed, ∆L (t) is a self-adjoint operator on L, bounded by 0 and 1. As a function of t, ∆L (t) is strongly measurable, and generates by (∆L v)(t) = ∆L (t)v(t) (v ∈ L2 (L)) a self-adjoint operator ∆L on L2 (L), also bounded by 0 and 1. For f ∈ M(L) we have

2. F UNCTIONAL MODELS FOR A GIVEN CONTRACTION

249

k(I − PL∗ ) f k2 = k f k2 − kPL∗ f k2 = kΦ L f k2 − kΦ L∗ PL∗ f k2 = kΦ L f k2 − kΘLΦ L f k2

=

=

1 2π

Z2π

1 2π

Z2π

0

0

[k(Φ L f )(t)k2L − kΘL(eit ) · (Φ L f )(t)k2L∗ ] dt k∆L (t)(Φ L f )(t)k2L dt = k∆L Φ L f k2 .

Using (2.5) we deduce from this that there exists a unitary operator

ΦR : R → ∆L L2 (L), such that Consequently,

ΦR (I − PL∗ ) f = ∆L Φ L f

for

(2.6)

f ∈ M(L).

Φ = Φ L∗ ⊕ ΦR

(2.7)

is a unitary operator from the space

K = M(L∗ ) ⊕ R to the functional space

K = L2 (L∗ ) ⊕ ∆L L2 (L).

Because U commutes with (V.3.2)), we obtain from (2.6):

P L∗

and because of the relation

(2.8)

Φ LU

=

eit Φ L

(cf.

ΦRU(I − PL∗ ) f = ΦR (I − PL∗ )U f = ∆L Φ LU f = ∆L · eit Φ L f = eit · ∆L Φ L f = eit · ΦR (I − PL∗ ) f

for f ∈ M(L). All the operators occurring are continuous, and therefore we have ΦRUg = eit · ΦR g for every g ∈ R. On the other hand, Φ L∗ Uh = eit · Φ L∗ h for h ∈ M(L∗ ), thus Φ U = UΦ , (2.9) where U denotes the unitary operator on K defined by U(v∗ ⊕ v) = eit v∗ (t) ⊕ eit v(t) (v∗ ∈ L2 (L∗ ), v ∈ ∆L L2 (L)).

(2.10)

According to our convention of identifying the spaces of type L2+ (A) with the corresponding spaces H 2 (A) of analytic functions u(λ ) on D, it follows from the

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C HAPTER VI. F UNCTIONAL M ODELS

first of the decompositions (2.1) of K+ that Φ maps K+ onto the space K+ = H 2 (L∗ ) ⊕ ∆L L2 (L).

(2.11)

The operator U+ is thereby represented by the operator U+ on K+ defined by U+ (u∗ ⊕ v) = eit u∗ (eit ) ⊕ eit v(t) (u∗ ∈ H 2 (L∗ ),

v ∈ ∆L L2 (L)).

(2.12)

The adjoint U+∗ will be represented by U∗+ , which is given by the formula U∗+ (u∗ ⊕ v) = e−it [u∗ (eit ) − u∗(0)] ⊕ e−it v(t).

(2.13)

Let us find the image of the space H by the representation Φ . Because H = K+ ⊖ M+ (L) (cf. (2.1)), we have Φ H = K+ ⊖ Φ M+ (L). Now, for g ∈ M+ (L) we have

Φ g = Φ [PL∗ g + (I − PL∗ )g] = Φ L∗ PL∗ g ⊕ ΦR (I − PL∗ )g = ΘL Φ L g ⊕ ∆LΦ L g, and hence

Φ M+ (L) = {ΘL u ⊕ ∆Lu : u ∈ H 2 (L)}.

Thus Φ H = H, where

H = [H 2 (L∗ ) ⊕ ∆ L2 (L)] ⊖ {ΘLu ⊕ ∆Lu : u ∈ H 2 (L)}.

(2.14)

Returning to the contraction T , recall that it is connected with its isometric dilation U+ by the relation T ∗ = U+∗ |H; see (I.4.2). It follows that the transform of T by Φ , which we denote by T, is connected with U+ by the relation T∗ = U∗+ |H.

(2.15)

We study in more detail the contractive analytic function {L, L∗ , ΘL (λ )}, using Lemma V.3.2 in the case R+ = M+ (L),

U+ = U|R+ ;

R′+ = M+ (L∗ ),

U+′ = U|R′+ ;

Q = PL∗ |R+ .

We show first that ΘL (λ ) is purely contractive. Indeed, if PL∗ denotes orthogonal projection onto L∗ , we have kPL∗ PL∗ lk < klk for every l ∈ L, l 6= 0. Otherwise there would exist an l ∈ L, l 6= 0, such that l = PL∗ PL∗ l ( i.e. l ∈ L∗ ), and this contradicts the relation L ∩ L∗ = {0} proved in Sec. II.2. By virtue of point (b) of the same lemma, ΘL (λ ) is an inner function if and only if the operator PL∗ |M+ (L) is isometric. Because PL∗ is the orthogonal projection onto M(L∗ ), this condition means that M+ (L) ⊂ M(L∗ ), or (which amounts to the same thing because M(L∗ ) reduces U) that M(L) ⊂ M(L∗ ). By (2.4) this is equivalent to the condition M(L∗ ) = K, and hence also to the condition T ∗n → O (n → ∞); see Theorem II.1.2. We conclude that ΘL (λ ) is inner if and only if T ∈ C·0 . We have proved the following result.

2. F UNCTIONAL MODELS FOR A GIVEN CONTRACTION

251

Proposition 2.1. Let T be a completely nonunitary contraction on the space H. Let U be the minimal unitary dilation of T on the space K and let L and L∗ be the wandering subspaces for U defined by (2.2). Then there exists a purely contractive analytic function {L, L∗ , ΘL (λ )}, satisfying condition (2.3). This function generates by (2.7) a unitary transformation Φ from K to the functional space K = L2 (L∗ ) ⊕ ∆LL2 (L),

where

∆L (t) = [I − ΘL (eit )∗ΘL (eit )]1/2 .

By means of this transformation, called the “Fourier representation” of K, U is represented by the operator U of multiplication by the function eit in the space K, and the subspace K+ of K is represented by the subspace K+ = H 2 (L∗ ) ⊕ ∆LL2 (L) of K. Finally, the space H and the contraction T are represented by the subspace H of K and the operator T on H, defined by H = [H 2 (L∗ ) ⊕ ∆L L2 (L)] ⊖ {ΘLu ⊕ ∆Lu : u ∈ H 2 (L)} and

T∗ (u∗ ⊕ v) = e−it [u∗ (eit ) − u∗(0)] ⊕ e−it v(t) (u∗ ⊕ v ∈ H).

If the function ΘL (λ ) is inner (i.e. if ∆L (t) = O a.e.), the above formulas for K, H, and T simplify to K = L2 (L∗ ), H = H 2 (L∗ ) ⊖ ΘLH 2 (L), 1 (T∗ u∗ )(λ ) = [u∗ (λ ) − u∗(0)] (u∗ ∈ H). λ This is the case if and only if T ∈ C·0 .

2. Next we to establish a connection between the function ΘL (λ ) and the characteristic function of T . Proposition 2.2. The function ΘL (λ ) occurring in Proposition 2.1 coincides with the characteristic function ΘT (λ ) of T . Proof. Let us begin by observing that there is a unitary operator ϕ from L to DT and a unitary operator ϕ∗ from L∗ to DT ∗ such that

ϕ (U − T )h = DT h and ϕ∗ (I − UT ∗ )h = DT ∗ h (h ∈ H).

(2.16)

As to ϕ , this has been proved in Sec. II.1 (cf. (II.1.6)); for ϕ∗ the proof is analogous. We shall prove that

ϕ∗Θ2 (λ )ϕ −1 = ΘT (λ )

(λ ∈ D),

so that ΘL (λ ) and ΘT (λ ) coincide (in the sense defined in Sec. V.2.4).

(2.17)

252

C HAPTER VI. F UNCTIONAL M ODELS

n If ΘL (λ ) = ∑∞ 0 λ Θn (λ ∈ D) is the power series expansion of ΘL (λ ), with bounded operators Θn from L into L∗ as coefficients, then we have

(Θn l, l∗ )L∗ =

1 2π

Z 2π 0

e−int (ΘL (eit )l, l∗ )L∗ dt = (ΘL Φ L l, eint Φ L∗ l∗ )L2 (L∗ )

for l ∈ L, l∗ ∈ L∗ . Now relation (2.3) defining ΘL (λ ) implies that ΘL Φ L l = Φ L∗ PL∗ l; on the other hand we have eint · Φ L∗ l∗ = Φ L∗ U n l∗ from (V.3.2). As Φ L∗ is unitary, it follows that (Θn l, l∗ )L∗ = (PL∗ l,U n l∗ )K , and as

PL∗ U n l∗ = U n PL∗ l∗ = U n l∗

we conclude that (Θn l, l∗ )L∗ = (U ∗n l, l∗ )K

(l ∈ L; l∗ ∈ L∗ ; n = 0, 1, . . .).

Thus, denoting by PL∗ the orthogonal projection of K onto L∗ , we have

Θn = PL∗ U ∗n |L

(n = 0, 1, . . .).

(2.18)

We now show by a straightforward calculation that for l = (U − T )h (h ∈ H) we have Θn l = ln , with l0 = −(I − UT ∗ )T h and ln = (I − UT ∗ )hn , hn = T ∗n−1 (I − T ∗ T )h

(n ≥ 1).

These elements ln (n ≥ 0) obviously belong to L∗ . Therefore, by (2.18) we only have to show that U ∗n l − ln ⊥ L∗ (n ≥ 0). (2.19) For n = 0 this is immediate. Indeed,

l − l0 = U(I − T ∗ T )h ∈ UH and UH ⊥ UL∗ = L∗ . For n ≥ 1, (2.19) follows from the fact that, for every h′ ∈ H, (U ∗n l − ln , (I − UT ∗ )h′ ) = ((U ∗n−1 − U ∗nT )h − (I − UT ∗ )hn , (I − UT ∗ )h′ ) =((U ∗n−1 − U ∗nT )h − (I − UT ∗ )hn , h′ )

− ((U ∗n − U ∗n+1T )h − (U ∗ − T ∗ )hn , T ∗ h′ )

=((T ∗n−1 − T ∗n T )h − (I − T T ∗ )hn , h′ )

− ((T ∗n − T ∗n+1 T )h − (T ∗ − T ∗ )hn , T ∗ h′ )

=((I − T T ∗ )T ∗n−1 (I − T ∗ T )h, h′ ) − ((I − T T ∗ )hn , h′ ) = 0.

2. F UNCTIONAL MODELS FOR A GIVEN CONTRACTION

253

Comparing these results with relations (2.16) we see that  (n = 0), −DT ∗ T h = −T DT h ϕ∗Θn ϕ −1 DT h = ϕ∗Θn l = ϕ∗ ln = DT ∗ hn = DT ∗ T ∗n−1 DT DT h (n ≥ 1). The elements DT h (h ∈ H) being dense in DT , we conclude that  (n = 0), −T |DT ϕ∗Θn ϕ −1 = DT ∗ T ∗n−1 DT |DT (n ≥ 1). Comparing these formulas with (1.1)′ we obtain (2.17). 3. The above results allow us to construct functional models for c.n.u. contractions, in which the characteristic functions occur explicitly. In fact, (2.17) implies ΘT (λ )∗ΘT (λ ) = ϕΘL (λ )∗ΘL (λ )ϕ −1 , and thus, setting

∆T (t) = [I − ΘT (eit )∗ΘT (eit )]1/2 , we have

ϕ∆L (t)ϕ −1 = ∆T (t).

(2.20)

In this manner the unitary transformations

ϕ : L → DT , generate, by

ϕ∗ : L∗ → DT ∗

ϕb : u(eit ) ⊕ v(t) → ϕ∗ u(eit ) ⊕ ϕ v(t),

a unitary transformation

ϕb : H 2 (L∗ ) ⊕ ∆L L2 (L) → H 2 (DT ∗ ) ⊕ ∆T L2 (DT ),

commuting with multiplication by eit and such that

ϕb(ΘL u ⊕ ∆Lu : u ∈ H 2 (L)} = {ϕ∗ΘL u ⊕ ϕ∆Lu : u ∈ H 2 (L)} = {ΘT ϕ u ⊕ ∆T ϕ u : u ∈ H 2 (L)}

= {ΘT w ⊕ ∆T w : w ∈ H 2 (DT )}.

We proved the following result. Theorem 2.3. Every c.n.u. contraction T on the (separable) Hilbert space H (6= {0}) is unitarily equivalent to the operator T on the functional space H = [H 2 (DT ∗ ) ⊕ ∆T L2 (DT )] ⊖ {ΘT u ⊕ ∆T u : u ∈ H 2 (DT )} defined by T∗ (u ⊕ v) = e−it [u(eit ) − u(0)] ⊕ e−it v(t) (u ⊕ v ∈ H).

254

C HAPTER VI. F UNCTIONAL M ODELS

If T ∈ C·0 , and only in this case, ΘT (λ ) is inner, and then this model of T reduces to H = H 2 (DT ∗ ) ⊖ ΘT H 2 (DT ),

T∗ u(λ ) =

1 [u(λ ) − u(0)] (u ∈ H). λ

Interchanging the roles of T and T ∗ we obtain the following dual model. Theorem 2.3*. Under the conditions of Theorem 2.3, T is unitarily equivalent to the operator T′ on the functional space H′ = [H 2 (DT ) ⊕ ∆T ∗ L2 (DT ∗ )] ⊖ {ΘT ∗ u ⊕ ∆T ∗ u : u ∈ H 2 (DT ∗ )} defined by T′ (u ⊕ v) = e−it [u(eit ) − u(0)] ⊕ e−it v(t) (u ⊕ v ∈ H′ ). If T ∈ C0· , and only in this case, ΘT (λ ) is ∗-inner, and then this model of T reduces to H′ = H 2 (DT ) ⊖ ΘT ∗ H 2 (DT ∗ ),

T′ u(λ ) =

1 [u(λ ) − u(0)] (u ∈ H′ ). λ

3 Functional models for analytic functions 1. The above theorems raise the problem of whether every contractive analytic function {E, E∗ , Θ (λ )} generates, by analogous constructions, some c.n.u contractions T and T′ . As T and T′ interchange roles if one passes to the function {E∗ , E, Θ ˜(λ )}, it suffices to consider the case of T. Suppose we are given a contractive analytic function {E, E∗ , Θ (λ )} with ∞

Θ (λ ) = ∑ λ nΘn , 0

and set K = L2 (E∗ ) ⊕ ∆ L2 (E),

2

K+ = H 2 (E∗ ) ⊕ ∆ L2 (E) (⊂ K),

G = {Θ w ⊕ ∆ w : w ∈ H (E)}

(⊂ K+ ),

(3.1) (3.2)

where ∆ (t) = [IE − Θ (eit )∗Θ (eit )]1/2 . By virtue of the relation Z

1 2π ([Θ (eit )∗Θ (eit ) + ∆ (t)2 ]v(t), v(t))E dt 2π 0 Z 1 2π (v(t), v(t))E dt = kvk2L2 (E) (v ∈ L2 (E)), = 2π 0

kΘ vk2L2 (E∗ ) + k∆ vk2L2 (E) =

Ω : v → Θ v ⊕ ∆ v (v ∈ L2 (E))

(3.3)

3. F UNCTIONAL MODELS FOR ANALYTIC FUNCTIONS

255

is an isometry from L2 (E) into K. As G is the image under Ω of the subspace H 2 (E) of L2 (E), G is also a subspace of K+ . Finally, set H = K+ ⊖ G. (3.4) Denote by U the multiplication by eit on K ( i.e. simultaneous multiplication by on the two component spaces). Clearly, U is a unitary operator on K, and K+ is invariant for U; let us set U+ = U|K+ . eit

G is also invariant for U+ , and consequently H is invariant for U∗+ . One shows easily that U∗+ (u ⊕ v) = e−it [u(eit ) − u(0)] ⊕ e−it v(t) (u ⊕ v ∈ K+ ).

The operator U+ is isometric, thus U∗+ is a contraction on K+ , and so is the operator T on H defined by T∗ = U∗+ |H. (3.5)

Denote by P the orthogonal projection from K onto H, and by P+ the orthogonal projection from K+ onto H; then P+ = P|K+ . From (3.5) it follows that T∗n = U∗n + |H,

(3.6)

and for h, h′ ∈ H, ′ n ′ n ′ (Tn h, h′ )H = (h, T∗n h′ )H = (h, U∗n + h )K+ = (U+ h, h )K+ = (P+ U+ h, h )H ;

hence

Tn = P+ Un+ |H = PUn |H

(n ≥ 0);

that is,U is a unitary dilation of T.

(3.7)

2. Next we show that T is completely nonunitary. To this end we consider an element u ⊕ v ∈ H such that for n ≥ 0 (a)

kTn (u ⊕ v)k = ku ⊕ vk,

(b) kT∗n (u ⊕ v)k = ku ⊕ vk.

As an element of H 2 (E∗ ), u has an expansion ∞

u(λ ) = ∑ λ k ak 0

  ∞ ak ∈ E∗ , ∑ kak k2 < ∞ . 0

By virtue of the relation

2

∞

kT∗n (u ⊕ v)k2 = ∑ ei(k−n)t ak 2 ∞

k=n

L (E∗ )

+ kvk2L2(E)

= ∑ kak k2E∗ + kv||2L2(E) → kvk2L2 (E) n

(3.8) (n → ∞),

256

C HAPTER VI. F UNCTIONAL M ODELS

assumptions (b) imply u = 0. On the other hand it follows from (3.7) that assumptions (a) mean that Un+ (u ⊕ v) is contained in H (n ≥ 0). Because u = 0, we have Un+ (u ⊕ v) = 0 ⊕ eint v, and this has to be orthogonal to G for n ≥ 0; that is, 0 = (0 ⊕ eint v, Θ w ⊕ ∆ w) =

1 2π

Z 2π 0

eint (v(t), ∆ (t)w(eit ))E dt

for every w ∈ H 2 (E), in particular for w = eimt f ( f ∈ E; m ≥ 0); thus Z 2π 0

ei(n−m)t (∆ (t)v(t), f )E dt = 0

(n, m ≥ 0).

This implies (∆ (t)v(t), f )E = 0 a.e and, as E is separable, ∆ (t)v(t) = 0 a.e.; thus ∆ v = 0, v ⊥ ∆ L2 (E). On the other hand, v ∈ ∆ L2 (E), therefore v = 0. This shows that (a) and (b) imply u ⊕ v = 0; thus T is c.n.u. 3. We assume from now on that Θ (λ ) is purely contractive, that is, kΘ (0) f k < k f k

for

f ∈ E, f 6= 0,

(3.9)

and we show that the characteristic function of T then coincides with Θ (λ ). To this end we show first that U is the minimal unitary dilation of T, that is, K=

∞ W

Un H.

(3.10)

−∞

From the definition of K and K+ it follows immediately that K = to establish (3.10) it suffices to show that K+ =

W0

−∞ U

nK

∞ ∞ W W Un+ H = Un H. 0

+ ; hence

(3.11)

0

Suppose u ⊕ v is an element of K+ orthogonal to Un+ H (n = 0, 1, . . .), thus belongs to G for n = 0, 1, . . .:

U∗n + (u ⊕ v)

(n) ⊕ ∆ w(n) U∗n + (u ⊕ v) = Θ w

(w(n) ∈ H 2 (E); n = 0, 1, . . .).

The recursive relation U∗+ (Θ w(n) ⊕ ∆ w(n) ) = Θ w(n+1) ⊕ ∆ w(n+1)

(n ≥ 0)

gives e−it [Θ w(n) − (Θ w(n) )(0)] ⊕ e−it ∆ w(n) = Θ w(n+1) ⊕ ∆ w(n+1) thus

Θ ω (n) = Θ (0)w(n) (0),

∆ ω (n) = 0 (n ≥ 0),

(n ≥ 0); (3.12)

3. F UNCTIONAL MODELS FOR ANALYTIC FUNCTIONS

with

257

ω (n) (λ ) = w(n) (λ ) − λ w(n+1)(λ ) ∈ H 2 (E).

Now relations (3.12) imply

∞

ω (n) = Θ ∗Θ ω (n) = Θ ∗Θ (0)w(n) (0) = ∑ e−ikt Θk∗Θ0 w(n) (0). k=0

Because ω (n) ∈ H 2 (E), this is possible only if

ω (n) (λ ) = Θ0∗Θ0 w(n) (0)

(3.13)

for every λ , in particular for λ = 0. Because ω (n) (0) = w(n) (0) we obtain that w(n) (0) = Θ0∗Θ0 w(n) (0),

kw(n) (0)k = kΘ0 w(n) (0)k.

As the function Θ (λ ) is purely contractive, this implies w(n) (0) = 0, and on account of (3.13) also ω (n) (λ ) ≡ 0; hence w(n) (λ ) = λ · w(n+1) (λ ). This being true for n ≥ 0, we obtain w(0) (λ ) = λ n · w(n) (λ ) (n ≥ 0). Hence w(0) (λ )/λ n belongs to H 2 (E) for every n ≥ 0, which is impossible unless w(0) (λ ) ≡ 0. So we have u ⊕ v = Θ w(0) ⊕ ∆ w(0) = 0. This proves (3.11) and hence (3.10) also; that is the unitary dilation U of T is minimal. Using this fact we could continue our study of T by using Theorem II.2.1. However, we prefer to follow a more explicit analytic approach. 4. Our next step is to describe L∗ = (I − UT∗ )H, where I denotes the identity operator on K. It is obvious that for u ⊕ v ∈ H we have (I − UT∗ )(u ⊕ v) = [u − (u − u(0))] ⊕ [v − v] = u(0) ⊕ 0,

(3.14)

where u(0) is considered as a constant function in L2 (E∗ ). Let us choose in particular u( ¯ λ ) = (I − Θ (λ )Θ0∗)g,

v(t) ¯ = −∆ (t)Θ0∗ g

with g ∈ E∗ . It is obvious that u¯ ⊕ v¯ ∈ K+ ; we show that actually u¯ ⊕ v¯ ∈ H. In fact, we have for w ∈ H 2 (E), (u¯ ⊕ v, ¯ Θ w ⊕ ∆ w) =

1 2π

Z 2π 0

¯ it ) + ∆ (t)v(t), ¯ w(eit ))E dt = 0, (Θ (eit )∗ u(e

because ∞

Θ ∗ u¯ + ∆ v¯ = Θ ∗ g − Θ ∗ΘΘ0∗ g − ∆ 2Θ0∗ g = (Θ ∗ − Θ0∗)g = ∑ e−int Θn∗ g ⊥ H 2 (E). 1

By virtue of (3.14) we have (I − UT∗ )(u¯ ⊕ v) ¯ = (I − Θ0Θ0∗ )g ⊕ 0.

(3.15)

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C HAPTER VI. F UNCTIONAL M ODELS

Now the elements of the form (I − Θ0Θ0∗ )g (g ∈ E∗ ) are dense in E∗ . Otherwise there would exist a g′ ∈ E∗ , g′ 6= 0, such that (IE∗ − Θ0Θ0∗ )g′ = 0,

(3.16)

and hence kΘ0∗ g′ k = kg′ k = kΘ0Θ0∗ g′ k. By virtue of (3.9) this implies Θ0∗ g′ = 0, and by (3.16) g′ = 0. This contradiction proves our assertion. This result, combined with (3.14) and (3.15), yields L∗ = E∗ ⊕ {0}

(3.17)

(one identifies, as usual, constant functions in L2 (E∗ ) with their values in E∗ ). We also have therefore M(L∗ ) = L2 (E∗ ) ⊕ {0}. (3.17′)

In other words, denoting by PL∗ the orthogonal projection from K onto M(L∗ ), we have for u ⊕ v ∈ K, ( L P ∗ (u ⊕ v) = u ⊕ 0, (3.18) Φ L∗ PL∗ (u ⊕ v) = Φ L∗ (u ⊕ 0) = Φ E∗ u ⊕ 0 = u ⊕ 0. 5. The condition that the element u ⊕ v ∈ K+ should belong to H can be expressed in a detailed form as (u ⊕ v, Θ w ⊕ ∆ w)K+ = 0

or (Θ ∗ u + ∆ v, w)L2 (E) = 0,

for every w ∈ H 2 (E). Thus our condition means that the function Θ ∗ u + ∆ v (which obviously belongs to L2 (E)) should be orthogonal to H 2 (E), so that their Fourier series is of the following form,

Θ (eit )∗ u(eit ) + ∆ (t)v(t) = e−it f1 + · · · + e−int fn + · · · , where fn ∈ E,

∞

∑ k fn k2 < ∞. 1

We calculate the explicit form of T. By virtue of (3.7) we have T(u ⊕ v) = PU(u ⊕ v) = P(eit u ⊕ eit v) that is,

(u ⊕ v ∈ H);

T(u ⊕ v) = (eit u ⊕ eit v) − (Θ w ⊕ ∆ w),

where the function w ∈ H 2 (E) is defined by the condition (eit u ⊕ eit v) − (Θ w ⊕ ∆ w) ⊥ Θ w′ ⊕ ∆ w′

for all

w′ ∈ H 2 (E),

(3.19)

3. F UNCTIONAL MODELS FOR ANALYTIC FUNCTIONS

259

or equivalently, by the condition

Θ ∗ [eit u − Θ w] + ∆ [eit v − ∆ w] = eit (Θ ∗ u + ∆ v) − w ⊥ H 2 (E)

(3.20)

(in L2 (E)). Using (3.19) we derive from (3.20) that w must equal f1 . Thus the explicit form of T is T(u ⊕ v) = (eit u(eit ) − Θ (eit ) f1 ) ⊕ (eit v(t) − ∆ (t) f1 ) where f1 =

1 2π

Hence we obtain

Z 2π 0

(u ⊕ v ∈ H),

eit (Θ (eit )∗ u(eit ) + ∆ (t)v(t)) dt.

(U − T)(u ⊕ v) = Θ (eit ) f1 ⊕ ∆ (t) f1 .

(3.21)

(3.22)

(3.23)

When u ⊕ v varies over H the corresponding elements f1 vary over a set E1 dense in E. To show this let us first observe that, on account of (3.9), the elements of the form f = (I − Θ0∗Θ0 )g (g ∈ E, Θ0 = Θ (0))

are dense in E, and setting

u ⊕ v = e−it [Θ (eit ) − Θ0]g ⊕ e−it ∆ (t)g we have

Θ (eit )∗ u(eit ) + ∆ (t)v(t) = e−it [Θ (eit )∗Θ (eit ) − Θ (eit )∗Θ0 + ∆ (t)2 ]g = e−it [I − Θ (eit )∗Θ0 ]g

= e−it (I − Θ0∗Θ0 )g − e−2it Θ1∗Θ0 g − · · · ;

this shows that u ⊕ v belongs to H (cf. (3.19)), and the corresponding element f1 is equal to the element f with which we started. Let us recall definition (3.3) of the isometry Ω from L2 (E) into K. Let ω be the restriction of Ω to the subspace E of L2 (E) (formed by the constant functions); that is, ω: f →Θ f ⊕∆ f ( f ∈ E). (3.24)

From (3.23), (3.24), and the fact that E1 is dense in E, we deduce that

L = (U − T)H = {Θ f1 ⊕ ∆ f1 : f1 ∈ E1 } = ω E1 = ω E1 = ω E; thus

L = {Θ f ⊕ ∆ f : f ∈ E}.

(3.25)

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C HAPTER VI. F UNCTIONAL M ODELS

Hence ω is a unitary transformation from E to L. On the other hand, we deduce from (3.17) that ω∗ : f∗ → f∗ ⊕ 0∆ L2 (E) ( f∗ ∈ E∗ ) (3.26) is a unitary transformation from E∗ to L∗ . If we set

Θ (λ ) = ω∗Θ (λ )ω −1

(λ ∈ D),

(3.27)

we obtain a contractive analytic function {L, L∗ ,Θ (λ )} coinciding with the function {E, E∗ , Θ (λ )}. We show that it satisfies the relation

Φ L∗ P L∗ l = Θ Φ L l

for l ∈ M(L).

(3.28)

We first consider the case when l = Un ln ,

ln = ω fn

( fn ∈ E; n = 0, ±1, . . .).

On account of (3.18) we have then PL∗ l = Un PL∗ ln = Un (Θ fn ⊕ 0), and hence [Φ L∗ PL∗ l](τ ) = (einτ Θ (eiτ ) fn ⊕ 0) = ω∗Θ (eiτ )einτ fn

= ω∗Θ (eiτ )ω −1 einτ ω fn = Θ (eiτ )[Φ L Un ω fn ](τ )

= Θ (eiτ )[Φ L l](τ )

a.e.,

which proves (3.28) in the case l ∈ Un L. The general case follows readily from this. Comparing (3.28) with (2.3) we obtain, by virtue of Proposition 2.2, that the characteristic function of T coincides with Θ (λ ) and hence with the initial function Θ (λ ) as well. We have thus proved the following result. Theorem 3.1. Given an arbitrary contractive analytic function {E, E∗ , Θ (λ )} and setting ∆ (t) = [I − Θ (eit )∗Θ (eit )]1/2 , the operator T defined on the Hilbert space H = [H 2 (E∗ ) ⊕ ∆ L2 (E)] ⊖ {Θ w ⊕ ∆ w : w ∈ H 2 (E)} by

T∗ (u∗ ⊕ v) = e−it [u∗ (eit ) − u∗(0)] ⊕ e−it v(t)

(u∗ ⊕ v ∈ H),

(3.29) (3.30)

will be a completely nonunitary contraction. If the function {E, E∗ , Θ (λ )} is purely contractive, then it coincides with the characteristic function of T. In this case, if we consider H in the natural way as a subspace of the spaces K = L2 (E∗ ) ⊕ ∆ L2 (E)

and K+ = H 2 (E∗ ) ⊕ ∆ L2 (E),

3. F UNCTIONAL MODELS FOR ANALYTIC FUNCTIONS

261

then the operators U and U+ defined by U(u∗ ⊕ v) = eit u∗ ⊕ eit v

(u∗ ⊕ v ∈ K) and U+ = U|K+

are the minimal unitary and minimal isometric dilations of T, respectively. 6. We complete this theorem in some respects. Proposition 3.2. (a) In order that the space H defined by (3.29) be 6= {0} it is necessary and sufficient that the function Θ (λ ) not be a unitary constant.2 (b) The characteristic function of the c.n.u contraction T on H, defined by (3.30), coincides with the purely contractive part {E0 , E0∗ , Θ 0 (λ )} of the function {E, E∗ , Θ (λ )} (cf Proposition V.2.1).

Proof. If Θ (λ ) ≡ Θ0 , where Θ0 is a unitary operator from E to E∗ , then ∆ (t) ≡ 0 and hence H = H 2 (E∗ ) ⊖ Θ0H 2 (E) = H 2 (E∗ ) ⊖ H 2(Θ0 E) = {0}. If the function Θ (λ ) is not a unitary constant, then assertion (b) implies that dim DT = dim E0

and

dim DT∗ = dim E0∗ ,

where dim E0 and dimE0∗ are not both 0 (because Θ (λ ) has a nontrivial purely contractive part). Because DT and DT∗ are subspaces of H we conclude that dim H > 0. Therefore it only remains to prove assertion (b). Let us denote by ∆ 0 (t) the function analogous to ∆ (t) but formed from Θ 0 (λ ) instead of Θ (λ ). The decomposition E = E′ ⊕ E0 obviously implies the decompositions L2 (E) = L2 (E′ ) ⊕ L2 (E0 ) and H 2 (E) = H 2 (E′ ) ⊕ H 2 (E0 ). Because Θ ′ (λ ) is a unitary constant, we have ∆ (t)v(t) = 0 ⊕ ∆ 0(t)v0 (t) for any v = v′ ⊕ v0 ∈ L2 (E). Consequently, {Θ w ⊕ ∆ w : w ∈ H 2 (E)} = {Θ ′ w′ ⊕ Θ 0 w0 ⊕ ∆ 0 w0 : w′ ∈ H 2 (E′ ), w0 ∈ H 2 (E0 )}, and hence {Θ w ⊕ ∆ w : w ∈ H 2 (E)} = H 2 (E′∗ ) ⊕ {Θ 0w0 ⊕ ∆ 0 w0 : w0 ∈ H 2 (E0 )}, because

Θ ′ H 2 (E′ ) = H 2 (Θ ′ E′ ) = H 2 (E′∗ ).

It follows that for u∗ ⊕ v ∈ H we have u∗ ⊥ H 2 (E′∗ ), and hence u∗ ∈ H 2 (E0∗ ). Consequently, the space H can be identified in a natural way with the space H0 = [H 2 (E0∗ ) ⊕ ∆ 0L2 (E0 )] ⊖ {Θ 0w0 ⊕ ∆ 0 w0 : w0 ∈ H 2 (E0 )}, and the operator T on H with the operator T0 on H0 defined by T0∗ (u0∗ ⊕ v0 ) = e−it [u0∗ (eit ) − u0∗(0)] ⊕ e−it v0 (t) 2

(u0∗ ⊕ v0 ∈ H0 ).

We also admit the trivial transformation 0 → 0 from {0} to {0} as unitary.

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Now, because the function {E0 , E0∗ , Θ 0 (λ )} is purely contractive, we have by Theorem 3.1 that the characteristic function of T0 —and hence the characteristic function of T as well—coincide with {E0 , E0∗ , Θ 0 (λ )}. The proof is complete. Proposition 3.3. If the contractive analytic functions {E, E∗ , Θ (λ )}

and {E′ , E′∗ , Θ ′ (λ )}

coincide, then the contractions T and T′ which they generate in the sense of Theorem 3.1 are unitarily equivalent. In fact, if

τ : E → E′

and τ∗ : E∗ → E′∗

are unitary operators such that Θ ′ (λ ) = τ∗Θ (λ )τ −1 (λ ∈ D), then

τˆ : u(eit ) ⊕ v(t) → τ∗ u(eit ) ⊕ τ v(t) is a unitary operator from H to H′ such that T′ = τˆ Tτˆ −1 . The proof is just the same as in the particular case considered in Sec. 2.3. We apply this result to characteristic functions. In conjunction with Secs. 1.2 and 2.2, it yields the following theorem. Theorem 3.4. Two completely nonunitary contractions are unitarily equivalent if and only if their characteristic functions coincide. 7. We have already seen (Theorem 2.3) that a c.n.u. contraction T is of class C·0 if and only if its characteristic function is an inner function. Theorems 2.3 and 3.1 allow us to complete this result as follows. Proposition 3.5. For a c.n.u contraction T we have (i) T ∈ C·0 ,

(ii) T ∈ C·1 ,

(iii) T ∈ C0· ,

if and only if the characteristic function of T is (i) inner, (ii) outer, (iii) ∗-inner,

(iv) T ∈ C1· ,

(iv) ∗-outer, respectively.

(For the definitions, cf. Secs. II.4 and V.2.3.)

Proof. The cases (iii) and (iv) reduce to the cases (i) and (ii) if one replaces T by T ∗ and recalls (1.5). Case (i) is contained in Theorem 2.3. It remains therefore to consider the case (ii). Because the classes C·0 , and so on, are obviously invariant under unitary equivalence, and because the property of a contractive analytic function of being outer (or inner) does not change if this function is replaced by a coinciding one, it suffices to prove our assertion for the functional model of T . Accordingly, let

3. F UNCTIONAL MODELS FOR ANALYTIC FUNCTIONS

263

T be the contraction defined by a purely contractive analytic function {E, E∗ , Θ (λ )} in the sense of Theorem 3.1. By (3.8), we have lim kT∗n (u ⊕ v)k = kvk for u ⊕ v ∈ H.

n→∞

This shows that T ∈ C·1 if and only if u ⊕ 0 ∈ H implies u = 0. Now u ⊕ 0 ∈ H means that u ⊥ Θ H 2 (E), and the latter condition implies u = 0 if and only if Θ (λ ) is an outer function. 8. In our functional models the minimal isometric dilations appear in an explicit form and this enables us to derive, by using Theorem II.2.3, explicit forms for the commutants also. For the sake of simplicity we only consider contractions of class C·0 , that is, we assume the functions Θ (λ ) to be inner. Thus suppose {E, E∗ , Θ (λ )} is a purely contractive inner function, and let T be the contraction on the space H = H 2 (E∗ ) ⊖ Θ H 2(E) defined by (T∗ u)(λ ) =

1 [u(λ ) − u(0)] λ

(3.31)

(u ∈ H);

(3.32)

then, by Theorem 3.1, the operator U+ defined on K+ = H 2 (E∗ ) by (U+ u)(λ ) = λ · u(λ ) is a minimal isometric dilation of T. Let H′ , T′ , and so on, correspond similarly to a function {E′ , E′∗ , Θ ′ (λ )} of the same kind. Because U+ and U′+ are unilateral shifts with E∗ and E′∗ as generating subspaces, Lemma V.3.2 implies that every bounded operator Y from H 2 (E′∗ ) to H 2 (E∗ ) such that U+ Y = YU′+ , can be represented in the form (Yu)(λ ) = Y (λ )u(λ ) (u ∈ H 2 (E′∗ )) by means of some bounded analytic function {E′∗ , E∗ ,Y (λ )} such that kY k∞ = kYk.3 Combining this fact with Theorem II.2.3 we obtain the following result. Theorem 3.6. Every bounded operator X from H′ to H satisfying TX = XT′

(3.33)

can be represented in the form Xu = P+ (Yu)

(u ∈ H′ ),

(3.34)

3 For a bounded analytic function {A, A , A(λ )} we define kAk as the supremum of the operator ∗ ∞ norm kA(λ )k on D.

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where P+ denotes the orthogonal projection from H 2 (E∗ ) onto H, and where {E′∗ , E∗ ,Y (λ )} is a bounded analytic function satisfying (3.35)

Y Θ ′ H 2 (E′ ) ⊂ Θ H 2 (E)4

and kY k∞ = kXk.

(3.36)

Conversely, every bounded analytic function {E′∗ , E∗ ,Y (λ )} satisfying (3.35) yields by (3.34) a solution X of (3.33) with kXk ≤ kY k∞ .

Observe that (3.35) is obviously satisfied if Θ (λ ) and Θ ′ (λ ) are scalar-valued and coincide; then Y (λ ) is also scalar-valued (i.e., we have Y ∈ H ∞ ), and in this case the right-hand side of (3.34) is equal to Y (T)u; see Chapter III, in particular Theorem III.2.1(g). This result is obviously invariant under unitary equivalence so we get the following corollary. Corollary 3.7. Let T be a contraction of class C00 and with defect indices dT = dT ∗ = 1. Then every bounded operator X commuting with T is a function of T , say X = y(T ), where y ∈ H ∞ . Moreover, we can choose y such that kyk∞ = kXk. From the many possible applications of Theorem 3.6 we only mention one. Proposition 3.8. Let T be given by (3.31) and (3.32) and let ϕ be a function in H ∞ . In order that the operator ϕ (T) be boundedly invertible it is necessary and sufficient that there exist a bounded analytic function {E∗ , E∗ ,Y (λ )} such that for every u ∈ H 2 (E) we have Y Θ u ∈ Θ H 2 (E) and u − ϕ Yu ∈ Θ H 2 (E). The details of the proof may be supplied by the reader; we only remark that the inverse of ϕ (T) and the function Y (λ ) are connected by the relation ϕ (T)−1 u = P+ (Yu), u ∈ H.

4 The characteristic function and the spectrum 1. The following theorem establishes relations between the characteristic function of a c.n.u contraction T , and the spectrum σ (T ) or point spectrum σ p (T )5 of T . As before, C denotes the unit circle and D denotes the open unit disc. 4

This condition is equivalent to the condition that there exists a bounded analytic function {E′ , E, Z(λ )} such that Y (λ )Θ ′ (λ ) = Θ (λ )Z(λ ). (3.35)′ This follows easily by applying Lemma V.3.2. 5 That is, the set of eigenvalues.

4. T HE CHARACTERISTIC FUNCTION AND THE SPECTRUM

265

Theorem 4.1. Let T be a c.n.u contraction on H. Denote by ST the set of points µ ∈ D for which the operator ΘT (µ ) is not boundedly invertible, together with those µ ∈ C not lying on any of the open arcs of C on which ΘT (λ ) is a unitary operatorvalued analytic function of λ . Furthermore, denote by ST0 the set of points µ ∈ D for which ΘT (µ ) is not invertible at all. Then

σ (T ) = ST and σ p (T ) = ST0 . Proof. Because T is a c.n.u contraction, it cannot have eigenvalues on C. Thus if a ∈ σ p (T ) then a ∈ D; setting Ta = (T −aI)(I − aT ¯ )−1 we have therefore 0 ∈ σ p (Ta ). Consequently we have Ta f = 0 and hence (I − Ta∗ Ta ) f = f for some nonzero f , which shows that f ∈ DTa and, by virtue of (1.1),

ΘTa (0) f = −Ta f = 0. As ΘTa (0) differs from ΘT (a) by unitary factors only (cf. Sec. 1.3) it follows that ΘT (a)g = 0 for some nonzero g, and hence a ∈ ST0 . Thus we have σ p (T ) ⊂ ST0 . The opposite inclusion can be proved by the same reasoning in reverse order. Thus σ p (T ) = ST0 . Now we consider the problem for σ (T ) and show first that

σ (T ) ∩ D = ST ∩ D;

(4.1)

that is, that a point a ∈ D belongs either to both σ (T ) and ST , or to neither of them. It suffices to show this for the point 0 because the case of a nonzero a can be reduced to this as in the above reasoning, replacing T by Ta . For a = 0 we proceed as follows. First, we observe that Z = T |(H ⊖ DT ) is a unitary operator from H ⊖ DT to H ⊖ DT ∗ (see Sec. I.3.1). Next, from definition (1.1) of the characteristic function we infer that T |DT = −ΘT (0). Thus T is the orthogonal sum of −ΘT (0) and the unitary operator Z. In order that T be boundedly invertible it is therefore necessary and sufficient that ΘT (0) be boundedly invertible. This proves that 0 belongs to σ (T ) if and only if it belongs to ST . This proves (4.1). It remains to prove the equality

σ (T ) ∩C = ST ∩C.

(4.2)

The inclusion ρ (T ) ∩C ⊂ C\ST was proved at the end of Sec. 1.1. Taking complements with respect to C we obtain σ (T ) ∩ C ⊃ ST ∩ C. Thus to prove (4.2) it only remains to establish the reverse inclusion. We have to prove, therefore, that if α is an arc of C on which the function ΘT (λ ) is analytic and unitary operator-valued, then α ⊂ ρ (T ). Clearly it suffices to show this fact for our functional model for T . Thus let T be the c.n.u contraction generated in the sense of Theorem 3.1 by some purely contractive analytic function {E, E∗ , Θ (λ )} such that Θ (λ ) is analytic and unitary valued on some open arc α of C. We first prove the following result.

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Lemma. For every element u ⊕ v of H = [H 2 (E∗ ) ⊕ ∆ L2 (E)] ⊖ {Θ w ⊕ ∆ w : w ∈ H 2 (E)}, the function u (∈ H 2 (E∗ )) is analytic on α . Proof. By (3.19), the condition u ⊕ v ∈ H is equivalent to the condition that the function f (t) = Θ (eit )∗ u(eit ) + ∆ (t)v(t) (∈ L2 (E)) (4.3) be orthogonal to H 2 (E), thus f (t) = e−it f1 + e−2it f2 + · · ·

∞

∑ k fk k2 < ∞.

with

1

(4.4)

Now the latter condition implies that the function

ϕ (λ ) = λ f1 + λ 2 f2 + · · ·

(λ ∈ D)

belongs to H 2 (E). By virtue of Sec. V.1 we have

ϕ (re−it ) → ϕ (e−it ) = f (t) a.e. (convergence in E) and

Z 2π 0

kϕ (re−it ) − f (t)k2E dt → 0,

(4.5)

(4.6)

as r → 1 − 0. On the other hand, u(λ ) ∈ H 2 (E∗ ), thus u(reit ) → u(eit ) a.e. (convergence in E∗ ) and

Z 2π 0

ku(reit ) − u(eit )k2E∗ dt → 0.

(4.7)

(4.8)

Observe that, because Θ (eit ) is unitary for eit ∈ α , (4.3) implies

Θ (eit ) f (t) = u(eit ) for almost every point eit ∈ α .

(4.9)

Let G denote the domain where Θ (λ ) is analytic; thus G ⊃ D ∪ α . Let G+ be the part of G exterior to C, and let G− = D. Set

ϕ− (λ ) = u(λ ) for λ ∈ G− and

ϕ+ (λ ) = Θ (λ )ϕ (1/λ ) for λ ∈ G+ .

We deduce from relations (4.5)–(4.9) that Lemma V.6.6 applies to these functions and to each arc α1 with closure in α . It follows that ϕ− (λ ) and ϕ+ (λ ) are analytic continuations of each other through the arc α1 , and as α1 was arbitrary, through the

4. T HE CHARACTERISTIC FUNCTION AND THE SPECTRUM

267

whole arc α . This means in particular that u(λ ) is analytic on α , as asserted in the lemma. Now we are able to complete the proof of Theorem 4.1. Let us observe that if u(eit ) ⊕ v(t) ∈ H and if ν is any point of D then we also have uν ⊕ vν ∈ H, where uν (λ ) = moreover,

1 [λ u(λ ) − ν u(ν )], λ −ν

(I − ν T∗ )(uν ⊕ vν ) = u ⊕ v

vν (t) =

1 eit v(t); eit − ν

(I = IH ).

(4.10)

All this can be verified easily from the definition of H and T, and from Theorem 3.1, also using the characterization (3.19) of the elements of H. Now let ν tend to a point ε of the arc α . Because Θ (λ ) is analytic and unitary-valued on α , it follows from our lemma that u(λ ) is analytic on α and hence in particular at λ = ε . On the other hand, we have ∆ (t) = O for t ∈ (α ), and consequently v(t) = 0 for t ∈ (α ) (because v ∈ ∆ L2 (E)). Hence, by observing that uν (λ ) is bounded when both ν and λ are in a small disc centered at ε , we easily conclude that uν ⊕ vν converges in H 2 (E∗ ) ⊕ ∆ L2 (E) to a limit, which we denote by uε ⊕ vε and which also belongs to H; relation (4.10) gives in the limit: (I − ε T∗ )(uε ⊕ vε ) = u ⊕ v.

(4.10′)

Because T∗ is c.n.u., it has no eigenvalue on C; consequently I − ε T∗ is invertible. By (4.10′) its inverse is defined everywhere on H; thus I − ε T∗ , and hence ε I − T also, are boundedly invertible. This proves that every point of α belongs to the resolvent set ρ (T). This ends the proof of Theorem 4.1. 2. We consider an example demonstrating the usefulness of our functional model in obtaining contractions with prescribed properties. By Proposition II.3.5(iii) every contraction T of class C11 is quasi-similar to a unitary operator U, and by Proposition II.5.1 this quasi-similarity preserves in a certain sense the hyperinvariant subspaces. However, quasi-similarity does not generally preserve the spectrum. Indeed, we exhibit a contraction T ∈ C11 whose spectrum consists of the closed disc D. To this end recall the example treated in Sec V.4.5, namely the constant function Θ (λ ) ≡ A, where A is a self-adjoint operator satisfying the inequalities O ≤ A ≤ I and such that 0 and 1 are not eigenvalues of A. This function is analytic, purely contractive, and outer from both sides, so that the c.n.u contraction T which it generates in the sense of Theorem 3.1 is of class C11 . Applying Theorem 4.1 we obtain that the spectrum of T coincides either with the unit circle C or with the closed unit disc D, accordingly as the operator A is boundedly invertible or not. A complete description of the spectra of C11 -contractions is given in Chap. IX. 3. Theorem 4.1 suggests a further study of the relations between the resolvent (λ I − T )−1 and the characteristic function of T when the spectrum of T does not cover the entire unit disc D.

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C HAPTER VI. F UNCTIONAL M ODELS

As we already observed in 4.1, T is the orthogonal sum of −ΘT (0) (from DT to DT ∗ ) and of a unitary operator Z (from H ⊖ DT to H ⊖ DT ∗ ). Hence, if T is boundedly invertible, then T −1 is the orthogonal sum of the operator −ΘT (0)−1 and of the unitary operator Z −1 . As a consequence, kT −1 k is then equal to the larger one of the values kΘT (0)−1 k and kZ −1 k = 1. Because ΘT (0) is a contraction, its inverse has norm not less than 1. Therefore we have kT −1 k = kΘT (0)−1 k.

(4.11)

Consider now any point a ∈ D\σ (T ) and the operator Ta = (T − aI)(I − aT ¯ )−1 ;

(4.12)

Ta is a contraction and is boundedly invertible, with Ta−1 = (I − aT ¯ )(T − aI)−1.

(4.13)

Applying (4.11) to Ta in place of T we obtain kTa−1 k = kΘTa (0)−1 k.

(4.14)

Now from 1.3 we know that ΘTa (0) and ΘT (a) are equal up to unitary factors; hence we have kΘTa (0)−1 k = kΘT (a)−1 k. (4.15) From (4.13) we obtain

¯ )−1 Ta−1 , (T − aI)−1 = (I − aT and this implies that k(T − aI)−1 k ≤ k(I − aT ¯ )−1 kkTa−1 k −1

≤ (1 − |a|)

(4.16)

kTa−1 k.

Again from (4.13) we deduce Ta−1 = −aI ¯ + (1 − |a|2)(T − aI)−1, and this implies that kTa−1 k ≤ |a| + (1 − |a|2)k(T − aI)−1 k ≤ 1 + 2(1 − |a|)k(T − aI)−1k.

Relations (4.13)–(4.17) prove the following result. Proposition 4.2. If λ ∈ D\σ (T ), then kΘT (λ )−1 k = k(λ I − T )−1 (I − λ¯ T )k

(4.17)

4. T HE CHARACTERISTIC FUNCTION AND THE SPECTRUM

269

and (1 − |λ |)k(λ I − T )−1 k ≤ kΘT (λ )−1 k ≤ 1 + 2(1 − |λ |)k(λ I − T )−1 k. Let λ0 be an isolated point of σ (T ) in D and let λ be variable in D\σ (T ), tending to λ0 . From the above inequalities it follows that if there exists an integer p ≥ 1 such that one of the values lim sup k(λ − λ0) p (λ I − T )−1 k, λ →λ0

lim sup k(λ − λ0 ) pΘT (λ )−1 k λ →λ0

is finite then so is the other. This remark proves that if the point λ0 ∈ D is a pole of order p for one of the functions (λ I − T )−1 and ΘT (λ )−1 then it is a pole of order p for both. 4. Consider now a c.n.u contraction T whose characteristic function ΘT (λ ) admits a scalar multiple δ (λ ) 6≡ 0, that is, for which there exists a contractive analytic function {DT ∗ , DT , Ω (λ )} such that

Ω (λ )ΘT (λ ) = δ (λ )IDT , Then we have

ΘT (λ )−1 =

ΘT (λ )Ω (λ ) = δ (λ )IDT ∗ . 1 Ω (λ ) δ (λ )

at every point λ ∈ D, where δ (λ ) 6= 0, and hence ΘT (λ )−1 is a meromorphic function on D; the order of a pole a is at most equal to the multiplicity of a as a zero of δ. By virtue of Theorem 4.1 and what we have just proved on the poles, we can state the following proposition. Proposition 4.3. If the characteristic function of a c.n.u contraction T admits a scalar multiple, then (λ I − T )−1 is a meromorphic function on D.

Let us make the additional assumption that T ∈ C·1 . As ΘT (λ ) is then an outer function its scalar multiple δ (λ ) can be chosen to be outer also. In this case δ (λ ) 6= 0 on D, and thus ΘT (λ ) is boundedly invertible for every λ ∈ D. Consequently, σ (T ) is situated on the circle C. Suppose α is an arc of C such that ΘT (eit ) is isometric at almost every point eit ∈ α . By virtue of Proposition V.6.7, ΘT (λ ) then extends analytically through α to the whole exterior of C and, by virtue of Proposition V.6.5, ΘT (eit ) is, for eit ∈ α , even unitary. So it follows from Theorem 4.1 that α is contained in the resolvent set of T . Collecting these results we obtain the following corollary. Proposition 4.4. Let T be a c.n.u contraction of class C·1 whose characteristic function admits a scalar multiple. The spectrum σ (T ) is then the complement on C of the union of the open arcs of C on which ΘT (eit ) is isometric a.e. We conclude this section by characterizing the similarity of a contraction T to a unitary operator in terms of the characteristic function of T .

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Theorem 4.5. In order that a contraction T on a (separable) Hilbert space H be similar to a unitary operator, it is necessary and sufficient that ΘT (λ ) be boundedly invertible at every point λ ∈ D and kΘT (λ )−1 k have a bound independent of λ . Equivalently, it is necessary and sufficient that ΘT (λ ) satisfy the conditions kΘT (λ )gk ≥ ckgk ΘT (λ )DT = DT ∗

(g ∈ DT , λ ∈ D) and (λ ∈ D),

(4.18) (4.19)

the second at least at one (and then at every) point of D. Under these conditions, T is similar in particular to the residual part R of its minimal unitary dilation. Moreover, the least upper bound of kΘT (λ )−1 k on D is equal to the minimum of kSkkS−1k for affinities S such that ST S−1 is unitary. Proof. Taking into account the obvious fact that T is similar to a unitary operator if and only if its c.n.u part T1 satisfies this property, and by recalling that ΘT1 (λ ) = ΘT (λ ), we can assume that T is a c.n.u. contraction. By virtue of Proposition II.3.6 similarity of T to a unitary operator is equivalent to the condition that the transformation Q = PL∗ |M+ (L) from M+ (L) to M+ (L∗ ) be boundedly invertible, and then kQ−1 k is the minimum of kSkkS−1k for affinities S such that ST S−1 is unitary. Applying the Fourier representations Φ+L and Φ+L∗ , we obtain that the contractive analytic function {L, L∗ , ΘL (λ )} associated with T in the sense of Proposition 2.1 satisfies the intertwining relation Φ+L∗ Q = (ΘL )+ Φ+L . Recalling part (e) of Lemma V.3.2 we infer that Q is boundedly invertible if and only if ΘL (λ ) is boundedly invertible at every point λ ∈ D and kΘL (λ )−1 k has a bound independent of λ ; furthermore kQ−1 k is the upper bound of kΘL (λ )−1 k on D. By virtue of Proposition 2.2 the function ΘL (λ ) coincides with the characteristic function ΘT (λ ), thus the previous conditions hold for ΘL (λ ) and ΘT (λ ) at the same time. It remains to prove that if (4.18) is true for every λ ∈ D and (4.19) is valid for at least one λ0 ∈ D then ΘT (λ ) is boundedly invertible on D. To show this, we begin by defining Λ as the set of points λ ∈ D at which ΘT (λ ) is boundedly invertible. Because Λ is obviously an open set, and nonempty because λ0 ∈ Λ , we have proved Λ = D if we show that the assumptions

λn ∈ Λ ,

λn → λ ∈ D

imply λ ∈ Λ . Now, this can be seen from the relation

ΘT (λ ) = ΘT (λn )[I + ΘT (λn )−1 [ΘT (λ ) − ΘT (λn )]], if we recall that kΘT (λn )−1 k ≤ 1/c by (4.18), and that kΘT (λ ) − ΘT (λn )k < c for λn sufficiently close to λ . Thus we have indeed Λ = D; that is, (4.19) holds for every λ ∈ D. We recall (cf. (2.4) and the following discussion up to (2.10)) that if T is c.n.u., then R is unitarily equivalent to the operator of multiplication by eit on the function

5. T HE CHARACTERISTIC AND THE MINIMAL FUNCTIONS

space

∆T L2 (DT ),

where

∆T (t) = [I − ΘT (eit )∗ΘT (eit )]1/2 .

271

(4.20)

We obtained the following corollary.

Corollary 4.6. If the contraction T is similar to a unitary operator, then its c.n.u part is similar to multiplication by eit on the space (4.20). It is possible to give Theorem 4.5 a form in which the characteristic function is replaced by the resolvent of T . One just has to recall the inequalities between kΘT (λ )−1 k and (1 − |λ |)k(λ I − T )−1 k established in Proposition 4.2. Thus Theorem 4.5 has the following consequence. Corollary 4.7. In order that the contraction T on H be similar to a unitary operator, it is necessary and sufficient that the open unit disc D be contained in the resolvent set of T and that there exist a constant b such that k(λ I − T )−1 k ≤

b for λ ∈ D. 1 − |λ |

(4.21)

Actually, it suffices to assume that at least one point of D belongs to the resolvent set and that we have 1 k(λ I − T )hk ≥ (1 − |λ |)khk for λ ∈ D, h ∈ H. b

(4.22)

The last statement can be proved in the same way as the analogous one for ΘT (λ ) in Theorem 4.5.6

5 The characteristic and the minimal functions 1. In Sec III.4 we have introduced the class C0 of c.n.u. contractions T such that δ (T ) = O for some function δ ∈ H ∞ , δ (λ ) 6≡ 0 (δ depending on T ), and we have shown that C0 ⊂ C00 ; that is, T ∈ C0 implies T n → O and T ∗n → O (strongly). Our aim in this section is to characterize the class C0 by means of the characteristic function. Theorem 5.1. Let T be a contraction of class C00 . In order that T belong to the class C0 it is necessary and sufficient that its characteristic function ΘT (λ ) admit a scalar multiple. To be more specific, a function δ ∈ H ∞ (with δ (λ ) 6≡ 0 and |δ (λ )| ≤ 1) satisfies the equation δ (T ) = O if and only if it is a scalar multiple of ΘT (λ ). Proof. We may consider instead of T its model T constructed with the aid of a contractive analytic function {E, E∗ , Θ (λ )} coinciding with ΘT (λ ); see Theorem 6

Using the last statement in Theorem 4.5 it is easy to extend Theorem 4.5 and its Corollary 4.7 to nonseparable spaces.

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C HAPTER VI. F UNCTIONAL M ODELS

2.3. Because T ∈ C00 , the function Θ (λ ) is inner (from both sides), and the model is of the simple form H = H 2 (E∗ ) ⊖ Θ H 2(E),

T∗ u∗ (λ ) =

1 [u∗ (λ ) − u∗(0)]. λ

Let us assume that δ (λ ) is a scalar multiple of Θ (λ ), namely that there exists a contractive analytic function {E∗ , E, Ω (λ )} for which Ω (λ )Θ (λ ) = δ (λ )IE and Θ (λ )Ω (λ ) = δ (λ )IE∗ . Then we have for u∗ ∈ H,

δ (U)u∗ = δ u∗ = Θ Ω u∗ ∈ Θ H 2 (E) ⊥ H; by virtue of the relation

δ (T) = PH δ (U)|H (cf. Theorem III.2.1(g))

(5.1)

this yields δ (T) = O. Conversely, let us assume that δ is any function in H ∞ with |δ (λ )| ≤ 1, δ (λ ) 6≡ 0, and δ (T) = O. Let δ (λ ) = δe (λ )δi (λ ) be the canonical factorization of δ (λ ), with inner factor δi (λ ) and outer factor δe (λ ); because δe (T) is invertible (cf. Sec. III.3) we also have δi (T) = O. From relation (5.1) applied to δi we obtain δi (U)H ⊥ H, and hence δi H = δi (U)H ⊂ K+ ⊖ H = Θ H 2 (E).

On the other hand δiΘ H 2 (E) = Θ δi H 2 (E) ⊂ Θ H 2 (E), therefore we conclude:

δi H 2 (E∗ ) = δi [H ⊕ Θ H 2(E)] ⊂ Θ H 2 (E). From this inclusion we infer that to each u∗ ∈ H 2 (E∗ ) there corresponds a unique u ∈ H 2 (E) such that δi u∗ = Θ u. (5.2) Because δi (λ ) and Θ (λ ) are inner functions, (5.2) implies ku∗ k = kuk; thus u = Qu∗ × defines an isometry Q from H 2 (E∗ ) into H 2 (E). It is obvious that if U∗+ and U+× it 2 2 denote the operators of multiplication by e on the spaces H (E∗ ) and H (E), re× spectively, then QU∗+ = U+× Q. Applying Lemma V.3.2 to the case

R+ = H 2 (E∗ ),

× U+ = U∗+ ;

R′+ = H 2 (E),

U+′ = U+× ;

Q,

and observing that in this case the Fourier transformations Φ E∗ and Φ E are the identity transformations, we obtain that there exists an inner function {E∗ , E, Ω (λ )} such that Qu∗ = Ω u∗ . So we have

δi u∗ = Θ Ω u∗

(u∗ ∈ H 2 (E∗ )).

(5.3)

5. T HE CHARACTERISTIC AND THE MINIMAL FUNCTIONS

273

Setting u∗ = Θ u with u ∈ H 2 (E) we derive Θ (δi u − ΩΘ u) = 0; the operator Θ is an isometry, therefore this relation implies

δi u = ΩΘ u

(u ∈ H 2 (E)).

(5.4)

Relations (5.3) and (5.4) say that Θ (λ )Ω (λ ) = δi (λ )IE∗ and Ω (λ )Θ (λ ) = δi (λ )IE ; thus δi (λ ) is a scalar multiple of Θ (λ ). The same is then true for δ (λ ) = δe (λ )δi (λ ), because δe (λ )Ω (λ ) is also contractive (note that kδ k∞ ≤ 1 implies kδe k∞ ≤ 1). This concludes the proof of Theorem 5.1. An important particular case is as follows. Theorem 5.2. Let T be a contraction on a space H 6= {0}, of class C00 , and with finite defect indices dT and dT ∗ . Then T ∈ C0 . Moreover, we have in this case dT = dT ∗ = n ≥ 1 and dT (T ) = O, where dT (λ ) denotes the determinant of the matrix of ΘT (λ ) corresponding to some orthonormal bases in the defect spaces DT and DT ∗ . The function dT (λ ) is inner. If n > 1, the minimal function mT (λ ) equals the quotient of dT (λ ) by the greatest common inner divisor of the minors of order n − 1 of the matrix of ΘT (λ ); if n = 1, then mT (λ ) = dT (λ ). On the other hand, dT (λ ) is always a divisor of (mT (λ ))n . Proof. According to Sec. 3.7, our assumption T ∈ C00 implies that ΘT (eit ) is unitary a.e., and hence dT = dim DT = dim DT ∗ = dT ∗ . Because the defect indices were assumed finite, they must be equal to some integer n ≥ 1 (n = 0 would mean that T is unitary, a case which is excluded by our assumptions). Let ϑ (λ ) be the matrix of the operator ΘT (λ ) for some orthonormal bases in the defect spaces; the entries of ϑ (λ ) are obviously functions of class H ∞ , and so is dT (λ ) = det ϑ (λ ). Under changes of the orthonormal bases, ϑ (λ ) changes by constant unitary factors, and therefore its determinant changes by a constant numerical factor of modulus 1. Thus dT (λ ) is determined by T up to a constant numerical factor of modulus 1. Moreover, because the operator ΘT (eit ) is unitary a.e., so is its matrix, and hence we have |dT (eit )| = 1 a.e. Thus dT (λ ) is an inner function. At every point λ ∈ D such that dT (λ ) 6= 0, the operator ΘT (λ ) is (boundedly) invertible. By virtue of Proposition V.6.1, dT (λ ) is a scalar multiple of ΘT (λ ), and the corresponding contractive analytic function Ω (λ ) has the matrix ϑ A (λ ), namely the algebraic adjoint of the matrix ϑ (λ ); if n = 1 then ϑ A (λ ) consists of the single entry 1. Obviously the entries of ϑ A (λ ) are also in H ∞ . Let us denote by k(λ ) the greatest common inner divisor (cf. Sec III.1) of the entries of ϑ A (λ ) (i.e., of the inner factors of the nonzero entries). Then we have Ω (λ ) = k(λ )M(λ ) and dT (λ ) = k(λ )m(λ ), where m(λ ) is a (scalar-valued) inner

274

C HAPTER VI. F UNCTIONAL M ODELS

function and M(λ ) is an (operator-valued) contractive7 analytic function the entries of whose matrix µ (λ ) have no nonconstant common inner divisor (in H ∞ ). (For n = 1 we have k(λ ) = 1, and hence Ω (λ ) = M(λ ), dT (λ ) = m(λ ).) The relations Ω (λ )ΘT (λ ) = dT (λ )IDT , ΘT (λ )Ω (λ ) = dT (λ )IDT ∗ imply M(λ )ΘT (λ ) = m(λ )IDT ,

ΘT (λ )M(λ ) = m(λ )IDT ∗ ;

(5.5)

thus m(λ ) is a scalar multiple of ΘT (λ ), and consequently we have m(T ) = O; cf. Theorem 5.1. Thus T ∈ C0 . By the same theorem, the minimal function mT (λ ) is also a scalar multiple of T ; thus there exists a contractive analytic function {DT ∗ , DT , ΩT (λ )} satisfying

ΩT (λ )ΘT (λ ) = mT (λ )IDT ,

ΘT (λ )ΩT (λ ) = mT (λ )IDT ∗ .

(5.6)

Because m(λ ) is inner and m(T ) = O, we have m(λ ) = p(λ )mT (λ ) for some inner function p(λ ). From (5.5) and (5.6) we deduce [p(λ )ΩT (λ ) − M(λ )]ΘT (λ ) = [p(λ )mT (λ ) − m(λ )]IDT = O

(λ ∈ D).

Because ΘT (λ ) has a boundedly invertible value at every point where d(λ ) 6= 0, and hence on a set dense in D, we infer that p(λ )ΩT (λ ) = M(λ )

(λ ∈ D),

(5.7)

and hence, if we denote the matrix of ΩT (λ ) by ω (λ ), p(λ )ω (λ ) = µ (λ ). This shows that the entries of the matrix µ (λ ) have p(λ ) as a common inner divisor in H ∞ , which is impossible unless p(λ ) is a constant (of modulus 1). Thus mT (λ ) coincides with m(λ ). Finally, it follows from (5.6) that

ω (λ )ϑ (λ ) = mT (λ )In = ϑ (λ )ω (λ );

(5.8)

taking determinants this implies [det ω (λ )] · dT (λ ) = [mT (λ )]n . This shows that dT (λ ) is a divisor of [mT (λ )]n , and hence concludes the proof of Theorem 5.2. Remark. In the case n = 1, mT (λ ) coincides with dT (λ ) and hence with ΘT (λ ). As the characteristic function of a c.n.u contraction T , considered up to coincidence, determines T up to unitary equivalence, it follows that the contractions of class C00 with defect indices equal to 1 are determined by their minimal functions up to unitary equivalence. By Proposition III.4.6, two contractions of class C0 , one of which is a quasi-affine transform of the other, have the same minimal function. We obtain the following corollary. 7

Consequence of relation (V.2.3).

5. T HE CHARACTERISTIC AND THE MINIMAL FUNCTIONS

275

Corollary 5.3. Two contractions of class C0 , with defect indices equal to 1, are unitarily equivalent if one of them is a quasi-affine transform of the other. 2. In order that a contraction belong to the class C0 it is not necessary that the corresponding defect operators be of finite rank, or even that they be compact. In fact, we give an example for a contraction T of class C0 , having the minimal function   λ +1 , (5.9) mT (λ ) = exp λ −1 and such that the defect operators DT and DT ∗ have no nonzero eigenvalue. To this end, consider the purely contractive analytic function {E, E, Θ (λ )}, where   λ +1 E = L2 (0, 1) and Θ (λ ) f (x) = exp x f (x) ( f ∈ E). λ −1 Let T be the corresponding contraction in the sense of Theorem 3.1. Because   exp x λ + 1 = 1 for 0 ≤ x ≤ 1, |λ | = 1, λ 6= 1, λ −1

Θ (λ ) is inner from both sides, and consequently T ∈ C00 . Moreover, the function {E, E, Ω (λ )} defined by   λ +1 · f (x) Ω (λ ) f (x) = exp (1 − x) λ −1 is also contractive and we have obviously

Ω (λ )Θ (λ ) = Θ (λ )Ω (λ ) = e1 (λ )IE , where we again use the notation   λ +1 es (λ ) = exp s λ −1

(s ≥ 0).

Thus e1 (λ ) is a scalar multiple of Θ (λ ). By Theorem 5.1 this implies e1 (T) = O. Because e1 (λ ) admits no inner divisors other than es (λ ), 0 ≤ s ≤ 1, the minimal function of T must be of the form mT (λ ) = es (λ ) with some s such that 0 < s ≤ 1. As mT (T) = O, we see by invoking Theorem 5.1 again that mT (λ ) is a scalar multiple of Θ (λ ), that is, there exists a contractive analytic function Ω ′ (λ ) such that

Ω ′ (λ )Θ (λ ) = Θ (λ )Ω ′ (λ ) = mT (λ )IE . Hence

kΘ (0) f k ≥ kΩ ′ (0)Θ (0) f k = |mT (0)|k f k;

276

that is,

C HAPTER VI. F UNCTIONAL M ODELS

Z

0

1

|e−x f (x)|2 dx

1/2

≥ e−s

Z

1 0

| f (x)|2 dx

1/2

(5.10)

for every function f ∈ E = L2 (0, 1). Choosing f (x) to be 0 except in a small neighborhood of x = 1, we conclude that s cannot be less than 1. Thus mT (λ ) = e1 (λ ), which proves (5.9). It remains to prove that the defect operators DT and DT ∗ have no nonzero eigenvalues. As we have, up to coincidence,

ΘT∗ (λ ) = Θ T˜ (λ ) = Θ ˜(λ ) = Θ (λ ) = ΘT (λ ), it follows from Proposition 3.3 that T and T∗ are unitarily equivalent, and hence so are DT and DT∗ . It suffices therefore to show that DT∗ (or equivalently D2T∗ ) has no nonzero eigenvalue. For this purpose we use formula (3.14); it allows us to write D2T∗ u = (I − TT∗ )u = P(I − UT∗ )u = Pu(0), where

H = H 2 (E) ⊖ Θ H 2 (E),

u ∈ H,

E = L2 (0, 1),

and P denotes the orthogonal projection from H 2 (E) onto H. Let us make it explicit that we have here ∞

u(λ ) = u(λ ; x) = ∑ λ n fn (x) (|λ | < 1; x ∈ (0, 1); fn ∈ L2 (0, 1)). 0

The element w = Pu(0) is determined by the conditions that w ∈ H and u(0) − w ∈ Θ H 2 (E); it follows that w(λ ) = [IE − Θ (λ )Θ (0)∗ ]u(0)

(u ∈ H).

Suppose we have (I − TT∗ )u = ρ u for some number ρ 6= 0 and some element u ∈ H. It follows from the above that then

ρ u(λ ) = [IE − Θ (λ )Θ (0)∗]u(0),

(5.11)

and hence in particular

ρ u(0; x) = (1 − e−2x )u(0; x) a.e on 0 ≤ x ≤ 1; therefore we have u(0; x) = 0 a.e., that is, u(0) = 0. Thus (5.11) implies u(λ ) = 0 for every λ ∈ D, and hence u = 0 as an element of H. This proves that ρ is not an eigenvalue of I − TT∗ . All our assertions concerning the example have been established.

6. S PECTRAL TYPE OF THE MINIMAL UNITARY DILATION

277

6 Spectral type of the minimal unitary dilation 1. Our functional model also allows us to solve completely the problem of finding the spectral type of the minimal unitary dilation U of an arbitrary c.n.u contraction T , namely determining the structure of U up to unitary equivalence. Let us recall Theorem II.7.4, which says that if at least one of the defect indices of T is infinite, then U is a bilateral shift with multiplicity equal to dmax = max{dT , dT ∗ }; in other words, U is then unitarily equivalent to an orthogonal sum of dmax copies of the operator of multiplication by eit on the space L2 (0, 2π ) of scalar-valued functions x(t). Note that this holds whenever kT k < 1 and the dimension of the Hilbert space is infinite. We are now going to consider the remaining case when both defect indices are finite. We are concerned with spaces L2 (S), where S is a measurable subset of (0, 2π ), and scalar products and norms have to be taken with respect to the normalized Lebesgue measure dt/2π . Theorem 6.1. Let T be a c.n.u contraction with finite defect indices: dT = n, dT ∗ = m. The minimal unitary dilation U of T is then unitarily equivalent to the operator of multiplication by eit on the space L2 (M1 ) ⊕ · · · ⊕ L2 (Mm ) ⊕ L2 (N1 ) ⊕ · · · ⊕ L2 (Nn ),

(6.1)

where M1 = · · · = Mm = (0, 2π ) and Nk = {t : t ∈ (0, 2π ), r(t) ≥ k}

(k = 1, . . . , n);

(6.2)

here r(t) denotes the rank of the operator ∆T (t) = [I − ΘT (eit )∗ΘT (eit )]1/2 . Proof. As the minimal unitary dilations of two unitarily equivalent contractions are obviously also unitarily equivalent, it suffices to consider our functional model T with the given defect indices n, m. Thus let {E n , E m , Θ (λ )} be a purely contractive analytic function and let T be the contraction it generates in the sense of Theorem 3.1. The characteristic function of T coincides with the given function Θ (λ ), so ∆T (t) is unitarily equivalent to ∆ (t) = [I − Θ (eit )∗Θ (eit )]1/2 (by a constant unitary transformation). Hence r(t) = rank ∆T (t) = rank ∆ (t) for every t. By virtue of Theorem 3.1 the minimal unitary dilation U of T is equal to the multiplication by eit on the space K = L2 (E m ) ⊕ ∆ L2 (E n ). Now it is obvious that L2 (E m ) is the orthogonal sum of m copies of the space L (0, 2π ) of scalar valued functions, with the multiplication by eit on L2 (E m ) corresponding to the multiplication by eit on each of the component spaces L2 (0, 2π ). 2

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C HAPTER VI. F UNCTIONAL M ODELS

We study next the residual part of U on ∆ L2 (E n ). For each value of t at which it is defined, ∆ (t) is a self-adjoint operator on E n bounded by 0 and 1, and hence there exists an orthonormal base {ψk (t)}n1 of E n composed of eigenvectors of ∆ (t), that is, ∆ (t)ψk (t) = δk (t)ψk (t) (k = 1, . . . n), where the eigenvalues δk (t) are arranged in nonincreasing order: 1 ≥ δ1 (t) ≥ δ2 (t) ≥ · · · ≥ δn (t) ≥ 0. Because ∆ (t) f is, for every f ∈ E n , a measurable function of t, the eigenvalues δk (t) are also measurable functions of t; and because we obviously have {t : t ∈ (0, 2π ), r(t) ≥ k} = {t : t ∈ (0, 2π ), δk (t) > 0}, the sets Nk defined in the theorem are also measurable. Moreover, the eigenvectors ψk (t) can also be chosen so as to be measurable functions of t. The measurability of the greatest eigenvalue δ1 (t) is an easy consequence of the relation δ1 (t) = sup(∆ (t) fk , fk ), where { fk } denotes a sequence dense on the unit sphere of E n ; for the other eigenvalues one can make use of the “minimax” theorem (cf. [Func Anal.] Sec. 95). The measurable choice of the eigenvectors is somewhat more involved; it can be established by manipulations with the minors of the matrix of ∆ (t). Setting xk (t) = (v(t), ψk (t))E n for v ∈ L2 (E n ), we have n

n

1

1

∆ (t)v(t) = ∆ (t) ∑ xk (t)ψk (t) = ∑ xk (t)δk (t)ψk (t), and hence

n

k∆ (t)v(t)k2E n = ∑ |xk (t)δk (t)|2 1

and k∆ vk2L2 (E n ) =

Z 2π

1 n ∑ 2π 1

0

|xk (t)δk (t)|2 dt =

Z

1 n ∑ 2π 1

Nk

|xk (t)δk (t)|2 dt,

which shows that the transformation

∆ v → {x1 (t)δ1 (t), . . . , xn (t)δn (t)}

(6.3)

maps ∆ L2 (E n ) isometrically into the space L2 (N1 ) ⊕ · · · ⊕ L2 (Nn ). The obvious relation

(eit v(t), ψk (t))E n = eit xk (t)

(6.4)

6. S PECTRAL TYPE OF THE MINIMAL UNITARY DILATION

279

shows that multiplication by eit on ∆ L2 (E n ) is carried over by (6.3) into multiplication by eit on each of the component spaces L2 (Nk ). Let us choose, in particular, v(t) = ε (t)ψk (t) for some fixed k (1 ≤ k ≤ n) and some scalar-valued, bounded, measurable ε (t); we have then v ∈ L2 (E n ), and the vector corresponding to ∆ v by the transformation (6.3) will have its kth component equal to ε (t)δk (t) and its other components equal to 0. As the functions of type ε (t)δk (t) are dense in L2 (Nk ), we conclude that the transformation (6.3), when extended by continuity to the whole space ∆ L2 (E n ), maps this space isometrically onto the space (6.4). The assertion concerning multiplication by eit will hold true, by continuity, after this extension also. Theorem 6.1 is proved. 2. It is obvious that N1 ⊃ N2 ⊃ · · · ⊃ Nn . It is possible for the essential supremum rmax of the function r(t) to be less than n; in this case the spaces L2 (Nk ) reduce, for k > rmax , to the trivial space {0}, and hence can be omitted from (6.1). There is an asymmetry in (6.1) with respect to the two defect indices, but this can easily be removed. Indeed, if we consider T ∗ instead of T we obtain that UT∗ (= UT ∗ ) is unitarily equivalent to multiplication by eit on a space of type (6.1), but with the roles of m and n interchanged. The same is then true for UT too (one has only to replace the sets occurring by their images under the transformation t → 2π − t).8 Thus Theorem 6.1 has the following consequence. Corollary 6.2. The minimal unitary dilation of a c.n.u contraction T with finite defect indices dT = n and dT ∗ = m is unitarily equivalent to multiplication by eit on a space L2 (P1 ) ⊕ L2 (P2 ) ⊕ · · · ⊕ L2 (Pn+m ),

where Pk (k = 1, . . . , n + m) are measurable subsets of (0, 2π ), with Pk ⊃ Pk+1 (k = 1, . . . , n + m − 1) and with Pk = (0, 2π ) for k = 1, . . . , max{n, m}.

3. Now there arises the question if, conversely, multiplication by eit on every such space is unitarily equivalent to the minimal unitary dilation of some c.n.u contraction. The answer is yes and we formulate it as follows. Theorem 6.3. Let n, m be nonnegative integers with ν = max{n, m} ≥ 1. Let P1 ⊃ P2 ⊃ · · · ⊃ Pn+m

8

In the sum (6.1), the number of the terms different from {0} is equal to m + rmax . The roles of m and n are only seemingly asymmetric. This follows directly from the following: if Θ is a contraction from N to M (where N and M are Hilbert spaces of finite dimensions), then dim M + dim (I − Θ ∗Θ )1/2 N = dim N + dim (I − ΘΘ ∗ )1/2 M. Indeed, the left-hand side is equal to dim M + dim N − dim N0 , and the right-hand side to dim N + dim M − dim M0 , where N0 = {h : h ∈ N, (I − Θ ∗Θ )h = 0},

M0 = {h : h ∈ M, (I − ΘΘ ∗ )h = 0};

but dim N0 = dim M0 because Θ maps N0 isometrically onto M0 .

280

C HAPTER VI. F UNCTIONAL M ODELS

be measurable subsets of (0, 2π ) with Pk = (0, 2π ) for k = 1, . . . , ν . Then there exists a c.n.u contraction T with dT = n and dT ∗ = m such that the minimal unitary dilation of T is unitarily equivalent to multiplication by eit on the space L2 (P1 ) ⊕ L2 (P2 ) ⊕ · · · ⊕ L2 (Pn+m ). (We also allow the possibility that Pk equal (0, 2π ) for some k > ν , as well as the possibility that some Pk is of measure 0; in the latter case the corresponding spaces L2 (Pk ) reduce to {0} and hence can be omitted.)

Proof. Let us assume first that n ≤ m so that ν = m. The case n = 0 (i.e., ν = n + m) is simple. Multiplication by eit is then a bilateral shift with multiplicity ν , and hence it is the minimal unitary dilation of a unilateral shift with the same multiplicity. In the case 1 ≤ n ≤ m, let us consider the matrix-valued function

Θ (λ ) = [ϑ jk (λ )]

( j = 1, . . . , m; k = 1, . . . , n; |λ | < 1)

with

ϑ jk (λ ) ≡ 0 for j 6= k, ϑkk (k) ≡ λ · uk (λ ) for k = 1, . . . , n, where the function uk (λ ) is determined in such a way that it belongs to H ∞ and satisfies 1 |uk (eit )|2 = 1 − χk (t) a.e., 2 χk (t) denoting the characteristic function of the set Pm+k . The existence of such a function uk (λ ) is guaranteed by the fact that log[1 − (1/2)χk (t)] ∈ L1 (0, 2π ) (cf. (III.1.14–16)). As we have |ϑkk (λ )| ≤ |uk (λ )| ≤ 1 and ϑkk (0) = 0

(k = 1, . . . , n),

{E n , E m , Θ (λ )} will be a purely contractive analytic function. The matrix ∆ (t) = [I − Θ (eit )∗Θ (eit )]1/2 is of order n and of diagonal form; the entries on the diagonal are   1/2 r 1 1 = (k = 1, . . . , n). δkk (t) = 1 − 1 − χk (t) χk (t) 2 2 Thus we have

n

rank ∆ (t) = ∑ χk (t). k=1

The sets Pm+k (k = 1, . . . , n) are arranged in nonincreasing order, thus the inequality rank ∆ (t) ≥ k is satisfied (up to a set of 0 measure) exactly for the points of the set Pm+k . Let T be the c.n.u contraction generated by the purely contractive analytic function {E n , E m , Θ (λ )} in the sense of Theorem 3.1. Applying Theorem 6.1 we see

6. S PECTRAL TYPE OF THE MINIMAL UNITARY DILATION

281

that the minimal unitary dilation of T is unitarily equivalent to multiplication by eit on the space hL i m L2 (0, 2π ) ⊕ L2 (Pm+1 ) ⊕ · · · ⊕ L2 (Pm+n ). 1

This proves Theorem 6.3 in the case n ≤ m. In the case n > m we first construct a c.n.u contraction S for which dS = m and dS∗ = n, and whose minimal unitary dilation is unitarily equivalent to multiplication by eit on the space ′ L2 (P1′ ) ⊕ L2 (P2′ ) ⊕ · · · ⊕ L2 (Pm+n ),

where Pk′ denotes the symmetric image of the set Pk with respect to the point t = π ; such a contraction S exists by what we have just proved. Then T = S∗ satisfies the assertions of the theorem. Theorem 6.3 is proved.

Remark. Corollary 6.2 and Theorem 6.3 (for the case of finite defect indices) and Theorem II.7.4 (for the case of at least one infinite defect index) give a complete solution of the problem of determining the spectral type of the minimal unitary dilations of c.n.u contractions. Note that the contraction T constructed in the proof of Theorem 6.3 is, in general, reducible. It is natural to ask whether we can also find an irreducible T for this purpose. However, if for a contraction T we have max{m, n} > 1 and min{m, n} = 0, then, T or T ∗ is necessarily a shift of multiplicity > 1 and therefore always reducible. In the remaining case min{m, n} ≥ 1, an appropriate modification of the previous proof will yield an irreducible contraction T with the required properties. To present such a modification, first observe that, due to the last part of the proof of Theorem 6.3, we can assume 1 ≤ n ≤ m. Also, observe that if b is the c.n.u contraction associated with {E m , E m , Ω (λ )} is an inner function and if T the purely contractive analytic function {E n , E m , Θˆ (λ )} with Θˆ (λ ) = Ω (λ )Θ (λ ), b where Θ (λ ) and T are as in the proof above, then the minimal unitary dilation of T b and T have the same spectral type. Thus it remains to construct Ω (λ ) such that T be irreducible. To this end, choose an m × m unitary matrix G = [g jk ] with nonzero ∞ l constant √ entries (e.g G = exp(iA) = ∑l=0 (iA) /l! where all the entries of A are equal to π 2) and define Ω (λ ) = [ω jk (λ )] by

ω jk (λ ) = λ ( j−1)m+(k−1) g jk

for

j, k = 1, 2, . . . , m.

With this choice, the entries ϑˆ jk (λ ) ( j = 1, . . . , m; k = 1, . . . , n) of Θˆ (λ ) are linearly b were reducible then its independent, a fact left to the reader to prove. Finally, if T characteristic function would be the orthogonal sum of the characteristic functions of the reducing components. Because it also coincides with Θˆ (λ ), there would exist nonzero vectors c = (c1 , . . . , cn ) in E n and d = (d1 , . . . , dm ) in E m such that m

n

0 = (Θˆ (λ )c, d) = ∑ ∑ ck d j ϑˆ jk (λ ). j=1 k=1

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C HAPTER VI. F UNCTIONAL M ODELS

This contradicts the linear independence of the functions ϑˆ jk (λ ). Thus the c.n.u b is irreducible. contraction T

4. In Secs. II.1 and 7 we obtained several conditions upon a c.n.u contraction T which ensure that the minimal unitary dilation U of T be a bilateral shift. The above Theorem 6.3 shows that there exist c.n.u contractions T for which U is not a bilateral shift; indeed, we can choose some of the sets Pk such that both Pk and its complement (0, 2π )\Pk have positive measure. However, it may be instructive to give here a concrete example based on Theorem 6.1. Consider a numerical function w(λ ), holomorphic on D and such that |w(λ )| ≤ 1 on D and |w(0)| < 1. Let T be the c.n.u contraction generated (in the sense of Theorem 3.1) by the purely contractive analytic function {E 1 , E 1 , w(λ )}. As the defect indices of T are equal to 1 it follows from Theorem 6.1 that the minimal unitary dilation of T is unitarily equivalent to multiplication by eit on the space L2 (0, 2π ) ⊕ L2 (N), where N = {t : t ∈ (0, 2π ), 1 − |w(eit )|2 > 0}. In particular, choose for w(λ ) the conformal map of D onto the half-disc {λ : |λ | < 1, Im λ > 0}. Then we have 0 < meas N < 2π , and hence, obviously, L2 (0, 2π ) ⊕ {0} ⊂ L2 (0, 2π ) ⊕ L2(N) ⊂ L2 (0, 2π ) ⊕ L2(0, 2π ),

with proper inclusions. Now multiplication by eit is a bilateral shift on the first as well as on the third space, with multiplicity 1 and 2, respectively. If it were a bilateral shift on the intermediate space L2 (0, 2π ) ⊕ L2(N) also, then, by Proposition I.2.1, this space would coincide with one of the two other spaces, which is impossible because the measure of N differs from 0 and 2π .

7 Notes 1. The analysis of the resolvent (T − λ I)−1 by methods of complex function theory has long since proved to be one of the principal means to study the structure of the operator T , at least if T is a normal operator or if the spectrum of T consists of several closed components. For other operators the analytical behavior of the resolvent yields but little information about the structure of T . More recently, the school of M. G. Kre˘ın in Odessa, inspired by Kre˘ın’s research on the extension of symmetric operators, began to associate with certain Hilbert space operators T a new type of operator-valued analytic function whose behavior reflects further details of the structure of T . The definition of these functions, called characteristic functions, developed gradually. First they appeared in L IV Sˇ IC [1] (for operators such that I − T ∗ T and I − T T ∗ have rank 1) and [2] (for the case that these operators have a finite rank). The first general definition was given in Sˇ MUL′ JAN [1]; in the case of a contraction T this definition coincides with our definition (1.1) or its equivalent form (1.2). The interest of this school shifted subsequently to bounded operators T such that T ∗ − T is of finite rank or at least of finite trace. For these operators T a characteristic function was defined in an analogous way, and by means of it a far-reaching theory of these operators was achieved; see L IV Sˇ IC [3]; M. S. B RODSKI˘I [1]–[3], and in particular B RODSKI˘I AND L IV Sˇ IC [1],

7. N OTES

283

where one also finds further references. Interesting applications to some physical problems appeared recently in the book L IV Sˇ IC [4]. One of the principal results is the construction of a concrete model for the operators considered, by means of Volterra-type integral operators and operators of multiplication by nondecreasing functions. Analogous results were obtained later for operators T such that I − T ∗ T and I − T T ∗ are of the same finite rank; see P OLJACKI˘I [1],[2]. Later the interest of this school was directed mainly to the representation of the operators by integral operators that are no longer in immediate connection with the characteristic function; see, for example, S AHNOVI Cˇ [2]–[4]; M. S. B RODSKI˘I [4],[7]; and G OHBERG AND K RE˘I N [1],[2],[6],[7]. The authors of this book came upon characteristic functions in 1962 in an entirely different way, namely as a result of the harmonic analysis of the minimal unitary dilation of c.n.u contractions T ; moreover, they obtained simultaneously a functional model of T depending explicitly and exclusively on the characteristic function of T ; see S Z .-N.–F. [2], [3], [VIII], and Secs. 2 and 3 of this chapter. 2. In the particular case of contractions T ∈ C·0 and T ∈ C0· the same functional model was also obtained by American mathematicians (cf. ROTA [1]; ROVNYAK [1]; and H ELSON [1]), who arrived at it essentially in the following direct way. Let T be a contraction on H with T ∗n → O, and let H be the space of the se2 2 2 quences h = {hn }∞ 0 with hn ∈DT ∗ and khk = ∑ khn k < ∞. As we have khk = 2 ∗ 2 kDT ∗ hk + kT hk , and consequently ∞

khk2 = ∑ kDT ∗ T ∗ j hk2 0

(h ∈ H),

(cf. Sec. I.10), we can embed H in H by identifying h ∈ H with h = {DT ∗ h, DT ∗ T ∗ h, DT ∗ T ∗2 h, . . .} ∈ H. Denoting by V the unilateral shift {h0 , h1 , . . .} → {0, h0 , h1 , . . .} on H, we have T ∗ = V ∗ |H. Take now the Fourier representation of H and V : identify H with H 2 (DT ∗ ) and V with the operator U × of multiplication by eit on this space. Then the subspace H 2 (DT ∗ ) ⊖ H is invariant for U × and hence, by Theorem V.3.3, we have H 2 (DT ∗ ) ⊖ H = Θ H 2 (F) for some inner function {F, DT ∗ , Θ (λ )}. Thus we see that T is unitarily equivalent to the operator T defined on H 2 (DT ∗ ) ⊖ Θ H 2(F) by T∗ = U ×∗ |[H 2 (DT ∗ ) ⊖ Θ H 2(F)].

(*)

This coincides with our functional model in the case T ∈ C·0 . However, in this nonelaborate form the model does not give any explicit information on Θ (λ ) as a function of T , and in particular it does not state the relation between Θ (λ ) and the characteristic function of T . 3. The functional model of Sec. 2 was originally given in S Z .-N.–F. [2], [VIII], and [IX] (the lemmas on Fourier representation were explicitly used in the third paper only). Section 3 reproduces part of the paper [VIII]. It should be noted that Theorem 3.4, which asserts that two c.n.u contractions are unitarily equivalent if and

284

C HAPTER VI. F UNCTIONAL M ODELS

only if their characteristic functions coincide, is contained in a result of Sˇ TRAUS [3],[4]. However, our Theorem 3.4 is an immediate consequence of Theorem 3.1, that is, of the functional model for c.n.u contractions, for which no analogue is at ˇ present known for the more general operators considered by Straus. Our Proposition 3.5 was first stated in S Z .-N.–F [3] and proved in [VIII]; it establishes a correspondence between the nature (inner, outer, etc.) of the characteristic function and the class Cαβ of the operator. This was made possible precisely by the manner in which we have extended the notions of inner and outer functions to operator-valued functions. Theorem 3.6 was first proved in S Z .-N.–F [11] (for Θ (λ ) and Θ ′ (λ ) inner from both sides) using entirely different methods. The special case of scalar valued functions considered in Corollary 3.7 was proved, again by different methods, in S ARA SON [3]; the investigations on the dilation of commutants started with this paper. For the case of not necessarily inner Θ (λ ) and Θ ′ (λ ), see S Z .-N.–F. [16]. In connection with Proposition 3.8 see also F UHRMANN [1],[2], where the special case of finite matrix-valued functions Θ (λ ) is considered. Theorem 4.1 on the relations between the characteristic function and the spectrum of T was first proved in S Z .-N.–F. [VIII]. In the particular case of finite rectangular matrix-valued functions these relations were found earlier by some Soviet authors; see, for example L IV Sˇ IC [1]–[3]; Sˇ MUL′ JAN [1]; B RODSKI˘I AND L IV Sˇ IC [1]; and P OLJACKI˘I [2]. In the case of operator-valued inner functions related to an operator T ∈ C·0 by the relations (∗) (and hence, by virtue of our Theorem 3.1, coinciding with the characteristic function of T ), Theorem 4.1 was found independently and almost simultaneously by Srinivasan, Wang, and Helson (cf. H ELSON [1] p. 74), and a little earlier for scalar valued inner functions by M OELLER [1]. Actually, the result of Moeller appears in the light of our Theorem 3.1 as a consequence of the corresponding result of L IV Sˇ IC [1]. Proposition 4.2 is due to G OHBERG AND K RE˘I N [5]; the present proof is new. The general criterion of Theorem 4.5 for a contraction to be similar to a unitary operator was originally published in S Z .-N.–F. [X]. This result illustrates the usefulness of both the characteristic functions and of the functional models related to them. Corollary 4.7 was found by G OHBERG –K RE˘I N as a consequence of Proposition 4.2. The results in Sec. 4.4 are new, and so is Theorem 5.1 on the contractions of class C0 . Theorem 5.2, which states that every contraction T of class C00 with finite defect indices is also of class C0 , and shows how to calculate the minimal function mT (λ ) from the characteristic function ΘT (λ ), was first given in S Z .-N.–F [VIII]. This theorem indicates a close analogy between the characteristic function ΘT (λ ) of such a contraction and the characteristic matrix T (λ ) of a finite square matrix T defined as in linear algebra, namely by T (λ ) = λ J − T . Indeed, the minimal function of the contraction T and the minimal polynomial of the matrix T have to be calculated from ΘT (λ ) and from T (λ ), respectively, in much the same way! Section 6 reproduces S Z .-N.–F [VIII] with the only difference that in Subsec. 4 we give a simpler example for a c.n.u contraction whose minimal unitary dilation is not a bilateral shift. The first example of such a contraction was given in S Z .-N.–F

7. N OTES

285

[V]; the construction there was based on a theorem of S AHNOVI Cˇ [2] (cf. B RODSKI˘I AND L IV Sˇ IC [1] p. 65). All the classes of c.n.u contractions which have been studied before yielded only examples where the minimal unitary dilation is a bilateral shift (cf. S CHREIBER [1]; DE B RUIJN [1]; and H ALPERIN [3]). 4. By virtue of the relation T (s) = es (T ) between a continuous semigroup of contractions {T (s)}s≥0 and its cogenerator T (cf. Sec. III.8), every model of T generates a model of {T (s)}. In particular, it follows from the results of Secs. 2 and 3 that the completely nonunitary continuous semigroups admit the functional model {H, T(s)} defined by ( H = [H 2 (E∗ ) ⊕ ∆ L2 (E)] ⊖ {Θ w ⊕ ∆ w : w ∈ H 2 (E)}, (a) T(s)(u∗ ⊕ v) = P(es u∗ ⊕ es v) (u∗ ⊕ v ∈ H), with P denoting orthogonal projection onto H. Here {E, E∗ , Θ (λ )} is a purely contractive, analytic, and otherwise arbitrary function on the unit disc; it coincides with the characteristic function of the cogenerator of the semigroup. For semigroups of class C·0 , that is, for which T (s)∗ → O as s → ∞, or T ∗n → O as n → ∞ (the equivalence of these conditions having been established by Proposition III.9.1), the model simplifies to the following, ( H = H 2 (E∗ ) ⊖ Θ H 2(E), (a′ ) T(s)u∗ = P(es u∗ ) (u∗ ∈ H);

Θ (λ ) is in this case an inner function. We can give to this model an alternative form in which the roles of the unit disc D and its boundary C are played by the upper half-plane and the real axis. In fact, for an arbitrary (separable) Hilbert space A, the space L2 (A) of functions u(eit ) (0 ≤ t < 2π ) is transformed unitarily onto the space A(A) = L2 (−∞, ∞; A) of functions f (x) (−∞ < x < ∞)9 by the transformation u → f , where f (x) =

1 x − i u . x+i x+i

The subspace L2+ (A) of L2 (A), consisting of the limits on C of the functions analytic on D and of class H 2 (A), is transformed thereby onto the subspace A+ (A) of A(A), consisting of the limits on the real axis of the functions f (z) that are analytic on the upper half-plane and for which sup

Z ∞

0
k f (x + iy)k2A dx < ∞;

by the theorem of Paley and Wiener the functions in A+ (A) can also be characterized as the Fourier transforms of the functions of class L2 (0, ∞; A). see, for example, H OFFMAN [1] Chaps. 7 and 8. 9

We take the measures dt/d π and dx/π , respectively.

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C HAPTER VI. F UNCTIONAL M ODELS

If we set S(z) = Θ

z − i z+i

and D(x) = [IE − S(x)∗ S(x)]1/2

the functional model of c.n.u continuous semigroups of contractions takes the following form  H = [A+ (E∗ ) ⊕ DA(E)] ⊖ {S f ⊕ D f : f ∈ A+ (E)}, (b) T (s)( f ⊕ g) = P [eisx f (x) ⊕ eisx g(x)] ( f ⊕ g ∈ H ), ∗ ∗ ∗

with P denoting orthogonal projection onto H . For semigroups of class C·0 the values of the function S(x) are a.e isometries on E, and the model (b) simplifies to  H = A+ (E∗ ) ⊖ SA+(E) (b′ ) T (s) f = P [eisx f (x)] ( f ∈ H ). ∗ ∗ ∗

The general model (b) was first formulated in F OIAS¸ [7]. The particular case (b′ ) for the class C·0 was obtained by L AX AND P HILLIPS [1]. These models can be used especially in “scattering theory”: the function S(x) appears there as the socalled “suboperator” of scattering. This was pointed out, in the case in which the particular model (b′ ) applies, in L AX AND P HILLIPS [1],[2], and in the general case in A DAMJAN AND A ROV [1], [2]. In the sequel, we continue using functional models associated with the unit disk. Our results can be transferred to the setting of the upper half-plane. In connection with this chapter see also A DAMJAN , A ROV AND K RE˘I N [1]; D OUGLAS [3]; Sˇ VARCMAN [1]; and M UHLY [1].

8 Further results 1. In the first appearance of the characteristic function, L IV Sˇ IC [1] studied a contraction operator with defect indices equal to 1 as a perturbation of a unitary operator. This also yields a model for the contraction on a function space which is useful in perturbation questions. This idea was developed independently in DE B RANGES [3]; note that the English translation of L IV Sˇ IC [1] only appeared in 1960. The case of defect indices equal to 1 was considered again in C LARK [2], where an explicit unitary equivalence is constructed between the functional model described in this chapter, and the model arising from the spectral decomposition of a unitary rank one perturbation of a given contraction. This approach was extended to general functional models in BALL AND L UBIN [1]. 2. Characteristic functions have been extended to commuting pairs of operators with finite rank imaginary parts. This extension and its function theoretical ramifications are discussed in detail in the monograph by L IV Sˇ IC ET AL . [1].

8. F URTHER RESULTS

287

3. In the context of noncommuting operators, characteristic functions were considered in F RAZHO [2]; see also P OPESCU [2],[8] for the analysis of this situation, and for further extensions. See also B HATTACHARYYA , E SCHMEIER , AND S ARKAR [1]. 4. For other approaches to functional models and their extensions see BALL [1],[2]; BALL AND K RIETE [1]; B UNCE [1]; M AKAROV AND VASYUNIN [1]; K RIETE [2]; N IKOLSKI˘I AND H RUS Sˇ Cˇ EV [1]; N IKOLSKI˘I AND T REIL [1]; and VASYUNIN [1]. 5. The similarity of operators to unitary or selfadjoint ones can be studied in terms of characteristic functions or in terms of the resolvent of the operator. For a variety of results in this direction, see VAN C ASTEREN [2]; NABOKO [1]; B ENAMARA AND N IKOLSKI˘I [1]; N IKOLSKI˘I AND T REIL [1]; and K UPIN AND T REIL [1].

Chapter VII

Regular Factorizations and Invariant Subspaces 1 The fundamental theorem 1. We continue our study of the geometric structure of the space of the minimal unitary (or isometric) dilation of a contraction T , given in Secs. II.1 and II.2.1. We now consider decompositions of this space induced by invariant subspaces of T . Thus, let T be a contraction on H, U the minimal unitary dilation of T on K (⊃ H), and U+ the minimal isometric dilation of T on K+ , where K+ =

∞ W U nH 0

and U+ = U|K+ . We recall that, according to (I.4.2), T ∗ = U+∗ |H.

(1.1)

Let us suppose, furthermore, that H1 is a subspace of H invariant for T ; H2 = H ⊖ H1 is then invariant for T ∗ , and by (1.1) invariant for U+∗ also. This implies in turn that the orthogonal complement of H2 in K+ , K′ = K+ ⊖ H2, is invariant for U+ . Let be the Wold decomposition of

(1.2)

K′ = M+ (F) ⊕ R1

K′

(1.3)

induced by the isometry U+

F = K′ ⊖ U+ K′

and R1 =

|K′ ;

here we have

∞ T U+n K′ . 0

B.Sz.-Nagy et al., Harmonic Analysis of Operators on Hilbert Space, Universitext, DOI 10.1007/978-1-4419-6094-8_7, © Springer Science + Business Media, LLC 2010

(1.4)

289

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C HAPTER VII. R EGULAR FACTORIZATIONS AND I NVARIANT S UBSPACES

As R1 reduces U+ to a unitary operator, R1 is necessarily included in the space R of the residual part of U+ ; in fact it follows from the relation K+ = M+ (L∗ ) ⊕ R,

T

see (II.2.7),

n that R = ∞ 0 U+ K+ , which implies that R is the maximal subspace of K+ on which U+ induces a unitary operator. Thus we have

R = R1 ⊕ R2 ;

(1.5)

as R and R1 reduce U+ to unitary operators, so does R2 . From (1.2), (1.3), and (II.2.7) we deduce H2 = K+ ⊖ K′ = [M+ (L∗ ) ⊕ R] ⊖ [M+(F) ⊕ R1]; by (1.5) this becomes H2 = [M+ (L∗ ) ⊕ R2] ⊖ M+ (F).

(1.6)

From (1.6) and from the analogous representation H = K+ ⊖ M+ (L) = [M+ (L∗ ) ⊕ R] ⊖ M+(L)

(1.7)

resulting from (II.2.5) and (II.2.7) we infer: H1 = H ⊖ H2 = {[M+ (L∗ ) ⊕ R] ⊖ M+(L)} ⊖ {[M+(L∗ ) ⊕ R2] ⊖ M+ (F)}. Hence

H1 = [M+ (F) ⊕ R1] ⊖ M+(L) = K′ ⊖ M+ (L).

(1.8)

These representations of H1 and H2 show that M+ (F) ⊂ M+ (L∗ ) ⊕ R2

and M+ (L) ⊂ M+ (F) ⊕ R1.

(1.9)

For every wandering subspace A for U we have M(A) =

W

n≥0

U −n M+ (A)

and furthermore U −n maps R1 and R2 onto themselves, thus (1.9) implies M(F) ⊂ M(L∗ ) ⊕ R2

and M(L) ⊂ M(F) ⊕ R1 .

(1.10)

Because R1 ⊂ R ⊥ M(L∗ ), it follows from the last relation that [M(L) ∨ M(L∗ )] ⊂ [M(F) ∨ M(L∗ )] ⊕ R1.

(1.11)

1. T HE FUNDAMENTAL THEOREM

291

If the contraction T is completely nonunitary, as we assume from now on, the left-hand side of (1.11) equals K; see (II.1.10). Thus (1.11) implies K = [M(F) ∨ M(L∗ )] ⊕ R1.

(1.12)

On the other hand K = M(L∗ ) ⊕ R (cf. (II.2.1), thus (1.12) and (1.5) yield R2 = [M(F) ∨ M(L∗ )] ⊖ M(L∗ ).

(1.13)

Let us return to the definition of F by (1.4). Because M+ (L) ⊂ K′ ⊂ K+ (cf. (1.9)), we obtain, using (1.6) as well, F = K′ ⊖ U+ K′ ⊂ K+ ⊖ U+ M+ (L) = [H ⊕ M+(L)] ⊖ U+ M+ (L), and thus (1.14)

F ⊂ H ⊕ L.

In analogy to the notations already adopted, let us denote by PL , PL∗ , PF , PR , PR1 , and PR2 the orthogonal projections of K onto M(L), M(L∗ ), M(F), R, R1 , and R2 , respectively. Because of (1.14) we infer from the decomposition (II.1.4) of K that F is orthogonal to U ν L and U −ν L∗ (ν ≥ 1), consequently U n L ⊥ U −m F

and U n F ⊥ U −m L∗

for n ≥ 0, m ≥ 1.

This implies PF M+ (L) ⊂ M+ (F) and PL∗ M+ (F) ⊂ M+ (L∗ ).

(1.15)

Observe now that (1.10) implies that for f ∈ M(F) and l ∈ M(L) we have f = PL∗ f + PR2 f ,

l = PF l + PR1 l.

(1.16)

Let us choose in particular f = PF l. The two relations (1.16) imply then l = PL∗ PF l + PR1 l + PR2 PF l

(l ∈ M(L)).

(1.17)

The first term of the right-hand side is an element of M(L∗ ), and the sum of the two other terms is an element of R1 ⊕ R2 = R. Because M(L∗ ) ⊥ R, we conclude for l ∈ M(L): P L∗ l = P L∗ P F l

(1.18) F

PR l = PR1 l + PR2 P l

(1.19)

On account of relation (II.2.1) defining R and of relations (1.10) we have PR k = (I − PL∗ )k

for k ∈ K,

(1.20)

292

and

Because

C HAPTER VII. R EGULAR FACTORIZATIONS AND I NVARIANT S UBSPACES

PR1 l = (I − PF )l for l ∈ M(L), PR2 f = (I − PL∗ ) f for f ∈ M(F).

(1.21)

PR M(L) = (I − PL∗ )M(L) = R (cf. (II.2.13)),

(1.22)

relation (1.19) implies PR1 M(L) = R1 and PR2 PF M(L) = R2 , and hence a fortiori PR1 M(L) = R1

and PR2 M(F) = R2 .

(1.23)

2. If the space H (and hence the space K also) is separable, we can give the above relations a functional form, by using the Fourier representations Φ L∗ , Φ L , and Φ F of the spaces M(L∗ ), M(L), and M(F) with respect to the bilateral shifts induced in these spaces by U. Namely, we choose the spaces and the contraction Q from one space into the other, in the three following ways (where, instead of the spaces themselves, we only indicate the generating spaces A and A′ ). (i) A = L, A′ = L∗ , Q = PL∗ |M(L) (case already considered in Sec. VI.2). (ii) A = L, A′ = F, Q = PF |M(L). (iii) A = F, A′ = L∗ , Q = PL∗ |M(F).

Q commutes with U in all three cases, because M(L∗ ), M(L), and M(F) reduce U. Thus condition (V.3.5) is fulfilled. By virtue of (II.2.12) and of relations (1.15), condition (V.3.6) is also fulfilled. Let {L, L∗ , ΘL (λ )}, {L, F, Θ1 (λ )}, {F, L∗ , Θ2 (λ )} (1.24) be the corresponding contractive analytic functions, as in Lemma V.3.1. By virtue of Sec. VI.2 (Proposition VI.2.2), the first of these functions coincides with the characteristic function of T . Relation (V.3.7) takes, respectively, the following forms:  L L Φ ∗ P ∗ l = ΘL Φ L l for l ∈ M(L),    (1.25) Φ F PF l = Θ1 Φ L l for l ∈ M(L),    L∗ L∗ Φ P f = Θ2 Φ F f for f ∈ M(F). Let us apply Φ L∗ to both sides of (1.18). Because of (1.25) we obtain

ΘL (eit )v(t) = Θ2 (eit )Θ1 (eit )v(t) (a.e.) for v ∈ L2 (L), in particular for every constant function v(t) ≡ l (l ∈ L); as L is separable this implies that ΘL (eit ) = Θ2 (eit )Θ1 (eit ) (a.e.), and consequently

ΘL (λ ) = Θ2 (λ )Θ1 (λ ) (λ ∈ D).

(1.26)

1. T HE FUNDAMENTAL THEOREM

293

Using relations (1.20) and (1.22) we obtain, just as in Sec. VI.2.1, that there exists a unique unitary operator (1.27) ΦR : R → ∆L L2 (L),

such that

ΦR PR l = ∆L Φ L l

(l ∈ M(L))

(cf. (VI.2.6)),

(1.28)

∆L denoting the operator generated on L2 (L) by the function ∆L (t) = [I − ΘL(eit )∗ΘL (eit )]1/2 . If we start with the relations (1.21) and (1.23) we obtain in the same way that there are two uniquely determined unitary operators,

ΦR1 : R1 → ∆1 L2 (L),

ΦR2 : R2 → ∆2 L2 (F),

(1.29)

such that

ΦR1 PR1 l = ∆1 Φ L l (l ∈ M(L)),

ΦR2 PR2 f = ∆2 Φ F f ( f ∈ M(F)),

(1.30)

where the operators ∆k are defined as multiplication by the functions

∆k (t) = [I − Θk (eit )∗Θk (eit )]1/2

(k = 1, 2).

Because R = R2 ⊕ R1 , the operator −1 Z = (ΦR2 ⊕ ΦR1 )ΦR

(1.31)

Z : ∆L L2 (L) → ∆2 L2 (F) ⊕ ∆1 L2 (L).

(1.32)

is also unitary; we have

Combining relations (1.28), (1.19), (1.30), and (1.25) we obtain for l ∈ M(L): Z ∆L Φ L l = Z ΦR PR l = (ΦR2 ⊕ ΦR1 )PR l = (ΦR2 ⊕ ΦR1 )(PR2 PF l ⊕ PR1 l) = ΦR2 PR2 PF l ⊕ ΦR1 PR1 l = ∆2 Φ F PF l ⊕ ∆1 Φ L l

= ∆2Θ1 Φ L l ⊕ ∆1 Φ L l;

as Φ L l runs over L2 (L) it follows that Z ∆ L v = ∆ 2Θ 1 v ⊕ ∆ 1 v

(v ∈ L2 (L)).

(1.33)

Because Z is unitary, we conclude that {∆2Θ1 v ⊕ ∆1v : v ∈ L2 (L)} = ∆2 L2 (F) ⊕ ∆1L2 (L).

(1.34)

Observe also that Z commutes with multiplication by eit . Indeed, it suffices to consider the elements ∆L v, which are dense in ∆L L2 (L), and for these elements we

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have by (1.33) Z(eit ∆L v) = Z ∆L eit v = ∆2Θ1 eit v ⊕ ∆1eit v = eit ∆2Θ1 v ⊕ eit ∆1 v = eit Z ∆L v. Now consider the unitary operator

Φ : K → K = L2 (L∗ ) ⊕ ∆L L2 (L)

(1.35)

defined in Proposition VI.2.1 (i.e. the Fourier representation of K). By (VI.2.7) and (1.31) we have

Φ = Φ L∗ ⊕ ΦR = Φ L∗ ⊕ Z −1 (ΦR2 ⊕ ΦR1 ).

(1.36)

By virtue of Proposition VI.2.1, Φ maps K+ onto K+ = H 2 (L∗ ) ⊕ ∆L L2 (L), M+ (L) onto G = {ΘL u ⊕ ∆Lu : u ∈ H 2 (L)}, and H = K+ ⊖ M+ (L) onto H = K+ ⊖ G = [H 2 (L∗ ) ⊕ ∆L L2 (L)] ⊖ {ΘLu ⊕ ∆Lu : u ∈ H 2 (L)},

(1.37)

the contraction T being transformed into the contraction T defined by T∗ (u∗ ⊕ v) = e−it [u∗ (eit ) − u∗(0)] ⊕ e−it v(t) (u∗ ⊕ v ∈ H). Let us find the images H1 and H2 of the subspaces H1 and H2 by the Fourier representation Φ = Φ L∗ ⊕ ΦR . We begin with H2 , for which we can use relation (1.6). Observe first that for r2 ∈ R2 we have

ΦR r2 = Z −1 (ΦR2 ⊕ ΦR1 )(r2 ⊕ 0) = Z −1 (ΦR2 r2 ⊕ 0),

(1.38)

and hence

Φ [M+ (L∗ ) ⊕ R2] = Φ L∗ M+ (L∗ ) ⊕ ΦRR2 = H 2 (L∗ ) ⊕ Z −1(∆2 L2 (F) ⊕ {0}). On the other hand, the first relation (1.16) implies that

Φ M+ (F) = {Φ L∗ PL∗ f ⊕ ΦR PR2 f : f ∈ M+ (F)}, whence, by virtue of (1.25), (1.30), and (1.38),

Φ M+ (F) = {Θ2u ⊕ Z −1(∆2 u ⊕ 0) : u ∈ H 2 (F)}.

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Therefore, substituting in (1.6), H2 =[H 2 (L∗ ) ⊕ Z −1 (∆2 L2 (F) ⊕ {0})] ⊖ {Θ2u ⊕ Z

−1

(1.39)

2

(∆2 u ⊕ 0) : u ∈ H (F)}.

Finally, H1 = H ⊖ H2 and ∆L L2 (L) = Z −1 (∆2 L2 (F) ⊕ ∆1 L2 (L)), thus (1.37) and (1.39) yield H1 = {Θ2 u ⊕ Z −1(∆2 u ⊕ v) : u ∈ H 2 (F), v ∈ ∆1 L2 (L)}

(1.40)

2

⊖ {ΘLw ⊕ ∆L w : w ∈ H (L)}.

We note also that because the restriction of Φ , given by (1.35), to H implements a unitary equivalence between the c.n.u. contraction T and the model operator T arising from the contractive analytic function {L, L∗ , ΘL (λ )}, it follows that every invariant subspace H1 of T is of the form (1.40) with an orthogonal complement H2 of the form (1.39), where ΘL (λ ) = Θ2 (λ )Θ1 (λ ) is a regular factorization as defined below. 3. In order to give these results a compact form, let us make some remarks and formulate some definitions. If Θ (λ ) = Θ2 (λ )Θ1 (λ ) is a factorization of a contractive analytic function {E, E∗ , Θ (λ )} into the product of contractive analytic functions {E, F, Θ1 (λ )} and {F, E∗ , Θ2 (λ )}, then we obviously have IE − Θ (eit )∗Θ (eit )

=Θ1 (eit )∗ [IF − Θ2(eit )∗Θ2 (eit )]Θ1 (eit ) + [IE − Θ1(eit )∗Θ1 (eit )], and introducing the corresponding functions ∆ (t), ∆1 (t), ∆2 (t) it follows that Z(t) : ∆ (t)g → ∆2 (t)Θ1 (eit )g ⊕ ∆1(t)g (g ∈ E)

(1.41)

is an isometry of ∆ (t)E into ∆2 (t)F ⊕ ∆1 (t)E a.e., indeed at every point such that the radial limits Θ (eit ), Θ1 (eit ), Θ2 (eit ) exist. As a consequence, Z : ∆ v → ∆ 2Θ 1 v ⊕ ∆ 1 v

(v ∈ L2 (E))

(1.42)

is an isometry of ∆ L2 (E) into ∆2 L2 (F) ⊕ ∆1 L2 (E). Completing by continuity we obtain isometries (denoted by the same letters): Z(t) : ∆ (t)E → ∆2 (t)F ⊕ ∆1 (t)E

a.e., and

Z : ∆ L2 (E) → ∆2 L2 (F) ⊕ ∆1L2 (E).

(1.41′) (1.42′)

Definition. The factorization Θ (λ ) = Θ2 (λ )Θ1 (λ ) will be said to be regular if the operator Z that it generates (in the sense of (1.42) and (1.42′)) is unitary, that is, if {∆2Θ1 u ⊕ ∆1u : u ∈ L2 (E)} = ∆2 L2 (F) ⊕ ∆1L2 (E).

(1.43)

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By this definition and by (1.34), the factorization (1.26) is regular. Now we formulate our results in terms of the functional model of c.n.u. contractions given by Theorem VI.3.1. Let H and T be the space and the c.n.u. contraction generated by the purely contractive analytic function {E, E∗ , Θ (λ )}. The function {L, L∗ , ΘL (λ )} corresponding to T coincides with {E, E∗ , Θ (λ )}; in fact, each of them coincides with the characteristic function of T, (cf. Proposition VI.2.2 and Theorem VI.3.1). One concludes readily that part (a) of the following theorem holds. Theorem 1.1. Let {E, E∗ , Θ (λ )} be a purely contractive analytic function and let T be the contraction in the space H = [H 2 (E∗ ) ⊕ ∆ L2 (E)] ⊖ {Θ w ⊕ ∆ w : w ∈ H 2 (E)}

(1.44)

T∗ (u∗ ⊕ v) = e−it [u∗ (eit ) − u∗ (0)] ⊕ e−it v(t) (u∗ ⊕ v ∈ H).

(1.45)

defined by

(a) To every subspace H1 of H, invariant for T, there corresponds a regular factorization Θ (λ ) = Θ2 (λ )Θ1 (λ ) (1.46) of {E, E∗ , Θ (λ )} into the product of contractive analytic functions, {E, F, Θ1 (λ )} and {F, E∗ , Θ2 (λ )}, such that if Z is the unitary operator (1.42′ ) corresponding to this factorization, then H1 and H2 = H ⊖ H1 have the representations H1 ={Θ2 u ⊕ Z −1(∆2 u ⊕ v) : u ∈ H 2 (F), v ∈ ∆1 L2 (E)}

(1.47)

H2 =[H 2 (E∗ ) ⊕ Z −1(∆2 L2 (F) ⊕ {0})]

(1.48)

2

⊖ {Θ w ⊕ ∆ w : w ∈ H (E)}, ⊖ {Θ2u ⊕ Z

−1

2

(∆2 u ⊕ 0) : u ∈ H (F)}.

(b) Every regular factorization of Θ (λ ) generates in this way a subspace H1 of H, invariant for T, and its orthogonal complement H2 . Proof. It only remains to prove (b). Observe that because Z is unitary, we have G2 ≡{Θ2 u ⊕ Z −1(∆2 u ⊕ v) : u ∈ H 2 (F), v ∈ ∆1 L2 (E)} ⊃ {Θ2Θ1 w ⊕ Z −1 (∆2Θ1 w ⊕ ∆1 w) : w ∈ H 2 (E)}

= {Θ w ⊕ ∆ w : w ∈ H 2 (E)} ≡ G and

[H 2 (E∗ ) ⊕ ∆ L2 (E)] ⊖ G2 = [H 2 (E∗ ) ⊕ Z −1 (∆2 L2 (F) ⊕ ∆1 L2 (E))] ⊖ G2

=[H 2 (E∗ ) ⊕ Z −1 (∆2 L2 (F) ⊕ {0})] ⊖ {Θ2u ⊕ Z −1(∆2 u ⊕ 0) : u ∈ H 2 (F)};

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the subspaces H1 , H2 defined by (1.47) and (1.48) can therefore be expressed as H1 = G2 ⊖ G,

H2 = [H 2 (E∗ ) ⊕ ∆ L2 (E)] ⊖ G2.

On account of (1.44) it follows that H = H1 ⊕ H2 . From (1.48), the elements of H2 are the orthogonal sums u∗ ⊕ Z −1 (v ⊕ 0) with u∗ ∈ H 2 (E∗ ) and v ∈ ∆2 L2 (F) such that Θ2∗ u∗ + ∆2 v ⊥ H 2 (F). The same conditions are then satisfied by u¯∗ ⊕ Z −1 (v¯ ⊕ 0) with u¯∗ (λ ) = [u∗ (λ ) − u∗ (0)]/λ and v(t) ¯ = e−it v(t), indeed we have

Θ2∗ u¯∗ + ∆2 v¯ = e−it (Θ2∗ u∗ + ∆2v) − e−it Θ2 (eit )∗ u∗ (0) ⊥ H 2 (F). Because Z commutes with multiplication by e±it , it follows that T∗ H2 ⊂ H2 . Hence TH1 ⊂ H1 . This concludes the proof of Theorem 1.1.

2 Some additional propositions 1. We begin with an addition to Theorem 1.1. Proposition 2.1. Under the conditions of Theorem 1.1, let   T X T= 1 O T2 be the triangulation of T corresponding to the decomposition H = H1 ⊕ H2 induced by the regular factorization Θ (λ ) = Θ2 (λ )Θ1 (λ ) of {E, E∗ , Θ (λ )}. Then the characteristic functions of T1 and T2 coincide with the purely contractive parts of Θ1 (λ ) and Θ2 (λ ), respectively. Proof. Let us note first that T∗2 = T∗ |H2

and T∗1 = P1 T∗ |H1 ,

where P1 denotes the orthogonal projection from H onto H1 . It is obvious that the operator Y defined by Y : u∗ ⊕ Z −1 (v ⊕ 0) → u∗ ⊕ v (u∗ ∈ H 2 (E∗ ), v ∈ ∆2 L2 (F))

(2.1)

maps the space H 2 (E∗ ) ⊕ Z −1 (∆2 L2 (F) ⊕ {0}) unitarily onto H 2 (E∗ ) ⊕ ∆2 L2 (F), and the subspace H2 of the first space onto the subspace H2 = [H 2 (E∗ ) ⊕ ∆2 L2 (F)] ⊖ {Θ2u ⊕ ∆2u : u ∈ H 2 (F)}

(2.2)

of the second one. As Z commutes with multiplication by eit , T∗ |H2 is transformed by Y to the operator T2∗ defined on H2 by T2∗ (u∗ ⊕ v2) = e−it [u∗ − u∗ (0)] ⊕ e−it v2

(u∗ ⊕ v2 ∈ H2 ).

(2.3)

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By Proposition VI.3.2(b) it follows from (2.2) and (2.3) that the characteristic function of T2 (hence that of T2 also) coincides with the purely contractive part of {F, E∗ , Θ2 (λ )}. As regards T1 , let us observe first that, because of the definition (1.47) of H1 and because Θ w ⊕ ∆ w = Θ2Θ1 w ⊕ Z −1 (∆2Θ1 w ⊕ ∆1 w), the elements of H belonging to H1 are those of the form Θ2 u ⊕ Z −1(∆2 u ⊕ v) with u ∈ H 2 (F), v ∈ ∆1 L2 (E), and

Θ1∗Θ2∗Θ2 u + Θ1∗∆22 u + ∆1v ⊥ H 2 (E);

(2.4)

the last condition obviously reduces to

Θ1∗ u + ∆1v ⊥ H 2 (E).

(2.5)

For the elements of H1 we have T∗ (Θ2 u ⊕ Z −1(∆2 u ⊕ v)) = e−it [Θ2 u − Θ2(0)u(0)] ⊕ Z −1(e−it ∆2 u ⊕ e−it v) = [Θ2 u1 ⊕ Z −1 (∆2 u1 ⊕ v1 )] + [u2 ⊕ Z −1 (v2 ⊕ 0)]

with 1 1 [u(λ ) − u(0)], u2 (λ ) = [Θ2 (λ ) − Θ2(0)]u(0), λ λ v1 (t) = e−it v(t), v2 (t) = e−it ∆2 (t)u(0).

u1 (λ ) =

On account of (2.5) we have

Θ1∗ u1 + ∆1 v1 = e−it (Θ1∗ u + ∆1v) − e−it Θ1∗ u(0) ⊥ H 2 (E), and hence Θ2 u1 ⊕ Z −1 (∆2 u1 ⊕ v1 ) ∈ H1 . On the other hand, u2 ⊕ Z −1 (v2 ⊕ 0) ∈ H2 because, obviously,

Θ2∗ u2 + ∆2 v2 = e−it u(0) − e−itΘ2∗Θ2 (0)u(0) ⊥ H 2 (F). We conclude that P1 T∗ [Θ2 u ⊕ Z −1(∆2 u ⊕ v)] = Θ2 u1 ⊕ Z −1 (∆2 u1 ⊕ v1).

(2.6)

It is obvious that the operator W defined by W : Θ2 u ⊕ Z −1(∆2 u ⊕ v) → u ⊕ v (u ∈ H 2 (F), v ∈ ∆1 L2 (E))

(2.7)

maps the set of elements Θ2 u ⊕ Z −1(∆2 u ⊕ v) unitarily onto H 2 (F) ⊕ ∆1 L2 (E), and that for w ∈ H 2 (E) we have W (Θ w ⊕ ∆ w) = W (Θ2Θ1 w ⊕ Z −1(∆2Θ1 w ⊕ ∆1w)) = Θ1 w ⊕ ∆1w.

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299

It follows that the space H1 defined by (1.47) is mapped by W onto the space H1 = [H 2 (F) ⊕ ∆1 L2 (E)] ⊖ {Θ1w ⊕ ∆1w : w ∈ H 2 (E)}.

(2.8)

By virtue of (2.6) the operator T∗1 (= P1 T∗ |H1 ) is transformed by W to the operator T1∗ defined on H1 by T1∗ (u ⊕ v) = u1 ⊕ v1 = e−it [u − u(0)] ⊕ e−it v

(u ⊕ v ∈ H1 ).

(2.9)

From (2.8) and (2.9) it follows by Proposition VI.3.2(b) that the characteristic function of T1 (hence that of T1 also) coincides with the purely contractive part of {E, F, Θ1 (λ )}. This concludes the proof of Proposition 2.1. We add that, as a consequence of formulas (2.8), (2.2), and of Proposition VI.3.2(a), H1 and H2 (and hence also H1 and H2 ) do not reduce to {0} unless Θ1 (λ ) or Θ2 (λ ), respectively, is a unitary constant. Thus Theorem 1.1 can be completed as follows. Theorem 2.2. In order that the invariant subspace H1 generated by the regular factorization Θ (λ ) = Θ2 (λ )Θ1 (λ ) be nontrivial, it is necessary and sufficient that this factorization be nontrivial (i.e., neither factor be a unitary constant). Every c.n.u. contraction T is unitarily equivalent to the contraction T generated by the characteristic function of T , thus the preceding two results yield the following theorem. Theorem 2.3. In order that the completely nonunitary contraction T on a (separable) Hilbert space have a nontrivial invariant subspace, it is necessary and sufficient that the characteristic function of T admit a nontrivial regular factorization. 2. As instructive examples, let us discuss the peculiar factorizations presented at the beginning of Sec. V.5. Recall that if {E, E∗ , Θ (λ )} is a contractive analytic function and V is a nonunitary isometric operator from E∗ into a Hilbert space F, then the factorization Θ (λ ) = Θ2 (λ )Θ1 (λ ), where Θ1 (λ ) = V Θ (λ ) and Θ2 (λ ) = V ∗ , is nontrivial. In this case, the operator-valued functions ∆1 and ∆2 defined in Sec. 1.3 are ∆ and I − VV ∗ , respectively. It follows that {∆2Θ1 u ⊕ ∆1u : u ∈ L2 (E)} = {0 ⊕ ∆ u : u ∈ L2 (E)} = {0} ⊕ ∆ L2(E) 6= L2 ((I − VV ∗ )F) ⊕ ∆ L2 (E) = = ∆2 L2 (F) ⊕ ∆1 L2 (E)

and thus the factorization is not regular. Recall also that if N(Θ ˜) 6= {0} (see again Sec. V.5) then Θ (λ ) = (I − P∗ + λ P∗ )Θ (λ ), where P∗ denotes the orthogonal projection of E∗ onto N(Θ ˜), is a nontrivial factorization. In this case Θ2 (λ ) = I − P∗ + λ P∗ , Θ1 (λ ) = Θ (λ ), ∆2 = 0 and ∆1 = ∆ . Clearly, this factorization is regular. It is worth observing that in view of Proposition 2.1 the restriction T1 of T to the invariant subspace H1 corresponding to this factorization is unitarily equivalent to T; moreover the compression T2 of T to

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the space H2 = H ⊖ H1 is the null operator. Note that the other two related examples in Sec. V.5 can be reduced to the ones already treated above by considering Θ ˜(λ ) instead of Θ (λ ) and by referring to the general Proposition 3.2 in the next section. 3. Another natural question related to Sec. V.5 is the relation between regular and strictly nontrivial factorizations. A partial answer is given by the following result. Proposition 2.4. (a) There exist nontrivial regular factorizations that are not strictly nontrivial. (b) There exist strictly nontrivial factorizations that are not regular. Proof. Part (a): Take E∗ = E ⊕ E and Θ (λ )e = e ⊕ 0, e ∈ E. Then      I I 0 I = Θ2 (λ )Θ1 (λ ) with Θ1 (λ ) = Θ (λ ). Θ (λ ) = = 0 0λ 0 Note that Θ (λ ) is a constant (in λ ) nonunitary isometry. Therefore, as we pointed out in Sec. V.5, Θ (λ ) has no strictly nontrivial factorization. Plainly, the above factorization is regular. Part (b): Let Θ (λ ) = Θ2 (λ )Θ1 (λ ) be the strictly nontrivial factorization presented in Theorem V.5.5. Recall that Θ and Θ1 are inner and that Θ2 is outer. Therefore ∆ = 0, ∆1 = 0 and consequently (see Sec. 1.3) {∆2Θ1 u ⊕ ∆1u : u ∈ L2 (E)} = Z{0} = {0} ⊕ {0},

∆2 L2 (E) ⊕ ∆1L2 (E) = ∆2 L2 (E) ⊕ {0}.

If this factorization is regular then we must have ∆2 L2 (E) = {0}, which is only possible if ∆2 (t) = 0 a.e., that is, if Θ2 is inner. Because Θ2 is also outer it follows that Θ2 is a constant unitary operator contradicting the fact that the factorization is strictly nontrivial. 4. Let us make a remark concerning the Beurling, Lax, and Halmos theorem (see Theorem V.3.3). Recall that this theorem states that if U+× is the unilateral shift on H 2 (E) (i.e., (U+× h)(λ ) = λ h(λ ), |λ | < 1, h ∈ H 2 (E)) then all its invariant subspaces H1 are of the form Θ2 H 2 (F) where {F, E, Θ2 (λ )} is some inner function. To derive this fact within the present approach observe first that the characteristic function of U+× is {{0}, E, Θ (λ )}, where Θ (λ ) ≡ 0. The regular factorizations of this function are of the form Θ (λ ) = 0 = Θ2 (λ )Θ1 (λ ), where {F, E, Θ2 (λ )} is inner and Θ1 (λ ) is the zero operator from {0} to F. It is easy to check that the formula (1.47) yields the invariant subspace H1 = Θ2 H 2 (F). 5. Finally, we point out the existence of strange factorizations. A factorization Θ (λ ) = Θ2 (λ )Θ1 (λ ) is said to be strange if it is not regular, but there exists a regular factorization Θ (λ ) = Θ2′ (λ )Θ1′ (λ ) such that the pure part of Θ j coincides with the pure part of Θ ′j for j = 1, 2. Consider a separable, infinite-dimensional Hilbert space E, and two isometries U,V on E with orthogonal ranges; thus V ∗U = O. We have then Θ (λ ) = Θ2 (λ )Θ1 (λ ), where {E, E, Θ (λ )}, {E, E, Θ1 (λ )}, and

3. R EGULAR FACTORIZATIONS

301

{E, E, Θ2 (λ )} are defined by Θ (λ ) = O, Θ1 (λ ) = U, and Θ2 (λ ) = V ∗ for λ ∈ D. The pure part of Θ1 is {{0}, E⊖UE, O}, and the pure part of Θ2 is {E⊖V E, {0}, O}. Observe that

∆2 L2 (E) ⊕ ∆1 L2 (E) = L2 ((I − VV ∗ )E) ⊕ {0} = L2 ((V E)⊥ ) ⊕ {0} and {∆2Θ1 u ⊕ ∆1u : u ∈ L2 (E)} = L2 ((I − VV ∗ )UE) ⊕ {0} = L2 (UE) ⊕ {0}. We conclude that this factorization is regular if and only if UE +V E = E. Note that the pure part of Θ1 coincides with {{0}, E, O}, and similarly the pure part of Θ2 does not depend on V up to coincidence. Thus, these pure parts coincide with the pure parts of the factors of a regular factorization of Θ obtained when UE+V E = E. Therefore the factorization is strange if this equality does not hold. In either case we can construct, as in the proof of Theorem 1.1, spaces H = H 2 (E) ⊕ [L2 (kerV ∗ ) ⊖ H 2 (UE)],

H1 = H 2 (E) ⊕ [H 2 (kerV ∗ ) ⊖ H 2(UE)],

H2 = {0} ⊕ [L2(kerV ∗ ) ⊖ H 2(kerV ∗ )] satisfying H = H1 ⊕ H2, and an operator T on H defined by T∗ ( f ⊕ g)(eit ) = e−it [ f (eit ) − f (0)] ⊕ e−it g(eit )

(eit ∈ C, f ⊕ g ∈ H).

It is easy to see that, relative to the decomposition H = H1 ⊕ H2 , we have the matrix representation   T1 X , T= O T2

where T1 and T∗2 are unilateral shifts of infinite multiplicity. Observe that the unitary summand of T is multiplication by eit on the space {0} ⊕ L2 (kerV ∗ ⊖UE). Hence T is completely nonunitary if and only if UE +V E = E, and in this case T = T1 ⊕ T2 . Even when UE + V E 6= E, the completely nonunitary summand of T is unitarily equivalent to T1 ⊕ T2 , but different from it because X 6= O in this case. (See also the notes concerning the general case of nonregular factorizations.)

3 Regular factorizations 1. Before continuing the investigation of the relations between invariant subspaces and regular factorizations, we need a closer look at the notion of regular factorization. Beside the notion of regularity of a factorization of a contractive analytic function, defined in Sec. 1, we also introduce the notion of regularity of a factorization of an individual contraction.

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Let A = A2 A1 be an arbitrary factorization of a contraction A from a space A into a space A∗ , as a product of a contraction A1 from A into a space B and of a contraction A2 from B into A∗ . From the obvious relation IA − A∗ A = A∗1 (IB − A∗2 A2 )A1 + (IA − A∗1A1 ) we infer that the transformation Z : Da → D2 A1 a ⊕ D1a (a ∈ A)

(3.1)

is isometric; D, D1 , and D2 denote here the defect operators: D = DA ,

D 1 = DA 1 ,

D 2 = DA 2 .

Completing by continuity we obtain an isometry (denoted by the same letter) Z : DA → D2 B ⊕ D1 A.

(3.2)

Definition. The factorization A = A2 A1 is said to be regular if the corresponding operator Z is (not only isometric, but also) unitary, that is, if {D2 A1 a ⊕ D1a : a ∈ A} = D2 B ⊕ D1 A.

(3.3)

Observe that there is an immediate relation between the two notions introduced above: regularity of the factorization Θ (λ ) = Θ2 (λ )Θ1 (λ ) for contractive analytic functions means regularity of the factorization Θ = Θ2Θ1 for the contractions on the respective L2 spaces. A less immediate relation is established in the following proposition (“local characterization” of regular factorizations of functions). Proposition 3.1. In order that the (functional) factorization

Θ (λ ) = Θ2 (λ )Θ1 (λ ) (|λ | < 1)

(3.4)

be regular it is necessary and sufficient that the (individual) factorization

Θ (eit ) = Θ2 (eit )Θ1 (eit ) (0 ≤ t ≤ 2π ) be regular a.e. Proof. We set and

B = {∆2Θ1 v ⊕ ∆1v : v ∈ L2 (E)} B(t) = {∆2 (t)Θ1 (eit )e ⊕ ∆1(t)e : e ∈ E}.

(3.5)

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Our proposition means then that the following two conditions are equivalent, B = ∆2 L2 (F) ⊕ ∆1L2 (E),

(3.6)

B(t) = ∆2 (t)F ⊕ ∆1 (t)E a.e.,

(3.7)

the closures being taken in L2 (F) ⊕ L2 (E) and in F ⊕ E, respectively. Let us suppose first that (3.6) holds. Considering in particular two constant functions: e(t) ≡ e, f (t) ≡ f (e ∈ E, f ∈ F), we deduce from (3.6) that there exists a sequence {vn } of elements of L2 (E) such that ∆2Θ1 vn ⊕ ∆1 vn tends to ∆2 f ⊕ ∆1 e in L2 (F) ⊕ L2 (E); replacing this sequence if necessary by a suitable subsequence, we can also require that ∆2 (t)Θ1 (eit )vn (t) ⊕ ∆1 (t)vn (t) tends to ∆2 (t) f ⊕ ∆1 (t)e a.e. (in the metric of F ⊕ E). Thus ∆2 (t) f ⊕ ∆1 (t)e ∈ B(t) a.e., the set of the exceptional points t depending on e and f . Let e and f run through two countable sets, say {en } and { fm }, dense in E and in F, respectively. Taking the union of the corresponding exceptional sets we obtain that ∆2 (t) fm ⊕ ∆1 (t)en ∈ B(t) holds for every point t with the possible exception of a set of zero measure, independent of m and n. For the nonexceptional t we have thus

∆2 (t)F ⊕ ∆1 (t)E ⊂ B(t), and this implies (3.7). Let us suppose now, conversely, that (3.7) holds a.e. Consider an element v2 ⊕ v1 ∈ ∆2 L2 (F) ⊕ ∆1 L2 (E)

(3.8)

that is orthogonal to B, thus (v2 , ∆2Θ1 v) + (v1 , ∆1 v) = 0 for each Then we have

v ∈ L2 (E).

Θ1 (eit )∗ ∆2 (t)v2 (t) + ∆1(t)v1 (t) = 0 a.e.,

and consequently, for the nonexceptional t, v2 (t) ⊕ v1(t) ⊥ B(t).

(3.9)

Now it is obvious that condition (3.8) implies v2 (t) ⊕ v1 (t) ∈ ∆2 (t)F ⊕ ∆1 (t)E a.e.

(3.10)

From (3.7), (3.9), and (3.10) we deduce that v2 (t) ⊕ v1 (t) = 0 a.e. and thus v2 ⊕ v1 = 0. This implies (3.6). 2. The following proposition lists some useful properties of regular factorizations of contractions.

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Proposition 3.2. Let A = A2 A1 be a factorization of the contraction A (from A into A∗ ) as a product of a contraction A1 (from A into B) and a contraction A2 (from B into A∗ ). (a) If this factorization is regular then so is the factorization A∗ = A∗1 A∗2 . (b) The factorization A = A2 A1 is regular whenever A2 or A∗1 is isometric. (c) If A is isometric (unitary) then the factorization A = A2 A1 is regular if and only if A1 and A2 are isometric (unitary) also. (d) We always have dim DA ≤ dim D2 B + dimD1 A. (3.11) Equality holds for regular factorizations and, if dim DA < ∞, only for regular factorizations.

Proof. Part (a): The proof of this assertion is rather laborious. Let Z∗ denote the (isometric) transformation generated by the factorization A∗ = A∗1 A∗2 (with the order of the component spaces reversed), that is, Z∗ : D∗ A∗ → D∗2 A∗ ⊕ D∗1 B

(3.12)

is obtained by completing the transformation Z∗ : D∗ a∗ → D∗2 a∗ ⊕ D∗1 A∗2 a∗

(a∗ ∈ A∗ ).

(3.13)

Here we have set D∗ = DA∗ , D∗1 = DA∗1 , D∗2 = D∗A2 . Let us set S = Z∗ AZ ∗ . We have then for a ∈ A: S(D2 A1 a ⊕ D1a) = Z∗ AZ ∗ ZDa = Z∗ ADa = Z∗ D∗ Aa = D∗2 Aa ⊕ D∗1A∗2 Aa = D∗2 A2 A1 a ⊕ D∗1A∗2 A2 A1 a = A2 D2 A1 a ⊕ [A1D1 a − D∗1D2 (D2 A1 a)]

and hence S(u2 ⊕ u1) = A2 u2 ⊕ (A1 u1 − D∗1 D2 u2 )

(3.14)

for u2 ⊕ u1 ∈ Z(DA). Suppose that the factorization A = A2 A1 is regular. The validity of (3.14) extends then by continuity to all elements u2 ⊕ u1 of D2 B ⊕ D1 A. We recall that the relation DT ∗ = T DT ⊕ NT ∗

(cf. (I.3.7))

(3.15)

holds for every contraction T from a space into itself or into some other space. This gives, for T = A, D∗ A∗ = ADA ⊕ NA∗ ;

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from this we get, using (3.14), Z∗ D∗ A∗ = Z∗ ADA ⊕ Z∗ NA∗ = SZDA ⊕ Z∗ NA∗ (3.16)

= S(D2 B ⊕ D1A) ⊕ Z∗ NA∗

= {A2 u2 ⊕ (A1 u1 − D∗1 D2 u2 ) : u1 ∈ D1 A, u2 ∈ D2 B} ⊕ Z∗ NA∗ . Let a′∗ ⊕ b′ be an element of D∗2 A∗ ⊕ D∗1 B orthogonal to Z∗ D∗ A∗ . As a consequence of (3.16), it is then orthogonal in particular (case u2 = 0) to the elements of the form 0 ⊕ A1 u1 with u1 ∈ D1 A. Applying (3.15) to T = A∗1 we obtain D1 A = A∗1 D∗1 B ⊕ NA1 and hence A1 D1 A = A1 A∗1 D∗1 B. We conclude that b′ ⊥ A1 A∗1 D∗1 B and hence A∗1 b′ ⊥ A∗1 D∗1 B. On the other hand A∗1 b′ ∈ A∗1 D∗1 B, thus A∗1 b′ = 0.

(3.17)

This relation implies D2∗1 b′ = b′ − A1A∗1 b′ = b′ and consequently D∗1 b′ = b′ .

(3.18)

By (3.16), a′∗ ⊕ b′ is also orthogonal to A2 u2 ⊕ (−D∗1 D2 u2 ) for every u2 ∈ D2 B (case u1 = 0). On account of (3.18) we deduce from this that A∗2 a′∗ − D2b′ ⊥ D2 B, and hence

D2 (A∗2 a′∗ − D2 b′ ) ⊥ B,

D2 A∗2 a′∗ − D22 b′ = 0.

Finally, (3.16) shows that a′∗ ⊕ b′ is orthogonal to Z∗ NA∗ , that D∗2 a∗ ⊕ D∗1 A∗2 a∗ with a∗ ∈ NA∗ . From this and from (3.18), D∗2 a′∗ + A2 b′ ⊥ NA∗ .

(3.19) is, to the elements (3.20)

But A∗ (D∗2 a′∗ + A2 b′ ) = A∗1 (A∗2 D∗2 a′∗ + A∗2A2 b′ ) = A∗1 (D2 A∗2 a′∗ − D22 b′ + b′ ) = 0 (cf. (3.17) and (3.19)); thus D∗2 a′∗ + A2 b′ ∈ NA∗ . This result and (3.20) imply

and consequently,

D∗2 a′∗ + A2b′ = 0,

(3.21)

A∗2 D∗2 a′∗ + A∗2A2 b′ = 0.

(3.22)

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Because A∗2 D∗2 = D2 A∗2 , (3.19) and (3.22) give b′ = D22 b′ + A∗2A2 b′ = 0. So we obtain, from (3.21), D∗2 a′∗ = 0; as a′∗ ∈ D∗2 A we conclude that a′∗ = 0. Thus the only element a′∗ ⊕ b′ of D∗2 A∗ ⊕ D∗1 B that is orthogonal to Z∗ D∗ A∗ , is zero. Hence Z∗ maps D∗ A∗ onto D∗2 A∗ ⊕ D∗1 B, showing that the factorization A∗ = A∗1 A∗2 is regular. Part (b): Due to the duality expressed by (a), it suffices to consider the case of an isometric A2 . Then D2 = O, and hence ZDA = ZDA = {D2 A1 a ⊕ D1a : a ∈ A} = {0 ⊕ D1a : a ∈ A} = {0} ⊕ D1A = D2 B ⊕ D1 A,

and this proves our assertion. Part (c): If A is isometric, then D = O. Now the transformation Z from DA = {0} into D2 B ⊕ D1 A is unitary if and only if D2 = O and D1 = O, that is, if A2 and A1 are isometric. The assertion for unitary A follows because of the duality expressed by (a). Part (d): Inequality (3.11) results immediately from the fact that the transformation Z generated by the factorization A = A2 A1 is an isometry. If Z is even unitary (i.e., if the factorization is regular), then we have equality in (3.11). On the other hand, if we have dim D2 B + dim D1 A = dim DA < ∞, (3.23) then Z is an isometric transformation from a finite dimensional space into a space of the same dimension, and hence it is necessarily unitary: the factorization is regular. This completes the proof of Proposition 3.2. Here are some immediate consequences of Propositions 3.1 and 3.2. Proposition 3.3. Let

(F) Θ (λ ) = Θ2 (λ )Θ1 (λ )

be a factorization of the contractive analytic function {E, E∗ , Θ (λ )} into the product of the contractive analytic functions {E, F, Θ1 (λ )} and {F, E∗ , Θ2 (λ )}, and let (F˜) Θ ˜(λ ) = Θ 1˜ (λ )Θ 2˜ (λ ) be its dual. (a) If (F) is regular then so is (F˜). (b) (F) is regular in each of the following cases. (i) Θ2 (λ ) is an inner function. (ii) Θ1 (λ ) is an ∗-inner function. (iii) At almost every point t where Θ2 (eit ) is not isometric, Θ1 (eit )∗ is isometric.

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(c) When Θ (λ ) is inner (inner from both sides), (F) is regular if and only if the factors are inner (inner from both sides) also. (d) For δ (t) = dim ∆ (t)E, δ1 (t) = dim ∆1 (t)E, δ2 (t) = dim ∆2 (t)F we have

δ (t) ≤ δ1 (t) + δ2(t) a.e.

(3.24)

Equality holds (a.e.) for regular (F) and, if δ (t) < ∞ a.e., only for regular (F). 3. By virtue of part (b) of this proposition, the canonical and the ∗-canonical factorizations of Θ (λ ) introduced in Sec. V.4.3, as well as the factorizations considered in Proposition V.4.3, are regular factorizations. In contrast, the factorization considered in Theorem V.5.5 is not regular; this follows from (c) and from Proposition V.2.3. The regular factorizations considered in Proposition V.4.3 are used in Sec. 5. Let us suppose now that Θ (λ ) coincides with the characteristic function of the c.n.u. contraction T , and let Θ (λ ) = Θi (λ )Θe (λ ) be the canonical factorization of Θ (λ ). This factorization is regular, therefore it induces, according to Theorem 1.1 and Proposition 2.1, a triangulation   T1 X T= O T2 of the functional model of T such that the characteristic functions of T1 and T2 coincide with the purely contractive parts of Θe (λ ) and Θi (λ ), respectively. Now it is obvious that the purely contractive part of an outer or inner function is also outer or inner, respectively. Thus by applying Proposition VI.3.5 we obtain that T1 ∈ C·1 and T2 ∈ C·0 . Because T is unitarily equivalent to its functional model T, this triangulation of T induces a triangulation of T of the same type. Analogous reasoning applies to the ∗-canonical factorization of Θ (λ ), which is also regular. We have proved the following proposition. Proposition 3.4. For a c.n.u. contraction T in the (separable) space H, the canonical and the ∗-canonical factorizations of the characteristic function of T induce the triangulations of type     C·1 ∗ C0· ∗ and , O C·0 O C1· respectively (cf. Sec. II.4).

4. Let us consider now, as an example, the case of a purely contractive analytic scalar function Θ (λ ), namely the case E = E∗ = E 1 . Then δ (t) = 0 or 1 according to |Θ (eit )| = 1 or < 1. Let Θ (λ ) = Θ2 (λ )Θ1 (λ ) be a factorization of Θ (λ ) as a product of the contractive analytic functions {E 1 , F, Θ1 (λ )} and {F, E 1 , Θ2 (λ )}. By virtue of Proposition 3.3(d), this factorization is regular if and only if

δ1 (t) + δ2 (t) = δ (t) (≤ 1) a.e.

(3.25)

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Let us observe that, for f ∈ F and for a.e. t, f = [IF − Θ2(eit )∗Θ2 (eit )] f + Θ2(eit )∗Θ2 (eit ) f it ∗

it

(3.26)

it ∗ 1

∈ [IF − Θ2(e ) Θ2 (e )]F ∨ Θ2 (e ) E

and that dim [IF − Θ2(eit )∗Θ2 (eit )]F = δ2 (t),

dim Θ2 (eit )∗ E 1 ≤ 1.

(3.27)

From (3.25)–(3.27) we infer that dim F ≤ 2, and hence, up to an isomorphism, F = E 0 (= {0}), or F = E 1 , or F = E 2 . The case F = E 0 can occur only if Θ (λ ) ≡ O, and then the factorization OE 1 →E 1 = OE 0 →E 1 · OE 1 →E 0

(3.28)

is obviously regular (δ (t) ≡ 1, δ1 (t) ≡ 1, δ2 (t) ≡ 0). (We suggest to the reader as an interesting exercise to find the contraction T and the subspace invariant for T , corresponding to the characteristic function {E 1 , E 1 , O} and to its regular factorization (3.28).) When F = E 1 the functions Θ1 (λ ) and Θ2 (λ ) are also scalar. Because δk (t) = 0 or 1 accordingly as |Θk (eit )| = 1 or < 1 (k = 1, 2), condition (3.25) means that, at almost every point t such that |Θ (eit )| = 1, we have |Θk (eit )| = 1 for k = 1 and k = 2, and at almost every point t such that |Θ (eit )| < 1, we have |Θk (eit )| = 1 for one of the indices k = 1, 2. Finally, if F = E 2 , the operator Θ2 (eit ) (from E 2 into E 1 ) cannot be isometric; thus we have δ2 (t) > 0. Hence in this case (3.25) means that δ2 (t) = 1, δ1 (t) = 0, δ (t) = 1, a.e. Thus Θ1 (eit ) is isometric a.e. If we represent Θ2 (λ ) by its matrix [ϑ21 (λ ) ϑ22 (λ )], the operator IF − Θ2(eit )∗Θ2 (eit ) is represented by the matrix   1 − |a|2 −ab ¯ with a = ϑ21 (eit ), b = ϑ22 (eit ). −ab¯ 1 − |b|2 As

dim[IF − Θ2(eit )∗Θ2 (eit )]F = δ2 (t) = 1 < dim F,

this matrix has determinant 0 and hence |a|2 + |b|2 = 1; consequently Θ2 (eit )∗ is isometric. Thus if F = E 2 , condition (3.25) implies that Θ1 (λ ) is inner and Θ2 (λ ) is ∗-inner. Moreover, in this case δ (t) = 1 and hence |Θ (eit )| < 1 a.e. Conversely, the factorizations of this type (i.e., with |Θ (eit )| < 1 a.e., Θ1 (λ ) inner and Θ2 (λ ) ∗-inner) are regular, because then δ (t) = 1, δ1 (t) = 0, and δ2 (t) = 1 a.e. So we have arrived at the results summarized below. Proposition 3.5. Let Θ (λ ) = Θ2 (λ )Θ1 (λ ) be a factorization of the scalar-valued contractive analytic function Θ (λ ) as a product of two contractive analytic functions. This factorization is regular if and only if one of the following cases occurs. (i) The trivial case (3.28).

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(ii) The factors are scalar-valued and, a.e., at least one of the values |Θk (eit )| (k = 1, 2) equals 1. (iii) |Θ (eit )| < 1 a.e., and the factors are of the form   ϑ (λ ) Θ1 (λ ) = 11 , Θ2 (λ ) = [ϑ21 (λ ) ϑ22 (λ )] ϑ12 (λ ) with scalar-valued functions ϑik (λ ) ∈ H ∞ such that |ϑ11 (eit )|2 + |ϑ12 (eit )|2 = 1

and |ϑ21 (eit )|2 + |ϑ22 (eit )|2 = 1 a.e.

In the last case, the factorization is of the form

Θ (λ ) = ϑ21 (λ )ϑ11 (λ ) + ϑ22(λ )ϑ12 (λ ). An example for case (iii) is given by   λ √  √  3 3 λ ·  √2  . λ=  2 3 2 2 2

5. We conclude the section with a proposition, which does not deal with regular factorizations, but presents an interesting consequence of Propositions 2.1 and 3.3(d). Proposition 3.6. Let T be a c.n.u. contraction in H, with (finite or infinite) defect indices dT and dT ∗ . Let H1 (6= {0}) be a subspace of H, invariant for T . Then the defect indices of the operator T1 = T |H1 satisfy the inequalities dT1 ≤ dT

and dT1∗ ≤ dT + dT ∗ .

Proof. Let {E, E∗ , Θ (λ )} be a contractive analytic function coinciding with the characteristic function of T . Let Θ (λ ) = Θ2 (λ )Θ1 (λ ) be the regular factorization of Θ (λ ) into the product of the contractive analytic functions {E, F, Θ1 (λ )} and {F, E∗ , Θ2 (λ )}, corresponding to the invariant subspace H1 in the sense of Theorem 1.1. According to Proposition 3.3(d) we have then dim ∆ (t)E = dim ∆2 (t)F + dim ∆1 (t)E a.e., and consequently dim ∆2 (t)F ≤ dim ∆ (t)E ≤ dim E = dT

a.e.

(3.29)

On the other hand, the relation IF = ∆2 (t)2 + Θ2 (eit )∗Θ2 (eit ) implies that, for almost every t, F is spanned by ∆2 (t)F and Θ2 (eit )∗Θ2 (eit )F, and hence dim F ≤ dim ∆2 (t)F + dim Θ2 (eit )∗Θ2 (eit )F.

(3.30)

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As Θ2 (eit )F ⊂ E∗ , we have dim Θ2 (eit )F ≤ dim E∗ = dT ∗ ; consequently, dim Θ2 (eit )∗Θ2 (eit )F ≤ dim Θ2 (eit )F ≤ dT ∗ .

(3.31)

From (3.29)–(3.31) it follows that dim F ≤ dT + dT ∗ . Now we know (cf. Proposition 2.1) that the characteristic function of T1 coincides with the purely contractive part {E0 , F0 , Θ10 (λ )} of {E, F, Θ1 (λ )}. Because E0 ⊂ E and F0 ⊂ F, we have dT1 = dim E0 ≤ dim E = dT ,

dT1∗ = dim F0 ≤ dim F ≤ dT + dT ∗ , and this completes the proof.

4 Arithmetic of regular divisors 1. We begin with a definition. Definition. Let A and A1 be contractions from the space A into the spaces A∗ and B, respectively. We say that A1 is a divisor of A if there exists a contraction A2 from B into A∗ such that A = A2 A1 . If A2 can be chosen so that this factorization is regular then we call A1 a regular divisor of A. This definition extends immediately to contractive analytic functions Θ (λ ), Θ1 (λ ), Θ2 (λ ) by considering the corresponding contractions Θ , Θ1 , Θ2 , in the respective L2 spaces. One of the first problems in this connection is that of the transitivity of regular divisors. This requires finding regularity relations between several factorizations. The most important case is that of the factorizations (F21 ) A21 = A2 A1 ,

(F3(21) ) A = A3 A21 ,

(F32 ) A32 = A3 A2 ,

(F(32)1 ) A = A32 A1

induced by a factorization (F321 ) A = A3 A2 A1 of a contraction A : A → A∗ into the product of three contractions: A1 : A → B1 ,

A2 : B1 → B21 ,

A3 : B21 → A∗ .

We need a preliminary result. Lemma 4.1. The factorizations (F21 ), (F3(21) ), (F32 ), (F(32)1 ) are all regular if one of the following conditions holds. (i) (F21 ) and (F3(21) ) are regular.

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(ii) (F32 ) and (F(32)1 ) are regular. (iii) (F3(21) ) and (F(32)1 ) are regular. Proof. For now let us denote by D, D1 , . . . , D32 the defect operators DA , DA1 , . . . , DA32 , and let us consider the isometric transformations Z21 , Z3(21) , and so on, corresponding to the factorizations (F21 ), (F3(21) ), and so on, in the sense of (3.1)– (3.2). Thus we have Z21 : D21 a → D2 A1 a ⊕ D1a Z3(21) : Da → D3 A21 a ⊕ D21a Z32 : D32 b → D3 A2 b ⊕ D2b Z(32)1 : Da → D32 A1 a ⊕ D1a

(a ∈ A), (a ∈ A), (b ∈ B1 ), (a ∈ A).

Similarly, let us attach to the factorization (F321 ) the transformation Z321 :

Da → D3 A21 a ⊕ D2A1 a ⊕ D1a

(a ∈ A),

which is also isometric. We use the same letters to denote the corresponding closed isometries. The relations between these isometric transformations are indicated by the following commutative diagram, the arrows denoting transformation from one space into the other. D3 B21 ⊕ D21 A

 I3(21) ⊕ Z21

ր Z3(21) 

DA −−−−−−−−−→ Z321

ց

D3 B21 ⊕ D2 B1 ⊕ D1 A



(I3(21) = ID3 B21 , I1 = ID1 A ).

ր

Z(32)1 ց

Z32 ⊕ I1 

D32 B1 ⊕ D1 A

Commutativity of this diagram means (I3(21) ⊕ Z21 )Z3(21) = Z321 = (Z32 ⊕ I1 )Z(32)1 ; these equations derive at once from the definitions. Let us assume first that (F3(21) ) and (F21 ) are regular; thus Z3(21) and Z21 (hence also I3(21) ⊕ Z21 ) are unitary. By the commutativity of the diagram, Z321 is then also unitary and so are Z(32)1 and Z32 ⊕ I1 , and hence Z32 too. This proves that (F(32)1 ) and (F32 ) are regular. The case when (F(32)1 ) and (F32 ) are regular can be treated similarly.

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Next we suppose that (F3(21) ) and (F(32)1 ) are regular. These assumptions imply the relations  (I3(21) ⊕ Z21 )Z3(21) DA = (I3(21) ⊕ Z21 )(D3 B21 ⊕ D21A)    = D3 B21 ⊕ Z21 D21 A, Z321 DA = (4.1) DA = (Z ⊕ I )Z (Z  32 ⊕ I1 )(D32 B1 ⊕ D1 A) 32 1 (32)1   = Z32 D32 B1 ⊕ D1 A, where

Z21 D21 A ⊂ D2 B1 ⊕ D1 A,

From (4.1) and (4.2) we deduce that

Z32 D32 B1 ⊂ D3 B21 ⊕ D2 B1 .

Z321 DA = D3 B21 ⊕ B2 ⊕ D1 A,

(4.2)

(4.3)

with B2 ⊂ D2 B1 and with D3 B21 ⊕ B2 = Z32 D32 B1 = {D3 A2 b ⊕ D2b : b ∈ B1 }. Hence we obtain B2 = D2 B1 , and thus, from (4.3), Z321 DA = D3 B21 ⊕ D2 B1 ⊕ D1 A, in other words, Z321 is unitary. This implies that all the isometric transformations occurring in our commutative diagram are unitary; hence all the factorizations are regular. This concludes the proof of Lemma 4.1. Alternatively, we can formulate this lemma in the following manner. Proposition 4.2. Let A, A21 , and A1 be contractions. (a) If A21 is a regular divisor of A and if A1 is a regular divisor of A21 , then A1 is a regular divisor of A. (b) If A21 and A1 are regular divisors of A, if A1 is a divisor of A21 , and if A1 has dense range, then A1 is a regular divisor of A21 . Proof. In fact, (a) corresponds to the case of Lemma 4.1 where the factorizations (F21 ) and (F3(21) ) are assumed to be regular. Under the conditions of (b) there exist contractions A2 , A3 , and A′ such that the factorizations A = A3 A21 , A = A′ A1 are regular, and A21 = A2 A1 . Because A′ A1 = A = A3 A21 = A3 A2 A1 and A1 has dense range, it follows that A′ = A3 A2 . Thus the factorizations (A3 A2 )A1 = A = A3 (A2 A1 ) of type (F(32)1 ) and (F3(21) ), respectively, are regular. Hence Lemma 4.1 implies that the factorization A21 = A2 A1 of type (F21 ) is also regular; that is, A1 is a regular divisor of A21 . We note that the density of the range of A1 cannot be omitted in (b). To construct an example we use the unilateral shift S on H 2 (i.e., the operator of multiplication by λ ). Choose A = S3 , A1 = A3 = S2 , and A21 = A23 = S. The factorizations A = A3 A21 = A23 A1 are obviously regular. Because A21 = S∗ A1 , A1 is a divisor of A21 .

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313

We claim that no factorization A21 = A′ A1 , with kA′ k ≤ 1, is regular. Assume to the contrary that such a factorization S = A′ S2 is regular. Proposition 3.2 implies that the factorization S∗ = S∗2 A′∗ is regular as well. The same proposition implies that 1 = dim DS∗ H 2 = dim DS∗2 H 2 + dimDA∗ H 2 ≥ 2, a contradiction. Thus A1 is not a regular divisor of A21 . 2. Let {E, E∗ , Θ (λ )} be a purely contractive analytic function, and let H and T be the space and the contraction corresponding to Θ (λ ) in the sense of (1.44) and (1.45). Definition. Let Θ1 (λ ) be a regular divisor of Θ (λ ). A subspace H1 of H is called an invariant subspace associated with Θ1 (λ ) if it derives by formula (1.47) from a regular factorization Θ2 (λ )Θ1 (λ ) of Θ (λ ), with the given factor Θ1 (λ ). Remark. We show later that there can exist different invariant subspaces associated with the same regular divisor Θ1 (λ ) of Θ (λ ). Here are the important relations between the regular divisors of Θ (λ ) and the invariant subspaces associated with them. Theorem 4.3. (a) Let Θ1 (λ ) and Θ1′ (λ ) be two regular divisors of Θ (λ ) such that we have H1 ⊂ H′1 for some invariant subspaces associated with Θ1 (λ ) and with Θ1′ (λ ), respectively. Then Θ1 (λ ) is a regular divisor of Θ1′ (λ ). If, moreover, H1 = H′1 , then Θ1 (λ ) and Θ1′ (λ ) can only differ by a constant unitary left-factor. (b) Let Θ1′ (λ ) be a regular divisor of Θ (λ ) and let Θ1 (λ ) be a regular divisor of Θ1′ (λ ) (hence of Θ (λ ) as well, cf. Proposition 4.2(a)). Every invariant subspace H′1 associated with Θ1′ (λ ) includes then an invariant subspace H1 associated with Θ1 (λ ) Proof. Part (a): Let us assume that the invariant subspaces H1 and H′1 derive from the regular factorizations Θ (λ ) = Θ2 (λ )Θ1 (λ ) and Θ (λ ) = Θ2′ (λ )Θ1′ (λ ) with the intermediate spaces F and F′ , respectively, and that H1 ⊂ H′1 . Denoting by Z and Z ′ the unitary operators generated by the above regular factorizations, we deduce from formula (1.47) that H1 ⊂ H′1 implies K+ ≡{Θ2u ⊕ Z −1(∆2 u ⊕ v) : u ∈ H 2 (F), v ∈ ∆1 L2 (E)} ⊂

{Θ2′ u′ ⊕ Z ′−1 (∆2′ u′ ⊕ v′ ) :

2

u ∈ H (F ), v ∈ ∆1′ L2 (E)} ′

′

′

(4.4) ≡ K+′ .

One of the consequences of the inclusion (4.4) is that to each u ∈ H 2 (F) there corresponds a u′ ∈ H 2 (F′ ) and a v′ ∈ ∆1 L2 (E) such that

Θ2 u ⊕ Z −1(∆2 u ⊕ 0) = Θ2′ u′ ⊕ Z ′−1 (∆2′ u′ ⊕ v′ ). Because kΘ2 u ⊕ Z −1(∆2 u ⊕ 0)k2 = kΘ2 uk2 + k∆2uk2 = kuk2

(4.5)

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and kΘ2′ u′ ⊕ Z ′−1 (∆2′ u′ ⊕ v′ )k2 = kΘ2′ u′ k2 + k∆2′ u′ k2 + kv′k2 = ku′ k2 + kv′k2 , (4.5) implies that

kuk2 = ku′ k2 + kv′k2 .

(4.6)

It follows that the operators Q and R defined by Qu = u′ ,

Ru = v′

are contractions from H 2 (F) into H 2 (F′ ) and ∆1′ L2 (E), respectively. All the operators figuring in formula (4.5) commute with multiplication by the function eit , therefore so does Q. As multiplication by eit is a unilateral shift on H 2 (F) as well as on H 2 (F′ ), with the respective generating subspaces consisting of the constant functions, it follows from Lemma V.3.2 that there exists a contractive analytic function {F, F′ , Ω (λ )} such that u′ (λ ) = (Qu)(λ ) = Ω (λ )u(λ ) (u ∈ H 2 (F)).

(4.7)

Another consequence of (4.4) is the inclusion T int T int ′ e K+ ⊂ e K+ .

n≥0

n≥0

(4.8)

Let W be the unitary operator from K+ to H 2 (F) ⊕ ∆1 L2 (E) defined by (2.7). It is obvious that W commutes with multiplication by eit , therefore we have hT i T int T int e K+ = W −1 eint H 2 (F) ⊕ e ∆1 L2 (E) n≥0

n≥0

=W

−1

n≥0

[{0} ⊕ ∆1L2 (E)] = {0 ⊕ Z −1(0 ⊕ v) : v ∈ ∆1 L2 (E)}

and analogously for K+′ . Thus (4.8) means that {0 ⊕ Z −1(0 ⊕ v) : v ∈ ∆1 L2 (E)} ⊂ {0 ⊕ Z ′−1(0 ⊕ v′) : v′ ∈ ∆1′ L2 (E)}.

(4.9)

It follows that for every v ∈ ∆1 L2 (E) there exists a v′ ∈ ∆1′ L2 (E) such that Z −1 (0 ⊕ v) = Z ′−1 (0 ⊕ v′).

(4.10)

Because (4.10) implies kvk = kv′ k, the operator V defined by V v = v′ is a well-defined isometry from ∆1 L2 (E) into ∆1′ L2 (E). Let w ∈ H 2 (E). Then

Θ w ⊕ ∆ w = Θ2′ Θ1′ w ⊕ Z ′−1 (∆2′ Θ1′ w ⊕ ∆1′ w),

(4.11)

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and making use of the operators Q, R,V defined above, and of (4.5) and (4.7),

Θ w ⊕ ∆ w = Θ2Θ1 w ⊕ Z −1 (∆2Θ1 w ⊕ ∆1 w)

= [Θ2Θ1 w ⊕ Z −1 (∆2Θ1 w ⊕ 0)] + [0 ⊕ Z −1(0 ⊕ ∆1w)]

(4.12)

= [Θ2′ ΩΘ1 w ⊕ Z ′−1 (∆2′ ΩΘ1 w ⊕ RΘ1w)] + [0 ⊕ Z ′−1(0 ⊕ V ∆1 w)] = Θ2′ ΩΘ1 w ⊕ Z ′−1 (∆2′ ΩΘ1 w ⊕ v), ¯ where v¯ = RΘ1 w + V ∆1 w ∈ ∆1′ L2 (E). Comparing (4.11) with (4.12) we get

Θ2′ Θ1′ w = Θ2′ ΩΘ1 w,

∆2′ Θ1′ w = ∆2′ ΩΘ1 w (w ∈ H 2 (E)).

(4.13)

The transformation u′ → Θ2′ u′ ⊕ ∆2′ u′ (u′ ∈ H 2 (F′ )) being isometric, (4.13) implies Θ1′ w = ΩΘ1 w and hence

Θ1′ (λ ) = Ω (λ )Θ1 (λ ) (λ ∈ D). In view of the equations (4.5) and (4.7) we readily obtain that

Θ2 u = Θ2′ u′ = Θ2′ Qu = Θ2′ Ω u whence

Θ2 (λ ) = Θ2′ (λ )Ω (λ )

follows. The factorizations

(u ∈ H 2 (F)), (λ ∈ D)

Θ (λ ) = (Θ2′ (λ )Ω (λ ))Θ1 (λ ) = Θ2′ (λ )(Ω (λ )Θ1 (λ )) of types (F(32)1 ) and (F3(21) ), respectively, are regular because Θ2′ (λ )Ω (λ ) = Θ2 (λ ) and Ω (λ )Θ1 (λ ) = Θ1′ (λ ). Thus Lemma 4.1 implies that the factorization Θ1′ (λ ) = Ω (λ )Θ1 (λ ) of type (F21 ) is also regular; that is, Θ1 (λ ) is a regular divisor of Θ1′ (λ ). If H1 = H′1 , we have equality in (4.4), (4.8), and (4.9) also. Now, equality in (4.4) implies (because of (4.9)) that if u runs over H 2 (F), then u′ = Qu runs over H 2 (F′ ), and thus QH 2 (F) = H 2 (F′ ). On the other hand, equality in (4.9) implies that if v runs over ∆1 L2 (E), then v′ = V v runs over ∆1′ L2 (E), and thus V is a unitary operator from ∆1 L2 (E) onto ∆1′ L2 (E). On account of (4.10) the right-hand side of (4.5) is then equal to [Θ2′ u′ ⊕ Z ′−1(∆2′ u′ ⊕ 0)] + [0 ⊕ Z ′−1(0 ⊕ v′)]

=[Θ2′ u′ ⊕ Z ′−1(∆2′ u′ ⊕ 0)] + [0 ⊕ Z −1(0 ⊕ V −1 v′ )]; hence, subtracting 0 ⊕ Z −1(0 ⊕ V −1 v′ ) from both sides of (4.5), we obtain

Θ2 u ⊕ Z −1(∆2 u ⊕ (−V −1 v′ )) = Θ2′ u′ ⊕ Z ′−1 (∆2′ u′ ⊕ 0), and taking norms,

kuk2 + kv′ k2 = ku′ k2 .

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Together with (4.6) this gives kuk = ku′ k; hence Q is isometric. Collecting the results obtained for Q, it follows that Q is unitary. By virtue of Lemma V.3.2(d) the function Ω (λ ) is therefore a unitary constant. Part (b): Let Θ1′ (λ ) also be denoted by Θ21 (λ ). Let H21 be an invariant subspace associated with the regular divisor Θ21 (λ ) of Θ (λ ) : H21 corresponds to some regular factorization (F3(21) ) Θ (λ ) = Θ3 (λ )Θ21 (λ ), where {F1 , F21 , Θ21 (λ )} is a contractive analytic function. By hypothesis, Θ1 (λ ) is a regular divisor of Θ21(λ ), and thus there exists a regular factorization (F21 ) Θ21 (λ ) = Θ2 (λ )Θ1 (λ ). By virtue of Lemma 4.1, the factorizations (F32 ) Θ32 (λ ) = Θ3 (λ )Θ2 (λ ),

(F(32)1 ) Θ (λ ) = Θ32 (λ )Θ1 (λ )

are then regular too. Consequently, the isometric transformations Z3(21) , and so on, corresponding to these factorizations are all unitary; applying our commutative diagram to the present situation we obtain −1 −1 −1 −1 −1 Z(32)1 (Z32 ⊕ I1) = Z321 = Z3(21) (I3(21) ⊕ Z21 ).

(4.14)

This is a unitary transformation

∆3 L2 (F21 ) ⊕ ∆2L2 (F) ⊕ ∆1 L2 (E) → ∆ L2 (E). Let H1 be the invariant subspace corresponding to the regular factorization Θ (λ ) = Θ32 (λ )Θ1 (λ ); the invariant subspace H21 corresponds to the regular factorization Θ (λ ) = Θ3 (λ )Θ21 (λ ), thus −1 (∆32 u ⊕ v) : u ∈ H 2 (F1 ), v ∈ ∆1 L2 (E)} ⊖ G, H1 = {Θ32 u ⊕ Z(32)1

(4.15)

−1 H21 = {Θ3 u′ ⊕ Z3(21) (∆3 u′ ⊕ v′ ) : u′ ∈ H 2 (F21 ), v′ ∈ ∆21 L2 (E)} ⊖ G,

with G = {Θ w ⊕ ∆ w : w ∈ H 2 (E)}; see (1.47). With every pair of functions u, v occurring in the definition of H1 let us associate the functions −1 u′ = Θ2 u, v′ = Z21 (∆2 u ⊕ v);

these are among those occurring in the definition of H21 . Because Θ32 (λ ) = Θ3 (λ )Θ2 (λ ), we have (4.16) Θ32 u = Θ3 u′ ;

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on the other hand (4.14) implies −1 −1 −1 Z(32)1 (Z32 ⊕ I1)((∆3Θ2 u ⊕ ∆2u) ⊕ v) (∆32 u ⊕ v) = Z(32)1

(4.17)

−1 −1 (I3(21) ⊕ Z21 )(∆3Θ2 u ⊕ (∆2u ⊕ v)) = Z3(21)

−1 (∆3 u′ ⊕ v′ ). = Z3(21)

From relations (4.15)–(4.17) it follows that H1 ⊂ H21 . This concludes the proof of Theorem 4.3. Notice that it turns out from the proof that the regular factorizations Θ (λ ) = Θ2 (λ )Θ1 (λ ) = Θ2′ (λ )Θ1′ (λ ) provide the same invariant subspace if and only if Θ1′ (λ ) = ZΘ1 (λ ) and Θ2′ (λ ) = Θ2 (λ )Z ∗ hold with a constant unitary operator Z from the intermediate space F onto the intermediate space F′ . 3. As already remarked, it is possible for more than one invariant subspace to be associated with the same regular divisor of Θ (λ ). For an example let us consider the simplest of the factorizations discussed in Sec. 2.4. Namely let {E, E∗ , Θ (λ )} be chosen with E = {0}, E∗ = E 1 , and Θ (λ ) ≡ O. Then the factorization

Θ (λ ) = Θ2 (λ )Θ (λ ) is regular for every inner function {E 1 , E 1 , Θ2 (λ )}. Now if we choose in particular Θ2 (λ ) equal to λ or λ 2 , the invariant subspaces H1 corresponding to the two factorizations are λ H 2 and λ 2 H 2 , respectively, hence different, Moreover, the characteristic function of T2 = P2 T|H2 coincides with {E 1 , E 1 , λ } in the first case and with {E 1 , E 1 , λ 2 } in the second case, and these functions clearly do not coincide. It is worthwhile to observe that in this example we have to do (up to unitary equivalence) with the unilateral shift S on H 2 , and with the subspaces SH 2 and S2 H 2 , respectively. This remark raises the problem of finding cases where uniqueness does hold. To this end we introduce a definition. Definition. A contractive analytic function {E, F, Θ1 (λ )} is called a strong regular divisor of {E, E∗ , Θ (λ )} if there exists a unique contractive analytic function {F, E∗ , Θ2 (λ )} such that (F) Θ (λ ) = Θ2 (λ )Θ1 (λ ) is a regular factorization of Θ (λ ). Observe that if in a factorization (F), regular or not, the factor Θ1 (λ ) is an outer function, then the factor Θ2 (λ ) is uniquely determined by Θ (λ ) and Θ1 (λ ). This follows immediately from the fact that we have then Θ1 (λ )E = F for every λ ∈ D; see Proposition V.2.4. Also, if the factor Θ1 (λ ) in (F) is a ∗-inner function, that is, if Θ 1˜(e−it ) = Θ1 (eit )∗ is isometric a.e., then we have Θ2 (eit ) = Θ2 (eit )Θ1 (eit )Θ1 (eit )∗ = Θ (eit )Θ1 (eit )∗ a.e.; thus Θ2 (eit ) is determined by Θ (eit ) and Θ1 (eit ) a.e., and hence the function Θ2 (λ ) is determined by Θ (λ ) and Θ1 (λ ). We have proved the following proposition.

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Proposition 4.4. Every outer or ∗-inner regular divisor of a contractive analytic function Θ (λ ), is a strong regular divisor of Θ (λ ). Thus, in particular, the outer factor in the canonical factorization of Θ (λ ), and the ∗-inner factor in the ∗-canonical factorization of Θ (λ ), are strong regular divisors of Θ (λ ). Somewhat less immediate is the following result. Proposition 4.5. Let {E, E∗ , Θ (λ )} be a contractive analytic function for which the conditions u∗ ∈ H 2 (E∗ ), v ∈ ∆ L2 (E), Θ ∗ u∗ + ∆ v = 0 (4.18) imply u∗ = 0, v = 0. Then every regular divisor of Θ (λ ) is strong.

Proof. We have to show that if Θ2 (λ )Θ1 (λ ) and Θ2′ (λ )Θ1 (λ ) are two regular factorizations of Θ (λ ) with the same second factor {E, F, Θ1 (λ )} then Θ2 (λ ) = Θ2′ (λ ). Let w ∈ H 2 (F). From the regularity property of the factorizations we deduce that there exist sequences {vn }, {v′n } of elements of L2 (E) such that

∆2Θ1 vn ⊕ ∆1 vn → ∆2 w ⊕ 0,

∆2′ Θ1 v′n ⊕ ∆1 v′n → ∆2′ w ⊕ 0 (n → ∞).

As ∆2Θ1 vn ⊕ ∆1 vn = Z ∆ vn and ∆2′ Θ1 v′n ⊕ ∆1 v′n = Z ′ ∆ v′n , where Z and Z ′ are the isometries corresponding to the two factorizations, we conclude that the limits l = lim ∆ vn and l ′ = lim ∆ v′n also exist (in L2 (E)). We have

∆ 2 vn = Θ1∗ ∆22Θ1 vn + ∆12 vn → Θ1∗ ∆22 w = Θ1∗ w − Θ1∗Θ2∗Θ2 w = Θ1∗ w − Θ ∗Θ2 w and similarly

∆ 2 v′n → Θ1∗ w − Θ ∗Θ2′ w (n → ∞).

On the other hand ∆ 2 vn → ∆ l and ∆ 2 v′n → ∆ l ′ , thus it follows that

Θ ∗ (Θ2 w − Θ2′ w) + ∆ (l − l ′ ) = 0, which implies by our hypothesis that Θ2 w − Θ2′ w = 0 and, because w was arbitrary, that Θ2 (λ ) = Θ2′ (λ ). Proposition 4.6. The hypothesis in Proposition 4.5 is satisfied in particular if Θ (eit ) is unitary at the points t of a set of positive measure. Proof. Condition (4.18) implies then that u∗ (eit ) = 0 on this set of positive measure. As the numerical function (u∗ (λ ), e∗ ) belongs to H 2 for every e∗ ∈ E∗ , we deduce that (u∗ (λ ), e∗ ) ≡ 0. It follows that u∗ (λ ) ≡ 0, and hence by (4.18) that ∆ (t)v(t) = 0 a.e. Because v ∈ ∆ L2 (E), this implies v(t) = 0 a.e. The cases considered in the last two propositions can be characterized in terms of contractions in the following manner: Theorem 4.7. Suppose Θ (λ ) coincides with the characteristic function of a c.n.u. contraction T on the space H. In order that Θ (λ ) satisfy the hypothesis of Proposition 4.5 it is necessary and sufficient that there exist no nonzero invariant subspace

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319

H1 for T such that T |H1 is isometric. In order that Θ (λ ) satisfy the hypothesis of Proposition 4.6, it suffices that the spectrum of T not contain the whole unit circle. Proof. The second assertion follows immediately from Theorem VI.4.1. As to the first one, let us observe that a subspace H1 is invariant for T and T |H1 is isometric if and only if H1 is invariant for any isometric dilation U+ of T . Thus, in order that no such H1 6= {0} exist, it is necessary and sufficient that there be no nonzero h in H all of whose images under U+n (n = 1, 2, . . .) belong to H. In terms of the functional model of T given by Theorem VI.3.1 this means that there exists no nonzero u∗ ⊕ v ∈ H 2 (E∗ ) ⊕ ∆ L2 (E) such that eint u∗ ⊕ eint v ∈ H

(n = 0, 1, 2, . . .),

that is, such that eint (Θ ∗ u∗ + ∆ v) ⊥ H 2 (E)

(n = 0, 1, 2, . . .);

(4.19)

but (4.19) is obviously equivalent to the condition Θ ∗ u∗ + ∆ v = 0. 4. Let {E, E∗ , Θ (λ )} be a purely contractive analytic function, and let H and T be the space and the contraction generated by (1.44)–(1.45). With each strong regular divisor Θ1 (λ ) of Θ (λ ) there is associated a unique invariant subspace for T; let us denote it by H(Θ1 ). It is easy to show that if Θ1 (λ ) is a strong regular divisor of Θ (λ ), then so is every function Θ1′ (λ ) which differs from Θ1 (λ ) in a constant unitary left-factor only, and in this case H(Θ1′ ) = H(Θ1 ). In particular, every constant unitary function {E, F, Θ0 } is a strong regular divisor of Θ (λ ); for these functions (and only for these; cf. Theorem 4.3(a)) we have H(Θ0 ) = {0}. On the other hand, assume the function Θ (λ ) itself is a strong regular divisor of Θ (λ ) (which is not always the case, as shown by the example given at the beginning of Subsec. 3); then we have H(Θ1 ) = H exactly for those functions Θ1 (λ ) that differ from Θ (λ ) only by a constant unitary left-factor. When all regular divisors of Θ (λ ) are strong (e.g., in the cases considered in Propositions 4.5 and 4.6), every set {Θγ (λ )} of regular divisors of ΘT (λ ) admits a greatest common regular divisor Θ∧ (λ ) and a least common regular multiple Θ∨ (λ ), determined up to coincidence by the following conditions. (d1 ) Θ∧ (λ ) is a regular divisor of every Θγ (λ ) (and consequently of Θ (λ )). (d2 ) If a function Θ ′ (λ ) is a regular divisor of every Θγ (λ ), it is also a regular divisor of Θ∧ (λ ). (m1 ) Every Θγ (λ ) is a regular divisor of Θ∨ (λ ). (m2 ) If a regular divisor Θ ′′ (λ ) of Θ (λ ) is such that every Θγ (λ ) is a regular divisor of Θ ′′ (λ ), then Θ∨ (λ ) is a regular divisor of Θ ′′ (λ ) (in particular, Θ∨ (λ ) is a regular divisor of Θ (λ )).

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To prove this, let us consider the subspaces H∧ =

T γ

H(Θγ ) and H∨ =

W γ

H(Θγ )

of H. They are obviously invariant for T and hence associated with some regular divisors of Θ (λ ), which we denote by Θ∧ (λ ) and Θ∨ (λ ), respectively. These functions are determined up to constant unitary left-factors. Properties (d1 ) and (m1 ) follow readily from Theorem 4.3(a). If Θ ′ (λ ) is of the type given in (d2 ), Θ ′ (λ ) is a regular divisor of Θ (λ ) also, and by virtue of Theorem 4.3(b), H(Θ ′ ) ⊂ H(Θγ ) for every Θγ ; hence H(Θ ′ ) ⊂ H∧ = H(Θ∧ ). By virtue of Theorem 4.3(a) this implies that Θ ′ (λ ) is a regular divisor of Θ∧ (λ ), proving property (d 2 ). Property (m2 ) of Θ∨ (λ ) can be established in an analogous manner.

5 Invariant subspaces for contractions of class C11 1. By Proposition II.3.5 every contraction T ∈ C11 is quasi-similar to a unitary operator, indeed to the residual part R = RT of the minimal unitary dilation U = UT of T . Using this fact, we were able to construct a large family of hyperinvariant subspaces for such a T . We now continue this study by applying the methods developed since Chapter V. Because for unitary operators we have spectral theory at our disposal, it does not essentially restrict generality if we only consider completely nonunitary contractions (of class C11 ). We begin by finding the form of R in the functional model. Thus let T be a c.n.u. contraction in a (separable) Hilbert space H and let {E, E∗ , Θ (λ )} coincide with the characteristic function of T . It follows from the results of Sec. VI.3 (cf. in particular (VI.3.1) and (VI.3.17′)) that R is unitarily equivalent to multiplication by eit in ∆ L2 (E). Consequently, if we denote by ER (α ) the spectral measure corresponding to the unitary operator R and to the Borel subset α of the unit circle C, then ER (α ) is unitarily equivalent to multiplication on the function space ∆ L2 (E) by the function  1 for t ∈ (α ), χα (t) = 0 for t ∈ (α ′ ), where α ′ = C\α and, according to our convention, (β ) = {t ∈ [0, 2π ) : eit ∈ β }. So we obtain readily that ER (α ) = O if and only if ∆ (t) = O a.e. on (α ). The resolvent set of R on C is therefore composed of the open arcs on which the radial limit of Θ (λ ) is isometric a.e. Definition 1. For a c.n.u. contraction T , let ε (T ) denote the set of points eit ∈ C at which the radial limit of the characteristic function of T exists and is not isometric. Definition 2. For any subset α of C, its essential support, denoted by ess supp α ,

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321

is defined as the complement, with respect to C, of the maximal open subset of C whose intersection with α is of zero Lebesgue measure. We can now express the above result as follows. Proposition 5.1. For a c.n.u. contraction T on a (separable) space H, we have σ (R) = ess supp ε (T ). Let us recall in this connection Proposition II.6.2, which implies that for every contraction T , σ (R) ⊂ σ (T ) ∩C,

and Proposition VI.4.4, which states that

σ (T ) = σ (T ) ∩C = ess supp ε (T ) at least for those c.n.u. contractions T ∈ C·1 whose characteristic function has a scalar multiple. For these contractions we have thus

σ (R) = σ (T ). We add one more definition. Definition 3. A Borel subset α of the unit circle C is said to be residual for the c.n.u. contraction T if ΘT (eit ) is isometric at almost every point of the complement α ′ of α . 2. After these preliminaries let us consider a c.n.u. contraction T of class C1· on the (separable) space H, and let {E, E∗ , Θ (λ )} be a function coinciding with the characteristic function of T . As this function is ∗-outer, we can apply to it Proposition V.4.3; see the Remark added to this proposition. Thus to every Borel subset α of C there corresponds a factorization

Θ (λ ) = Θ2α (λ )Θ1α (λ ) of {E, E∗ , Θ (λ )} as a product of two contractive analytic functions, with some intermediate space Fα , such that Θ1α (λ ) is outer and we have ( Θ (eit )∗Θ (eit ) a.e. on (α ), it ∗ it (5.1) Θ1α (e ) Θ1α (e ) = IE a.e. on (α ′ ),

Θ2α (eit )∗Θ2α (eit ) = IFα

a.e. on (α ).

(5.2)

We have already seen in Sec. 3.3 that this factorization is regular. Moreover, because Θ1α (λ ) is outer, it is a strong regular divisor of Θ (λ ). Thus if T is the functional model of T on the space H corresponding to Θ (λ ) in the sense of Theorem VI.3.1, then there exists a unique invariant subspace H(Θ1α ) associated with Θ1α (λ ); for

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the sake of brevity we denote it by Hα , and set Tα = T|Hα . Because Θ (λ ) is ∗outer, we have H 2 (E) = Θ ˜H 2 (E∗ ) = Θ 1˜α Θ 2˜α H 2 (E∗ ) ⊂ Θ 1˜α H 2 (Fα ) ⊂ H 2 (E), and hence we infer that Θ1α (λ ) is ∗-outer too. Thus Θ1α (λ ) is outer from both sides and so is its purely contractive part Θ10α (λ ), hence also the characteristic function of Tα . This implies Tα ∈ C11 . Moreover, Θ1α (eit ) is isometric (even unitary) a.e. in (α ′ ), therefore Θ10α (eit ) is also isometric a.e. in (α ′ ); thus α is residual for Tα . Now let H1 be any subspace of H, invariant for T and such that T1 = T|H1 ∈ C11 and α is residual for T1 . H1 is associated with a regular factor Θ1 (λ ) of Θ (λ ) (which is uniquely determined up to a constant unitary left-factor by Theorem 4.3(a)). Because T1 ∈ C11 , the purely contractive part Θ10 (λ ) of Θ1 (λ ) (coinciding with the characteristic function of T1 ) is outer from both sides. Consequently Θ1 (λ ) is also outer, and hence a strong regular divisor of Θ (λ ); thus H1 is the only invariant subspace associated with Θ1 (λ ), H1 = H(Θ1 ). On the other hand, the set α is residual for T1 , thus Θ10 (eit ) is isometric a.e. in (α ′ ), and the same is then true for Θ1 (eit ):

Θ1 (eit )∗Θ1 (eit ) = IE

a.e. in (α ′ ).

(5.3)

As Θ1 (λ ) is a divisor of Θ (λ ), we have obviously

Θ (eit )∗Θ (eit ) ≤ Θ1 (eit )∗Θ1 (eit )

a.e. on (0, 2π ).

(5.4)

Comparing (5.3) and (5.4) with (5.1) we obtain

Θ1α (eit )∗Θ1α (eit ) ≤ Θ1 (eit )∗Θ1 (eit ) a.e. on (0, 2π ). By Proposition V.4.1(a) it follows that Θ1 (λ ) is a divisor of Θ1α (λ ); and, taking into account that Θ1 (λ ) is outer, by Proposition 4.2(b), Θ1 (λ ) is even a regular divisor of Θ1α (λ ). Applying Theorem 4.3(b) to the strong regular divisors Θ1α (λ ) and Θ1 (λ ) of Θ (λ ) we obtain that H1 ⊂ Hα (because these are the only invariant subspaces associated with Θ1 (λ ) and Θ1α (λ ), respectively). This proves that Hα is the maximal subspace of H satisfying the conditions (i) THα ⊂ Hα ,

(ii) Tα = T|Hα ∈ C11 ,

(iii) α is residual for Tα .

As these properties are obviously invariant with respect to unitary equivalence, the existence of maximal subspaces Hα with these properties follows for every c.n.u. contraction T ∈ C1· on a (separable) Hilbert space H. Maximality clearly implies uniqueness of Hα . Let us show that Hα is even hyperinvariant for T , that is, invariant for every bounded operator X commuting with T . It suffices to consider such X that are boundedly invertible, because every other X could be replaced by X − µ I with

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323

|µ | > kX k. The space H′α = X Hα is then invariant for T , and Tα′ = T |H′α is similar to Tα = T |Hα : indeed we have Tα′ = STα S−1

with

S = X|Hα ,

S being an affinity from Hα onto Hα′ . Consequently, Tα ∈ C11 implies Tα′ ∈ C11 . Denoting by Rα and R′α the residual parts of the minimal unitary dilations of Tα and Tα′ , respectively, Rα is quasi-similar to Tα and Rα′ is quasi-similar to Tα′ ; as Tα and Tα′ are similar we conclude that Rα and R′α are quasi-similar and hence unitarily equivalent; see Proposition II.3.4. This implies that the set α is residual for Tα′ as well. By the maximality of Hα we have thus H′α ⊂ Hα , XHα ⊂ Hα . This proves that Hα is hyperinvariant for T . Let us return to the model T of T on H and find conditions for Hα1 ⊂ Hα2 . A necessary and sufficient condition follows from Theorem 4.3, namely that Θ1α1 (λ ) be a regular divisor of Θ1α2 (λ ). Now if a contractive analytic Θ1 (λ ) is a divisor of a contractive analytic Θ2 (λ ), then we have

Θ2 (eit )∗Θ2 (eit ) ≤ Θ1 (eit )∗Θ1 (eit ) a.e.

(5.5)

If Θ1 (λ ) is outer, this inequality implies, conversely, that Θ1 (λ ) is a divisor of Θ2 (λ ) (cf. Proposition V.4.1(a)), and if both of them are regular divisors of Θ (λ ) then Proposition 4.2(b) yields that Θ1 (λ ) is a regular divisor of Θ2 (λ ). When applied to the case under consideration, these facts imply that for Hα1 ⊂ Hα2 it is necessary and sufficient that

Θ1α2 (eit )∗Θ1α2 (eit ) ≤ Θ1α1 (eit )∗Θ1α1 (eit ) a.e. By (5.1), this condition is equivalent to the condition that almost every point of α1 ∩ α2′ belong to ε (T)′ , that is, that the complementary set, α1′ ∪ α2 , be residual for T. Thus for Hα1 ⊂ Hα2 it is necessary and sufficient that α1′ ∪ α2 be residual for T. This condition is satisfied in particular if α1 ⊂ α2 , because then α1′ ∪ α2 = C. We deduce from this result that Hα1 = Hα2 if and only if both α1′ ∪ α2 and α2′ ∪ α1 are residual for T. The intersection of these two sets being equal to the complement, in C, of the symmetric difference α1 △α2 , we conclude that Hα1 = Hα2 if and only if the complement of α1 △α2 in C is residual for T (i.e., if ΘT (eit ) is isometric a.e. in α1 △α2 ). Let us consider another problem. Let {αn } be a (finite or infinite) sequence of S Borel sets αn ⊂ C and let α = n αn . Because αn ⊂ α , we have Hαn ⊂ Hα and hence W M = H αn ⊂ H α . (5.6) n

The subspace M, being invariant for T, is associated with a regular divisor Ω (λ ) of Θ (λ ). Because Hαn ⊂ M, Theorem 4.3 a) implies that Θ1αn (λ ) is a divisor of Ω (λ ). So we have

Θ (eit )∗Θ (eit ) ≤ Ω (eit )∗ Ω (eit ) ≤ Θ1αn (eit )∗Θ1αn (eit ) a.e.;

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by virtue of (5.1) it follows that

Ω (eit )∗ Ω (eit ) = Θ (eit )∗Θ (eit )

(5.7)

a.e. in (αn ). This being true for n ≥ 1, we conclude that (5.7) holds a.e. in (α ). Using (5.1) again, we obtain

Ω (eit )∗ Ω (eit ) ≤ Θ1α (eit )∗Θ1α (eit ) a.e. This implies, by virtue of Proposition V.4.1(a), that Θ1α (λ ) is a divisor of Ω (λ ), and (because Θ1α (λ ) is outer) by virtue of Proposition 4.2 b), even a regular divisor of Ω (λ ). Applying Theorem 4.3 b) and recalling that Hα is the only invariant subspace associated with Θ1α (λ ), we obtain that Hα ⊂ M. Together with (5.6), this gives M = Hα . Finally, consider the particular cases where α is either empty or the whole circle C. For α empty, relation (5.1) implies that Θ1α (λ ) is an inner function. As it is also outer, it must be a unitary constant (cf. Proposition V.2.3). Direct application of formula (1.47) yields then Hα = {0}. For α = C, (5.1) implies

Θ1α (eit )∗Θ1α (eit ) = Θ (eit )∗Θ (eit ) a.e., and hence it follows that Θ1α (λ ) is the outer factor in the canonical factorization of Θ (λ ); see Proposition V.4.1(b). Consequently Hα is equal to the space of the C·1 part in the triangulation of T of type   C·1 ∗ ; O C·0 see Sec. 3.3. In particular, if T ∈ C11 we have necessarily Hα = H for α = C. We can summarize our results for T ∈ C11 as follows.

Theorem 5.2. Let T be a c.n.u. contraction of class C11 on a (separable) space H 6= {0}. Let α be a Borel subset of the unit circle C. Among the subspaces L of H, invariant for T and such that T |L ∈ C11

and α is residual for T |L,

there is a maximal one (i.e., containing all the others), which we denote by Hα ; Hα is even hyperinvariant for T . We have (i) Hα = {0} if α is empty; Hα = H if α = C. W S (ii) n Hαn = Hα for α = n αn . (iii) Hα1 ⊂ Hα2 if and only if α1′ ∪ α2 is residual for T , thus in particular if α1 ⊂ α2 .

From these properties it follows, moreover, that

(a) Hα1 = Hα2 if and only if C\(α1 △α2 ) is residual for T . (b) Hα = {0} if and only if C\α is residual for T . (c) Hα = H if and only if α is residual for T .

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325

We recall that T , being of class C11 , is quasi-similar to R; hence we infer in particular that the space R of R is different from {0}. On the other hand, because T is c.n.u., its minimal unitary dilation U, hence also R, has absolutely continuous spectra. It follows that the spectrum σ (R) is a nonempty perfect subset of C. If α and β are any two disjoint open arcs of C both having nonempty intersection with the perfect set σ (R), then (by virtue of Proposition 5.1) the corresponding subspaces Hα and Hβ are nontrivial (by (b) and (c)), different (by (a)), and neither of them includes the other (by (iii)). Thus, Theorem 5.2 implies in particular the following statement. For every c.n.u. T ∈ C11 there is an infinity of hyperinvariant subspaces, and T is not unicellular (cf. Sec. III.7.2).

6 Spectral decomposition of contractions of class C11 whose characteristic function admits a scalar multiple 1. For a c.n.u. contraction T ∈ C11 , whose characteristic function admits a scalar multiple, Theorem 5.2 can be given a more complete form. We begin with a lemma. Lemma 6.1. Let {E, E∗ , Θ (λ )} be a contractive analytic function admitting a scalar multiple δ (λ ), and let Θ (λ ) = Θ2 (λ )Θ1 (λ ) be a factorization of Θ (λ ) as the product of contractive analytic functions {E, F, Θ1 (λ )} and {F, E∗ , Θ2 (λ )} such that Θ2 (eit ) is isometric at the points t of a set of positive measure. Then Θ1 (λ ) admits the scalar multiple δ (λ ) also. Proof. By hypothesis, there exists a contractive analytic function {E∗ , E, Ω (λ )} such that

Ω (λ )Θ2 (λ )Θ1 (λ ) = δ (λ )IE ,

Θ2 (λ )Θ1 (λ )Ω (λ ) = δ (λ )IE∗ .

(6.1)

The second equation yields:

Θ2 (λ )Φ (λ ) ≡ O with Φ (λ ) = Θ1 (λ )Ω (λ )Θ2 (λ ) − δ (λ )IF . Hence Θ2 (eit )Φ (eit ) = O a.e., and consequently Φ (eit ) = O at almost every point t where Θ2 (eit ) is isometric. Thus Φ (eit ) = O on a set of positive measure. Because Φ (λ ) is a bounded analytic function, we have then necessarily Φ (λ ) ≡ O, and thus

Θ1 (λ )Ω (λ )Θ2 (λ ) = δ (λ )IF .

(6.2)

Equation (6.2) and the first equation (6.1) prove that δ (λ ) is a (scalar) multiple of Θ1 (λ ). Now we are able to prove the following theorem which together with Theorem 5.2 establishes a spectral decomposition of these operators. Theorem 6.2. For a c.n.u. contraction T on H of class C11 and such that the characteristic function of T admits a scalar multiple, the invariant subspaces Hα considered in Theorem 5.2 have the following additional properties,

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(iv) σ (Tα ) ⊂ α (the closure of α ). T T (v) Hα = n Hαn for α = αn .

Proof. Part (iv): By virtue of (5.2), Θ2α (eit ) is isometric a.e. in (α ). If α has positive measure, it follows from Lemma 6.1 that Θ1α (λ ) also admits a scalar multiple. The same is true if α is of zero measure. Indeed, in this case (5.1) implies Θ1α (eit )∗Θ1α (eit ) = IE a.e., so the outer function Θ1α (λ ) is also inner, and hence a unitary constant (cf. Proposition V.2.3), consequently it admits the scalar multiple δ (λ ) ≡ 1. Thus Θ1α (λ ) admits in every case a scalar multiple, and the same is then true for the purely contractive part of Θ1α (λ ) (cf. Proposition V.6.8). The latter function coincides with the characteristic function of Tα (observe that Tα is c.n.u. and of class C11 ); thus we have, by virtue of Proposition VI.4.4,

σ (Tα ) = ess supp ε (Tα ) (cf. Sec. 5.1). On the other hand, because α is residual for Tα (cf. Theorem 5.2), ε (Tα ) ∩ α ′ is of zero measure and a fortiori so is ε (Tα ) ∩ α ′ . Because α is an open set, this implies

α ′ = [ess supp ε (Tα )]′ ,

α ⊃ ess supp ε (Tα ),

and therefore σ (Tα ) ⊂ α . Part (v): In this proof we apply the functional model T of T on H. Because T α = n αn implies α ⊂ αn (n = 1, 2, . . .), it follows from Theorem 5.2 that Hα ⊂ Hαn (n = 1, 2, . . .), and hence H αm ⊃ M =

T n

H αn ⊃ H α .

(6.3)

As M is invariant for T, it is associated with a regular divisor Ω (λ ) of Θ (λ ). By virtue of Theorem 4.3(a), it follows from (6.3) that Θ1α (λ ) is a regular divisor of Ω (λ ) and Ω (λ ) is a regular divisor of Θ1αm (λ ) (m = 1, 2, . . .). This implies, a.e.,

Θ1α (eit )∗Θ1α (eit ) ≥ Ω (eit )∗ Ω (eit ) ≥ Θ1αm (eit )∗Θ1αm (eit ) (m = 1, 2, . . .); taking (5.1) into account we deduce that

Ω (eit )∗ Ω (eit ) = Θ1α (eit )∗Θ1α (eit ) a.e.

(6.4)

On the other hand, Ω (λ ) being a regular divisor of Θ1αm (λ ), the factors in the regular factorization Θ1αm (λ ) = Ωm (λ )Ω (λ ) must be such that Ωm (eit ) is isometric a.e. where Θ1αm (eit ) is isometric (cf. Propositions 3.1 and 3.2(c)), i.e. on (αm′ ). Suppose αm′ is of positive measure for at least one m. We have observed while proving assertion (iv) that the functions Θ1α (λ ) (for arbitrary α ) admit scalar multiples. Lemma 6.1 then tells us that Ω (λ ) also admits a scalar multiple. As Ω (λ ) is ∗-outer (because Ω ˜(λ )Ω m˜ (λ ) = Θ 1α˜m (λ ) and Θ 1α˜m (λ ) is outer), it follows that Ω (λ ) is outer from both sides (cf. Theorem V.6.2(c)). Now the relation (6.4) between the outer functions Ω (λ ) and Θ1α (λ ) implies that they are equal up to constant unitary left-factors (cf. Proposition V.4.1); hence M = Hα . This was proved under the

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327

assumption that at least one of the sets αm′ has positive measure. In the remaining case all the sets α1 , α2 , . . . as well as their intersection α are obviously residual for T; by Theorem 5.2(c) we have therefore Hα1 = Hα2 = · · · = H, Hα = H, and thus M = Hα holds in this case too. This completes the proof. 2. Let us consider now another problem, closely related to the previous one. This is to study c.n.u. contractions T of class C·1 , whose defect indices are not both infinite. The characteristic function ΘT (λ ) is then an outer function, thus dT ∗ = dim DT ∗ ≤ dim DT = dT (cf. the remark at the end of Sec. V.2.3). Therefore our hypothesis means that dT ∗ is finite. Theorem 6.3. Let T be a c.n.u. contraction of class C·1 such that dT ∗ < ∞. Then either every point of the open unit disc D is an eigenvalue of T , or no point of D belongs to the spectrum of T . In order that the second case occur it is necessary and sufficient that one of the following (equivalent) conditions be satisfied. (i) T ∈ C11 . (ii) dT = dT ∗ . (iii) ΘT (λ ) admits a scalar multiple. Remark. In Sec. VI.4.2 we constructed an example of a T ∈ C11 for which σ (T ) = D, but then we had dT = dT ∗ (= dim E) = ∞. Proof. No higher-dimensional space can be applied linearly, continuously, and oneto-one into a finite-dimensional space, so in the case dT > dT ∗ there is no λ ∈ D for which the operator ΘT (λ ) (from DT into DT ∗ ) is invertible. The characteristic function ΘT (λ ) is outer because T is of class C·1 . Hence, if dT = dT ∗ then ΘT (λ ) is invertible for every λ ∈ D, and so ΘT (λ ) admits a scalar multiple by Proposition V.6.1; that is, (ii) implies (iii). On the other hand, if (iii) holds then the outer function ΘT (λ ) is also ∗-outer by Theorem V.6.2(c), and hence T ∈ C11 . Finally, (i) implies that ΘT (λ ) is outer from both sides, whence dT = dT ∗ immediately follows. Examples. Both cases indicated in the theorem actually occur, as shown by the following examples. (1) The function {E 1 , E 1 , 12 (λ − 1)} is purely contractive analytic and, by virtue of Proposition III.1.3, outer (clearly from both sides). The c.n.u. contraction T generated by this function is therefore of class C11 , it has the defect indices dT = dT ∗ = 1, and σ (T ) is contained in the unit circle (in fact, σ (T ) = C). (2) The function {E 2 , E 1 , Θ (λ )} defined by   1 1 1 x Θ (λ ) 1 = √ (λ − 1)x1 + √ λ x2 x2 22 2 is evidently purely contractive and analytic. Moreover, it is outer, because   √ −2 2 u(λ ) = u(λ ) (u ∈ H 2 ), Θ (λ ) √ 2 u(λ )

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and hence

Θ H 2 (E 2 ) = H 2 = H 2 (E 1 ).

The contraction T generated by this function is thus of class C·1 and has dT = 2, dT ∗ = 1. So every point of D is an eigenvalue of T .

7 Notes 1. The fundamental Theorem 1.1, stating the relation between the invariant subspaces for a c.n.u. contraction T on the (separable) space H and the regular factorizations of the characteristic function ΘT (λ ), was first proved in S Z .-N.–F. [4] and [IX]. However, there is a difference in the presentation that needs some explanation. Here, as well as in the papers referred to, we make use of the functional model {H, T} of {H, T }, generated by a contractive analytic function {E, E∗ , Θ (λ )} coinciding with the characteristic function of T . Let Θ (λ ) = Θ2 (λ )Θ1 (λ ) be a factorization of Θ (λ ) as a product of two contractive analytic functions, {E, F, Θ1 (λ )} and {F, E∗ , Θ2 (λ )}, and let Z be the corresponding isometry Z : ∆ L2 (E) → ∆2 L2 (F) ⊕ ∆1L2 (E); see (1.42). If Z is unitary (i.e., if the factorization is regular), then we can identify the elements of the two spaces corresponding to Z and so we can pass from the model {H, T} to the (unitarily equivalent) model {H , T } defined by the formulas: H =[H 2 (E∗ ) ⊕ ∆2 L2 (F) ⊕ ∆1L2 (E)]

(1) 2

⊖ {Θ2Θ1 u ⊕ ∆2Θ1 u ⊕ ∆1u : u ∈ H (E)},

{T ∗ (u∗ ⊕ v2 ⊕ v1 ) =e−it [u∗ (eit ) − u∗(0)] ⊕ e−it v2 (t) ⊕ e−it v1 (t) (u∗ ⊕ v2 ⊕ v1 ∈ H );

(2)

this was done in the papers mentioned. To the decomposition H = H1 ⊕ H2 (cf. (1.47) and (1.48)) corresponds the decomposition H = H1 ⊕ H2 , where H1 ={Θ2 u ⊕ ∆2u ⊕ v1 : u ∈ H 2 (F), v1 ∈ ∆1 L2 (E)}

(3)

H2 =[H 2 (E∗ ) ⊕ ∆2 L2 (F) ⊕ {0}] ⊖ {Θ2u ⊕ ∆2u ⊕ 0 : u ∈ H 2 (F)}.

(4)

⊖ {Θ2Θ1 w ⊕ ∆2Θ1 w ⊕ ∆1w : w ∈ H 2 (E)},

When we have to do with a fixed regular factorization, the two manners of representation are equally useful. However, if we wish to study the relations between the invariant subspaces corresponding to different regular factorizations, then the model {H , T } cannot be used because it depends on the factorization chosen. This is why we have preferred in this book to construct the invariant subspaces in the model {H, T}. This enabled us to obtain the results stated in Theorem 4.3, and in particular those in Secs. 5 and 6. Nevertheless, the space H and its operator T

7. N OTES

329

have their own advantage: they can be constructed by the formulas (1) and (2) even if the factorization under consideration is not regular. T will always be a contraction on H , and the space H2 defined by (4) will be equal to H ⊖ H1 . Moreover one can show in analogy to Proposition 2.1 that if   T X T = 1 O T2 is the triangulation of T corresponding to the decomposition H = H1 ⊕ H2 , then the characteristic functions of T1 and T2 coincide with the purely contractive parts of Θ1 (λ ) and Θ2 (λ ), respectively. But if the factorization is not regular then T will not be unitarily equivalent to T (or to T , which amounts to the same thing). In fact, since Z is then nonunitary, the space Z = [∆2 L2 (F) ⊕ ∆1 L2 (E)] ⊖ Z · ∆ L2 (E) is nonzero. Thus it follows that T is the orthogonal sum of a contraction unitarily equivalent to T (and to T ), and of a unitary operator: the multiplication by the function eit on the space Z. We can formulate the following result. Theorem. To every factorization ΘT (λ ) = Θ2 (λ )Θ1 (λ ) of the characteristic function of a c.n.u. contraction T as a product of two contractive analytic functions, there corresponds an invariant subspace (if not for T , at least) for T ′ = T ⊕ U (where U is some unitary operator with absolutely continuous spectrum), such that if  ′ ′ T X ′ T = 1 ′ O T2 is the triangulation of T ′ induced by this invariant subspace and its orthogonal complement, then the characteristic functions of T1′ and T2′ coincide with the purely contractive parts of the factors Θ1 (λ ) and Θ2 (λ ), respectively. See S Z .-N.–F. [IX] p. 300. A different proof, entirely independent of dilation theory and using only the definition of the characteristic function, was given in S Z .N.–F. [3] and [8]. 2. The fact that to every nontrivial invariant subspace for an operator there corresponds a nontrivial factorization of the characteristic function of the operator, was discovered by Soviet authors including L IV Sˇ IC AND P OTAPOV [1], M. S. B ROD SKI˘I [1], and B RODSKI˘I AND L IV Sˇ IC [1], among others. The operators T considered by them were those that differ from a unitary or a self-adjoint one by an additive term of finite rank or at least of finite trace. These authors also noticed that, conversely, every factorization of the characteristic function generates an invariant subspace, if not of T at least of some operator T ′ = T ⊕ S, where S is a unitary or self-adjoint operator, respectively. They did not find criteria for the factorization to generate an invariant subspace of the given T itself, except in some particular cases; see B ROD ˇ MUL′ JAN [2] Theorem 2.7; and even in SKI˘I AND L IV Sˇ IC [1] Theorem 16, and S these particular cases their criteria are more involved than our general criterion of

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regularity of the factorization. Let us mention that Sˇ MUL′ JAN [2] (§4) obtained a result equivalent to case (ii) of our Proposition 3.5. (For the further development of this research see B RODSKI˘I AND Sˇ MUL′ JAN [1] and the monograph of M. S. B RODSKI˘I [9].) The research presented in this chapter, although influenced in part by the problems treated by the authors cited above, follows an entirely different path, which is related in a natural manner to the study of the structure of the unitary dilation and of the corresponding functional model. Sections 1 and 2 reproduce essentially parts of the papers S Z .-N.–F. [4] and [IX]. S Z .-NAGY [14] asked whether strange factorizations exist. The first examples of such factorizations were constructed in F OIAS¸ [10], where it was shown that ΘT has no strange factorizations if the defect indices of T are finite. The entire Sec. 4 constitutes an elaborated and precise form of some results sketched in S Z .-N.–F. [7]. Proposition 3.2 is new. The spectral decomposition of the operators of class C11 has been indicated in S Z .-N.–F. [IX] and [IX∗ ], but its study in Secs. 5 and 6 is much more complete. The reader may ask whether Theorem 6.2, which we have established for contractions of class C11 whose characteristic function admits a scalar multiple, does hold for every c.n.u. contraction of class C11 . As to property (iv) in this theorem, this is certainly not valid in the general case. Indeed, choose for T the contraction on H so that its characteristic function coincides with the constant function Θ (λ ) ≡ A, where A is a self-adjoint operator for which (a) O ≤ A ≤ I, (b) 0 and 1 are not eigenvalues of A, and (c) A−1 is not bounded. Let α be a proper subarc of C, of positive length. We have seen in Sec. V.4.5 that in the corresponding factorization Θ (λ ) = Θ2α (λ )Θ1α (λ ) the function Θ1α (λ ) has a unitary limit Θ1α (eit ) a.e. in α ′ , and Θ1α (λ ) cannot be extended analytically through α ′ . Thus by Theorem VI.4.1 we have σ (Tα ) ⊃ α ′ . (Actually σ (Tα ) = D.) This shows that in this case property (iv) does not hold. As to property (v) in the same theorem, the problem of its general validity is related to the following question. Does every contractive analytic function Θ (λ ), outer from both sides and such that Θ (eit ) is isometric on a set of positive measure, have the property that every regular divisor of Θ (λ ) is also outer from both sides? The negative answer to both these question is given in Chap. IX (see Sec. 4). Theorem 6.3 was stated in S Z .-N.–F. [IX] (Theorem 2) in a more general form, but the proof there was incomplete and it is still unknown whether in that general form the theorem holds. The restricted form given in the text was indicated in S Z .N.–F. [IX∗ ]. The fundamental problem, left open, is to elucidate the structure of the contractive analytic functions that are outer from one side and inner from the other side.

Chapter VIII

Weak Contractions 1 Scalar multiples 1. According to the usual definition, a self-adjoint operator A on a Hilbert space H, A ≥ O, is said to be of finite trace if A is compact (i.e., completely continuous) and the sum of its eigenvalues 6= 0 (each counted with the respective multiplicity) is finite. This sum is the trace of A and is denoted by tr(A). If A is of finite trace and if X is a bounded operator on H, then A′ = X ∗ AX is also of finite trace. In fact, A′ is compact, A′ ≥ O, and if Ah = ∑ µn (h, ϕn )ϕn ,

A′ h = ∑ µm′ (h, ϕm′ )ϕm′

n

m

(h ∈ H)

are the spectral decompositions of A and A′ with respect to the orthonormal systems {ϕn }, {ϕm′ } of eigenvectors (with the respective eigenvalues µn > 0, µm′ > 0), then we have ∑ µm′ = ∑(A′ ϕm′ , ϕm′ ) = ∑(AX ϕm′ , X ϕm′ ) = ∑ ∑ µn |(X ϕm′ , ϕn )|2 m

m

m

m n

= ∑ µn ∑ |(ϕm′ , X ∗ ϕn )|2 ≤ ∑ µn kX ∗ ϕn k2 ≤ kX ∗ k2 ∑ µn , n

so that

m

n

n

tr(X ∗ AX) ≤ kXk2 · tr(A).

It follows in particular that if PL is the orthogonal projection from H onto a subspace L, and if A is of finite trace, then PL APL and therefore AL = PL A|L are also of finite trace. Definition. A contraction T on Hilbert space H is called a weak contraction if (i) its spectrum σ (T ) does not fill the unit disc D, and (ii) I − T ∗ T is of finite trace.

Thus in particular all contractions T with finite defect index dT and with σ (T ) 6= D (among them all unitary operators) are weak contractions.

B.Sz.-Nagy et al., Harmonic Analysis of Operators on Hilbert Space, Universitext, DOI 10.1007/978-1-4419-6094-8_8, © Springer Science + Business Media, LLC 2010

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C HAPTER VIII. W EAK C ONTRACTIONS

Let us recall that if T is a contraction, then so is Ta = (T − aI)(I − aT ¯ )−1

(|a| < 1),

and we have I − Ta∗ Ta = S∗ (I − T ∗ T )S

with S = (1 − |a|2)1/2 (I − aT ¯ )−1 ;

see Sec. VI.1.3. On the other hand, σ (Ta ) is the image of σ (T ) by the homography λ → λa = (λ − a)/(1 − a¯λ ), we conclude that if T is a weak contraction, then so is Ta for every a ∈ D. Let T be a weak contraction and let us fix a point a ∈ D not belonging to σ (T ), that is, such that Ta is boundedly invertible. Let (I − Ta∗ Ta )h = ∑ µn (h, ϕn )ϕn

(1.1)

be the spectral representation of I − Ta∗ Ta by means of an orthonormal sequence of its eigenvectors ϕn corresponding to the respective eigenvalues µn 6= 0 (if there is no nonzero eigenvalue, the right-hand side of (1.1) should be taken to be 0). We have (Ta ϕm , Ta ϕn ) = (ϕm , ϕn ) − ((I − Ta∗ Ta )ϕm , ϕn ) = (1 − µm )(ϕm , ϕn ) = (1 − µm)δmn . As ϕn 6= 0 implies Ta ϕn 6= 0, we have 1 − µn > 0, and it follows that the vectors

ψn = (1 − µn)−1/2 Ta ϕn

(1.2)

also form an orthonormal sequence. Moreover, we have Ta∗ ψn = (1 − µn)−1/2 Ta∗ Ta ϕn = (1 − µn)−1/2 [ϕn − (I − Ta∗ Ta )ϕn ] = (1 − µn)1/2 ϕn , and hence

ϕn = (1 − µn)−1/2 Ta∗ ψn .

(1.3)

By (1.1)–(1.3) we have, for g ∈ H,

(I − Ta Ta∗ )Ta g = Ta (I − Ta∗ Ta )g = ∑ µn (g, ϕn )Ta ϕn

= ∑ µn (g, Ta∗ ψn )ψn = ∑ µn (Ta g, ψn )ψn .

As Ta H = H, this gives (I − Ta Ta∗ )h = ∑ µn (h, ψn )ψn

(1.4)

for every h ∈ H. This shows that I − Ta Ta∗ has the same nonzero eigenvalues as I − Ta∗ Ta ; hence tr(I − Ta∗ Ta ) = tr(I − Ta Ta∗ ). Moreover, σ (Ta∗ ) does not contain the point 0, thus Ta∗ is a weak contraction. Finally, the obvious relation T ∗ = (Ta∗ )−a¯

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333

implies that T ∗ is also a weak contraction. Thus, if T is a weak contraction, then so is T ∗ . An important property of weak contractions is established in the following theorem. Theorem 1.1. The characteristic function of every weak contraction T has a scalar multiple. Proof. Let us choose a point a as above and consider the characteristic function of Ta . The defect spaces of Ta are of the same dimension, because the sequences {ϕn } and {ψn } of eigenvectors of I − Ta∗ Ta and I − Ta Ta∗ , connected by (1.2) and (1.3), form orthonormal bases for these defect spaces. If this dimension is a finite number N, then the characteristic function ΘTa (λ ) has with respect to these bases the matrix [(ΘTa (λ )ϕi , ψ j )]

( j, i = 1, . . . , N).

(1.5)

As ΘTa (0) = −Ta |DT is invertible, we can apply Proposition V.6.1, and obtain that the determinant d(λ ) of the matrix (1.5) is a scalar multiple of ΘT (λ ). Moreover, we have (ΘTa (0)ϕi , ψ j ) = −(Ta ϕi , ψ j ) = −(1 − µi)1/2 (ψi , ψ j ) = −(1 − µi)1/2 δi j , and hence

N

|d(0)|2 = ∏(1 − µi) > 0.

(1.6)

1

(n)

If the defect spaces of Ta are infinite-dimensional, denote by P(n) and P∗ the orthogonal projections of these spaces onto the subspaces spanned by the vectors ϕ1 , . . . , ϕn and ψ1 , . . . , ψn , respectively. Define functions {DTa , DTa∗ , Θn (λ )} (n = 1, 2, . . .) by

∞

(n)

Θn (λ ) f = P∗ ΘTa (λ )P(n) f + ∑ ( f , ϕk )ψk n+1

( f ∈ DTa ).

These functions are contractive analytic and such that (n)

∞

Θn (λ )∗ g = P(n)ΘTa (λ )∗ P∗ g + ∑ (g, ψk )ϕk n+1

(g ∈ DTa∗ ).

As n → ∞, we have thus

Θn (λ ) → ΘTa (λ ),

Θn (λ )∗ → ΘTa (λ )∗

(strongly).

Let d (n) (λ ) be the determinant of the matrix [(ΘTa (λ )ϕi , ψ j )]

(i, j = 1, . . . , n).

(1.7)

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We have, for the same reason as (1.6), n

|d (n) (0)|2 = ∏(1 − µi),

(1.8)

1

(n)

and d (n) (λ ) is a scalar multiple of the contractive function {D(n) , D∗ , Θ (n) (λ )}, where D(n) = P(n) DTa ,

(n)

(n)

D∗ = P∗ DTa∗ ,

(n)

Θ (n) (λ ) = P∗ ΘTa (λ )|D(n) . (n)

Thus there exist contractive analytic functions {D∗ , D(n) , Ω (n) (λ )} (n = 1, . . .) such that

Ω (n) (λ )Θ (n) (λ ) = d (n) (λ )ID(n) ,

Θ (n) (λ )Ω (n) (λ ) = d (n) (λ )I

(n)

D∗

.

Setting ∞

(n)

Ωn (λ )g = Ω (n) (λ )P∗ g + d (n)(λ ) ∑ (g, ψk )ϕk n+1

(g ∈ DTa∗ )

we obtain contractive analytic functions {DTa∗ , DTa , Ωn (λ )} (n = 1, . . .) such that

Ωn (λ )Θn (λ ) = d (n) (λ )IDTa ,

Θn (λ )Ωn (λ ) = d (n) (λ )IDT ∗ . a

(1.9)

As the defect spaces of Ta are separable and the functions d (n) (λ ) and Ω (n) (λ ) are analytic on D with |d (n) (λ )| ≤ 1,

kΩn (λ )k ≤ 1

(λ ∈ D),

we know from the Vitali–Montel theorem that there exists a sequence of integers nq → ∞ for which d (nq ) (λ ) converges on D to an analytic function d(λ ), |d(λ )| ≤ 1, and Ωnq (λ ) converges on D (weakly) to a contractive analytic function Ω (λ ). Using also of (1.7) we deduce from (1.9) that

Ω (λ )ΘTa (λ ) = d(λ )IDTa , Moreover, (1.8) implies

ΘTa (λ )Ω (λ ) = d(λ )IDT ∗ . a

∞

|d(0)|2 = ∏(1 − µi ); 1

as 1 − µi > 0 (i = 1, . . .) and ∑ µi = tr(I − Ta∗ Ta ) < ∞, we have thus d(0) 6= 0. Hence d(λ ) 6≡ 0, and consequently d(λ ) is a scalar multiple of the characteristic function of Ta . The relation between the characteristic functions of T and Ta (cf. Sec. VI.1.3) implies that T also has equal defect indices and that δ (λ ) = d((λ − a)(1 − a¯λ )) is a scalar multiple of ΘT (λ ).

2. D ECOMPOSITION C0 –C11

335

2. Let the contraction T on H be such that I − T ∗ T is of finite trace, and let L be an invariant subspace for T . For TL = T |L we have then IL − TL∗ TL = PL (I − T ∗ T )|L, PL denoting the orthogonal projection onto L. Thus IL − TL∗ TL is also of finite trace. If σ (TL ) 6= D, then TL is also a weak contraction. But, in general, TL may not be a weak contraction if T is a weak contraction. An example is furnished by the regular factorization   # 1 "√ √ λ 3 3 1 2  (λ ∈ D) λ= λ ·√ (1.10) 3 2 2 2 2 √ of the scalar-valued function Θ (λ ) ≡ 3λ /2; see Sec. VII.3.4. Θ (λ ) is the characteristic function of a c.n.u. contraction T with defect indices equal to 1, and with σ (T ) ∩ D = {0}. Thus T is a weak contraction. The regular factorization (1.10) induces, in the sense of Theorem VII.1.1, an invariant subspace L for T . As the second factor in (1.10) is a purely contractive analytic function {E 1 , E 2 , Θ1 (λ )}, it coincides with the characteristic function of TL = T |L (cf. Proposition VII.2.1). Θ1 (λ ) maps, for every fixed λ , the space E 1 into the space E 2 , so it cannot have an inverse defined on the whole of E 2 , and hence (by virtue of Theorem VI.4.1) we have σ (TL ) = D; thus TL is not a weak contraction.

2 Decomposition C0 –C11 1. Let T be any contraction on the space H, and let T = T (0) ⊕ T (u) be its canonical decomposition into its unitary part T (u) and c.n.u. part T (0) . Then we have I − T ∗ T = [I (0) − T (0)∗ T (0) ] ⊕ O and σ (T ) ∩ D = σ (T (0) ) ∩ D. Hence we infer that if T is a weak contraction then so is T (0) , and conversely. Therefore it does not restrict generality if we suppose in the sequel that T is c.n.u. Thus let T be a c.n.u. weak contraction on H. Let   T0 X T= , H = H0 ⊕ H′1 (2.1) O T1′ be the triangulation of T of type   C0· ∗ . O C1·

(2.2)

This corresponds to the ∗-canonical factorization of ΘT (λ ) (cf. Sec. VII.3.3); thus the characteristic functions of T0 and T1′ coincide with the purely contractive parts of the ∗-inner and ∗-outer factors of ΘT (λ ), respectively. We have just proved that

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ΘT (λ ) admits a scalar multiple (Theorem 1.1); this implies, by virtue of Theorem V.6.2 and Proposition V.6.8, that the factors in the ∗-canonical factorization of ΘT (λ ), as well as their purely contractive parts, also admit scalar multiples. Hence it follows that these purely contractive parts are, respectively, inner and outer from both sides (cf. Theorem V.6.2). Consequently, T0 belongs to C00 and hence also to C0 , whereas T1′ belongs to C11 (cf. Proposition VI.3.5 and Theorem VI.5.1). Let mT0 (λ ) be the minimal function of T0 (mT0 (λ ) ≡ 1 if and only if H0 = {0}). By virtue of Theorem III.5.1, the spectrum of T0 in D consists of the zeros of mT0 (λ ) and hence is a discrete set in D. Hence T0 = T |H0 is also a weak contraction. Moreover, if L is an arbitrary subspace of H, invariant for T and such that TL = T |L ∈ C0 , then we have T n l = TLn l → 0 for l ∈ L, n → ∞ (because TL ∈ C0 ⊂ C00 ), and this implies L ⊂ H0 by virtue of the relation H0 = {h : h ∈ H, T n h → 0}; see (II.4.3). Hence H0 is the maximal invariant subspace for T on which T induces a contraction of class C0 . As to T1′ , because it belongs to C11 and its characteristic function has a scalar multiple, its spectrum must lie entirely on the circle C (cf. Proposition VI.4.4). Moreover, T1′∗ = T ∗ |H′1 and T ∗ is a weak contraction, thus we conclude that T1′∗ , and hence T1′ as well, are weak contractions. We now show that H′1 is maximal among those subspaces L∗ invariant for T ∗ for which T ∗ |L∗ ∈ C11 . In fact, if L∗ is such a subspace, set T∗ = (T ∗ |L∗ )∗

and L′∗ = {l : l ∈ L∗ , mT0 (T∗ )l = 0}.

L′∗ is invariant for T∗ and we have mT0 (T∗′ ) = O for T∗′ = T∗ |L′∗ . It follows that T∗n |L′∗ = T∗′n → O

as n → ∞.

Because T∗ ∈ C11 , this implies L′∗ = {0}. Hence l ∈ L∗

and mT0 (T∗ )l = 0 imply l = 0,

and this statement is obviously equivalent to the following, mT0 (T∗ )∗ L∗ = L∗ . Now we have for u ∈ H ∞ u(T∗ )∗ = u˜(T∗∗ ) = u˜(T ∗ |L∗ ) = u˜(T ∗ )|L∗ = u(T )∗ |L∗ ,

(2.3)

2. D ECOMPOSITION C0 –C11

337

and hence for l ∈ L∗ and h ∈ H0 (u(T∗ )∗ l, h) = (u(T )∗ l, h) = (l, u(T )h) = (l, u(T0 )h). In particular, taking u = mT0 this gives (mT0 (T∗ )∗ l, h) = 0 (l ∈ L∗ , h ∈ H0 ); thus mT0 (T∗ )∗ L∗ ⊥ H0 . By virtue of (2.3) this implies L∗ ⊥ H0 and hence L∗ ⊂ H′1 . In addition to the triangulation (2.1) of type (2.2) let us also consider the triangulation   T Y T = 1 ′ , H = H1 ⊕ H′0, (2.4) O T0 of type

  C·1 ∗ . O C·0

(2.5)

By arguments similar to the above, or simply observing that (2.4) is equivalent to the triangulation of T ∗ of type (2.2):  ′∗ ∗  T Y T ∗ = 0 ∗ , H = H′0 ⊕ H1, O T1 one sees that T1 and T0′ are weak contractions of classes C11 and C0 , respectively, that H1 includes all the subspaces M invariant for T such that T |M ∈ C11 , and H′0 includes all the subspaces M∗ invariant for T ∗ such that T ∗ |M∗ ∈ C0 . We call T0 the C0 part and T1 the C11 part of T . Obviously, the C0 and C11 parts of T ∗ are then T0′∗ and T1′∗ , respectively. The subspaces H0 and H1 are invariant for T , therefore so is H0 ∩ H1 . The restriction of T to this intersection (being at the same time a restriction of T0 ∈ C0 and a restriction of T1 ∈ C11 ) belongs to C0· and to C1· , which is impossible unless H0 ∩ H1 = {0}. Hence H0 ∩ H1 = {0}. This result, applied to T ∗ instead of T , yields H′0 ∩ H′1 = {0}; by virtue of the obvious relation H ⊖ (H0 ∨ H1 ) = H′0 ∩ H′1 , this implies H0 ∨ H1 = H. We turn now to the study of the relations between the spectra. We observe first that an operator matrix   X X M= 1 , O X2

whose entries are bounded operators, is boundedly invertible with the inverse   −1 X −X1−1X X2−1 M −1 = 1 (2.6) O X2−1

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provided X1 and X2 are boundedly invertible. From (2.4) follows the relation   Y T − λ I1 , T −λI = 1 O T0′ − λ I0′ which together with (2.6) gives

σ (T ) ⊂ σ (T1 ) ∪ σ (T0′ ).

(2.7)

We show next that the contractions T0 and T0′ , of class C0 , have the same minimal function, that is, mT0 (λ ) = mT ′ (λ ). (2.8) 0

This implies, by virtue of Theorem III.5.1, that σ (T0 ) = σ (T0′ ); thus (2.7) takes the form σ (T ) ⊂ σ (T1 ) ∪ σ (T0 ). (2.9)

To this end, let us consider the canonical and ∗-canonical factorizations of ΘT (λ ), ΘT (λ ) = Θ0′ (λ )Θ1 (λ ), ΘT (λ ) = Θ1′ (λ )Θ0 (λ ); (2.10) they correspond to the triangulations (2.4) and (2.1) of T , respectively. mT0 (λ ) is a scalar multiple of ΘT0 (λ ) (cf. Theorem VI.5.1) and hence of Θ0 (λ ) too (because the purely contractive part of Θ0 (λ ) coincides with ΘT0 (λ )). On the other hand, the purely contractive part of Θ1′ (λ ) coincides with ΘT ′ (λ ), which is outer 1 from both sides (because T1′ ∈ C11 ). ΘT ′ (λ ) admits a scalar multiple (because T1′ 1 is a weak contraction), and by virtue of Theorem V.6.2 it admits even an outer ′ scalar multiple δe (λ ). Thus Θ1 (λ ) also admits δe (λ ) as a scalar multiple. It follows from these results that the function ΘT (λ ) (= Θ1′ (λ )Θ0 (λ )) has the scalar multiple δ (λ ) = δe (λ )mT0 (λ ). Therefore the inner factor Θ0′ (λ ) in the canonical factorization ΘT (λ ) = Θ0′ (λ )Θ1 (λ ) will have as a scalar multiple the inner factor of δ (λ ), that is, mT0 (λ ) (cf. Proposition V.6.2). The purely contractive part of Θ0′ (λ ) coincides with ΘT ′ (λ ), therefore ΘT ′ (λ ) will also have the scalar multiple mT0 (λ ); 0 0 as T0′ ∈ C0 this implies mT0 (T0′ ) = O (cf. Theorem VI.5.1). Therefore mT ′ is an in0 ner divisor of mT0 . If we repeat these arguments, interchanging the roles of the two factorizations (2.10), we arrive at the result that mT0 is an inner divisor of mT ′ . The 0 two results together imply (2.8). The inclusion (2.9) being thereby established, we now prove its opposite, namely that σ (T0 ) ⊂ σ (T ) and σ (T1 ) ⊂ σ (T ). (2.11)

The points of σ (T0 ) in the interior of the unit circle C are eigenvalues of T0 (because T0 ∈ C0 ; see Proposition III.7.1) and hence also of T . On the other hand, σ (T1 ) is confined to C: this follows from Proposition VI.4.4 because T1 ∈ C11 and ΘT1 (λ ) admits a scalar multiple. Thus, in order to obtain (2.11), it remains only to consider the parts of the spectra on C.

2. D ECOMPOSITION C0 –C11

Now the inclusion

339

σ (T |L) ∩C ⊂ σ (T )

(2.12)

holds for every contraction T on H and for every invariant subspace L for T . In fact, we infer from the expansion ∞

(µ I − T )−1 = ∑ µ −n−1T n , 0

valid for |µ | > 1, that L is invariant for (µ I − T )−1 too and that (µ I − T )−1 |L = (µ IL − T |L)−1

(|µ | > 1).

(2.13)

If ε is a point of C belonging to the resolvent set of T , then (µ I − T )−1 converges in norm to (ε I − T )−1 as µ → ε , and thus the right-hand side of (2.13) also converges in norm to a limit, this being necessarily equal to (ε I − T |L)−1 . Thus every point ε ∈ C that belongs to the resolvent set of T , also belongs to the resolvent set of T |L. This implies (2.12). We conclude that the relations (2.11) are valid. Together with (2.9), they imply the equality σ (T ) = σ (T1 ) ∪ σ (T0 ). Summarizing, we have the following result. Theorem 2.1. (Decomposition C0 − C11 .) Let T be a completely nonunitary weak contraction on H. Among the subspaces L, invariant for T and such that T |L ∈ C0 , there exists a maximal one, denoted by H0 . Also, among the subspaces M, invariant for T and such that T |M ∈ C11 , there exists a maximal one, denoted by H1 . The contractions T0 = T |H0 and T1 = T |H1 , called the C0 part and the C11 part of T , are equal to those appearing in the triangulations     T0 X T1 Y T= and T = O T1′ O T0′ of type

  C0· ∗ O C1·

and

  C·1 ∗ , O C·0

respectively. T0′∗ and T1′∗ are then the C0 and C11 parts of T ∗ . All these contractions are weak ones; moreover, T0 and T0′ have the same minimal function. We have H0 ∨ H1 = H, H0 ∩ H1 = {0}, σ (T ) = σ (T0 ) ∪ σ (T1 ),

(2.14) (2.15)

σ (T1 ) lying on the unit circle C. Remark. Together with Theorem III.5.1, this implies that for every weak contraction (c.n.u. or not) the part of the spectrum interior to C is discrete.

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2. We have shown by an example in Sec. 1 that the restriction of a weak contraction to an invariant subspace need not be a weak contraction. But if it is, some important relations hold. Proposition 2.2. Under the conditions of Theorem 2.1, let L be an invariant subspace for T such that TL = T |L is also a weak contraction. For the spaces L0 and L1 of the C0 and C11 parts of TL we have then L0 = L ∩ H0 ,

L1 = L ∩ H1 .

(2.16)

Proof. As L0 is invariant for T , with T |L0 = TL |L0 ∈ C0 , it follows from the maximality of H0 that L0 ⊂ H0 . Similarly, we have L1 ⊂ H1 . Thus L0 ⊂ L ∩ H0 ,

L1 ⊂ L ∩ H1 .

(2.17)

On the other hand, as L0 is the space of the C0· part in the triangulation of TL of type (2.2), we have L0 = {l : l ∈ L, TLn l → 0 (n → ∞)}.

(2.18)

As TLn l = T n l for l ∈ L, it follows hence that L0 ⊃ L ∩ H0 . The first of the relations (2.16) is thereby proved. As regards the second one, we know (2.17) and it remains to prove that L1 ⊃ L ∩ H1 . Recalling the maximality property of L1 it suffices to prove to this effect that T∧ = T |H∧ ∈ C11 , where H∧ = L ∩ H1 . (2.19) Now because T∧ = (T |H1 )|H∧ = T1 |H∧ and T1 ∈ C11 , we have T∧ ∈ C1· . In order to prove (2.19) it suffices therefore to prove that T∧ is a weak contraction (indeed, its characteristic function then admits a scalar multiple, thus this characteristic function is not only ∗-outer, but outer from both sides, and hence T∧ is not only of class C1· , but also of class C11 ). Because T∧ is the restriction of the weak contraction T to the invariant subspace H∧ , we have proved that T∧ is also a weak contraction if we show that σ (T∧ ) does not include the whole disc D. To show this let us consider the triangulation of T∧ with respect to the invariant subspace L1 ,  ′  T X , H∧ = L1 ⊕ L′′ . (2.20) T∧ = O T ′′ As T ′ = T∧ |L1 = TL |L1 , T ′ is the C11 part of TL ; hence σ (T ′ ) ⊂ C. As to T ′′ , we observe that L′′ is contained in L ⊖ L1 (i.e., in the space of the C0 part of TL∗ ); let us denote this part of TL∗ simply by S. Let u(λ ) be any function in H ∞ . From the facts that (i) L′′ is invariant for T∧∗ , (ii) H∧ is invariant for TL , (iii) L ⊖ L1 is invariant for TL∗ , (iv) H∧ ⊃ L′′ , and (v) L ⊖ L1 ⊃ L′′ , we deduce: u(T ′′∗ ) = u(T∧∗ )|L′′ = [P∧ u(TL∗ )|H∧ ]|L′′ = P∧ u(TL∗ )|L′′ = [P∧ u(TL∗ )|L ⊖ L1 ]|L′′ = P∧ u(S)|L′′ ,

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341

where P∧ denotes the orthogonal projection of H onto H∧ . Choosing for u in particular the minimal function of S, we obtain mS (T ′′∗ ) = O,

m˜S (T ′′ ) = O.

Hence T ′′ ∈ C0 , and so the part of σ (T ′′ ) in D is a discrete set. Recalling (2.7), we see that

σ (T∧ ) ∩ D ⊂ (σ (T ′ ) ∪ σ (T ′′ )) ∩ D = σ (T ′′ ) ∩ D ⊂ σ (T ′′ ); hence σ (T∧ ) does not cover D. The proof is complete. 3. The following two propositions give useful complements to the preceding results. Proposition 2.3. If the spectrum of the weak contraction T does not include the whole unit circle C, then the restriction of T to any invariant subspace L is a weak contraction. Proof. We only have to show that the spectrum of TL = T |L does not include the whole disc D. To this end, we start with the relations [(µ I − T )−1 |L] · [µ IL − TL ] = [µ IL − TL ] · [(µ I − T )−1 |L] = IL ,

(2.21)

valid for |µ | > 1 by (2.13). Next we observe that our assumptions on T imply that the resolvent set of T is a domain G containing all the exterior of C, one or more arcs of C, and all the interior of C except a discrete set of points. Relations (2.21), valid on the exterior of C, extend by analyticity to the whole domain G (indeed, µ IL − TL is an analytic function of µ on the whole plane, whereas (µ I − T )−1 is analytic on G). Hence the resolvent set of TL includes G; consequently σ (TL ) ∩ D is a discrete set. Proposition 2.4. Under the assumptions of Theorem 2.1 we have H0 = {h : h ∈ H, mT0 (T )h = 0}

(2.22)

H1 = mT0 (T )H.

(2.23)

and As a consequence H0 and H1 are even hyperinvariant for T . Proof. Denoting the subspace defined by the right-hand side of (2.22) by L0 , it is obvious that H0 ⊂ L0 ; indeed we have mT0 (T )h = mT0 (T0 )h = 0 for h ∈ H0 . On the other hand, L0 is invariant for T and we have mT0 (T |L0 ) = O. Thus T |L0 ∈ C0 , which implies by the maximality property of H0 that L0 ⊂ H0 . This proves the equation (2.22).

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As regards (2.23), let us observe first that H ⊖ mT0 (T )H = {h : h ∈ H, mT0 (T )∗ h = 0},

(2.24)

and that, because T0 and T0′ have the same minimal function, mT0 (T )∗ = mT ′ (T )∗ = mT˜′ (T ∗ ) = mT ′∗ (T ∗ ). 0

0

0

Now, because T0′∗ is the C0 part of T ∗ , relation (2.22) applied to T ∗ instead of T shows that the right-hand side of (2.24) is equal to H′0 , that is, to H ⊖ H1 . This implies the validity of (2.23).

3 Spectral decomposition of weak contractions 1. Let T be a c.n.u. weak contraction on H, and let T0 and T1 be the C0 part and the C11 part of T (T0 = T |H0 , T1 = T |H1 ). Let   Z 2π it e +λ (3.1) d µt mT0 (λ ) = B(λ ) · exp − eit − λ 0 be the parametric representation of the minimal function as an inner function: B(λ ) is a Blaschke product and µ is a nonnegative, bounded, singular measure defined for the Borel subsets of (0, 2π ). With every Borel subset ω of the plane of complex numbers we associate the inner function   Z eit + λ (3.2) mω (λ ) = Bω (λ ) · exp − d µ t , it (C∩ω ) e − λ where Bω (λ ) means the product of those factors of B(λ ) whose zeros lie in ω . We define two new subspaces: H0 (ω ) = {h : h ∈ H0 , mω (T0 )h = 0},

(3.3)

and H1 (ω ) is the subspace of H1 associated with the contraction T1 (the C11 part of T ) and the Borel set α =C∩ω

in the sense of Theorem VII.5.2. The space H0 (ω ) is hyperinvariant for T0 (evident from (3.3)) and H1 (ω ) is hyperinvariant for T1 (by Theorem VII.5.2). In particular, H0 (ω ) is invariant for T0 , H1 (ω ) is invariant for T1 , and hence both are invariant for T . The subspace H(ω ) = H0 (ω ) ∨ H1 (ω ) (3.4) is then also invariant for T . We show that H(ω ) is even hyperinvariant for T . To this effect let us consider an arbitrary bounded operator X commuting with T . As H0 and H1 are hyper-

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343

invariant for T (cf. Proposition 2.4), we have X H j ⊂ H j , moreover X j = X |H j commutes with T j ( j = 0, 1). Because H j (ω ) is hyperinvariant for T j , we have XH j (ω ) = X j H j (ω ) ⊂ H j (ω ) ( j = 0, 1), and hence X H(ω ) ⊂ H(ω ), which proves the hyperinvariance of H(ω ) for T . Next we show that T (ω ) = T |H(ω ) is a weak contraction. To this effect let us also consider the operators T0 (ω ) = T0 |H0 (ω ) (= T |H0 (ω )) and T1 (ω ) = T1 |H1 (ω ) (= T |H1 (ω )). From the definition (3.3) of H0 (ω ) and from Theorem III.6.3 it follows that T0 (ω ) has its minimal function equal to mω . Because the zeros of mω in D are a subset of the zeros of mT0 , it follows from Theorem III.5.1 and from (2.15) that σ (T0 (ω )) ∩ D ⊂ σ (T0 ) ∩ D = σ (T ) ∩ D. (3.5) As to T1 (ω ), its spectrum lies entirely on the unit circle C (cf. Theorem VII.6.2, applied to T1 ∈ C11 and to α = C ∩ ω ). Let a be a point of D not belonging to σ (T ). Then it belongs neither to σ (T0 (ω )) nor to σ (T1 (ω )), so we have (T j (ω ) − aI j )H j (ω ) = H j (ω ) ( j = 0, 1; I j denoting the identity operator on H j ). This implies (T − aI)H(ω ) = (T − aI)[H0(ω ) ∨ H1 (ω )]

= (T0 (ω ) − aI0)H0 (ω ) ∨ (T1 (ω ) − aI1)H1 (ω ) = H0 (ω ) ∨ H1 (ω ) = H(ω ).

As T − aI is an affinity on H, we infer that T (ω ) − aIH(ω ) is an affinity on H(ω ): the point a does not belong to the spectrum of T (ω ). Therefore T (ω ) is a weak contraction. Let H(ω )0 and H(ω )1 , respectively, denote the subspaces of H(ω ) on which the C0 and C11 parts of T (ω ) act. We are going to prove that H(ω ) j = H j (ω )

( j = 0, 1).

(3.6)

To being with, we observe that, by virtue of Proposition 2.2, H(ω ) j = H(ω ) ∩ H j

( j = 0, 1).

(3.7)

We have H j (ω ) ⊂ H j by the definition of H j (ω ), and H j (ω ) ⊂ H(ω ) by virtue of the definition (3.4) of H(ω ), thus (3.7) implies H(ω ) j ⊃ H j (ω )

( j = 0, 1).

(3.8)

To establish (3.6) it remains to prove the opposite inclusion. Now, every h ∈ H(ω ) is the limit of a sequence {h0n + h1n} with h0n ∈ H0 (ω ) and h1n ∈ H1 (ω ), thus mω (T )h = lim mω (T )(h0n + h1n) = lim mω (T )h1n ∈ H1 (ω ) ⊂ H1 . n

n

If h ∈ H0 we have, on the other hand, mω (T )h = mω (T0 )h ∈ H0 . As H0 ∩ H1 = {0}, it follows that mω (T )h = 0 for h ∈ H(ω ) ∩ H0 , that is, h ∈ H0 (ω ). Hence we

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have H(ω ) ∩ H0 ⊂ H0 (ω ). Recalling (3.7) (case j = 0) we obtain H(ω )0 ⊂ H0 (ω ). Together with (3.8) (case j = 0) this yields (3.6) for j = 0. As regards the case j = 1, let us first recall that the minimal function of T0 (ω ) equals mω . Applying Proposition 2.4 to T (ω ) we obtain: H(ω )1 = mω (T (ω ))H(ω ) = mω (T )H(ω ) = mω (T )(H0 (ω ) ∨ H1 (ω )) = mω (T0 )H0 (ω ) ∨ mω (T1 )H1 (ω ) = mω (T1 )H1 (ω ) ⊂ H1 (ω );

together with (3.8) (case j = 1) this yields (3.6) for j = 1. We now show that the spaces H(ω ) enjoy some maximality properties. Let L ⊂ H be a hyperinvariant subspace for T and let TL = T |L. Then σ (TL ) ⊂ σ (T ) and hence TL is a weak contraction (see Sect. 1.2). Let L0 and L1 denote the spaces of the C0 part TL0 of TL and the C11 part TL1 of TL , respectively. By Proposition 2.2 we know that L0 ⊂ H0 and L1 ⊂ H1 . Assume that mTL (λ ) divides mω (λ ) and that the set 0 ω ∩ C is residual for TL1 (see Definition 3 in Sec. VII.5.1). Then due to Theorems III.6.3(ii) and VII.5.2, we have L0 ⊂ H0 (ω ) and L1 ⊂ H1 (ω ). By virtue of (2.14) (applied to TL and L) and of Definition (3.4) we have L = L0 ∨ L1 ⊂ H0 (ω ) ∨ H1 (ω ) = H(ω ). From the relations (3.6) established above we obtain, applying (2.15) to T (ω ) instead of T , that σ (T (ω )) = σ (T0 (ω )) ∪ σ (T1 (ω )).

Now from Theorem III.5.1 it follows that σ (T0 (ω )) ⊂ ω , and from Theorem VII.6.2 it follows that σ (T1 (ω )) ⊂ α ⊂ ω (where α = C ∩ ω ). So we obtain

σ (T (ω )) ⊂ ω

(3.9)

for every Borel subset ω of the set of complex numbers. If moreover ω is a closed set, we show that H(ω ) contains all the subspaces L of H which are invariant for T and such that (a) TL = T |L is a weak contraction, and (b) σ (TL ) ⊂ ω . If L0 and L1 are the spaces of the C0 and C11 parts of TL , we derive from (2.15) that

σ (TL0 ) ∪ σ (TL1 ) = σ (TL ) ⊂ ω , where TL j = TL |L j = T |L j ( j = 0, 1). (3.10) As TL0 ∈ C0 , we have L0 ⊂ H0 by the maximality property of H0 . This implies mT0 (TL0 ) = mT0 (T0 )|L0 = O; thus, denoting the minimal function of TL0 by l0 , l0 is a divisor of mT0 . On the other hand, (3.10) implies σ (TL0 ) ⊂ ω ; thus, by Theorem III.5.1, l0 has the following properties. (i) Its zeros in D belong to the set ω , and (ii) it is analytic on every open arc of C, disjoint from the closed set α = C ∩ ω . Now we deduce from (3.1) and (3.2) that the inner divisors of mT0 with these properties are also divisors of mω . Because L0 ⊂ H0 , we have for h ∈ L0 l0 (T0 )h = l0 (TL0 )h = 0;

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as l0 is a divisor of mω this implies mω (T0 )h = 0; as a consequence of (3.3) we have therefore h ∈ H0 (ω ). Thus L0 ⊂ H0 (ω ). (3.11)

On the other hand, because (3.10) also implies σ (TL1 ) ⊂ ω , the Borel set α = C ∩ ω is residual for the operator TL1 ∈ C11 (cf. Theorem VI.4.1); by virtue of the maximality property of the subspaces of type Hα (established in Theorem VII.5.2) we conclude that L1 ⊂ H1 (ω ). (3.12) Applying (2.14) to L (instead of H) and using (3.11) and (3.12), we obtain L = L0 ∨ L1 ⊂ H0 (ω ) ∨ H1 (ω ) = H(ω ), thus establishing the maximality property of H(ω ). The above arguments prove Parts (i) and (ii) of the following theorem. Theorem 3.1. (Spectral decomposition) Let T be a c.n.u. weak contraction on H. Then to every Borel subset ω of the plane of complex numbers there corresponds a unique subspace H(ω ) of H with the following properties. (i) H(ω ) is the maximal hyperinvariant subspace L for T satisfying the following conditions. The minimal function of TL0 divides mω and ω ∩ C is residual for TL1 (see the discussion above). (ii) T (ω ) = T |H(ω ) is a weak contraction and σ (T (ω )) ⊂ ω ; moreover, if ω is closed, then H(ω ) is maximal under these conditions, that is, H(ω ) includes all the subspaces L, invariant for T and such that TL = T |L is a weak contraction with σ (TL ) ⊂ ω . W T T S (iii) H( n ωn ( ) = n H(ωn ), H( n ωn ) = n H(ωn ) for any sequence {ωn }. {0} if ω is empty, (iv) H(ω ) = H if ω ⊃ σ (T ). (v) H(ω ) 6= {0} if ω is open and ω ∩ σ (T ) is not empty. Proof. It remains to prove Properties (iii), (iv) and (v). The first property (iii), respectively, property (iv) are simple, respectively, trivial consequences of the corresponding properties of the spaces H0 (ω ) and H1 (ω ) and the definition (3.4) of H(ω ). For the first one we notice that m∪ωn is the smallest common inner multiple of the functions mωn and then apply Theorem III.6.3(iii), as for the spaces H1 (ωn ) we apply Theorem VII.5.2. Concerning the second property (iii), we observe that m∩ωn is the largest common inner divisor of the functions mωn and thus by virtue of Theorem III.6.3(iii) we have   T T H0 ωn = H0 (ωn ). n

n

On the other hand by applying Theorem VII.6.2 (v) we also have   T T H1 ωn = H1 (ωn ). n

n

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It follows that T

T

L = H(ωn ) = (H0 (ωn ) ∨ H1 (ωn )) n n       T T T ⊃ H1 (ωn ) = H H0 (ωn ) ∨ ωn . n

n

n

T

But the space L satisfies the conditions in Property (i) with respect to ω = n ωn T hence L ⊂ H( n ωn ). This completes the proof of Property (iii). Let us now consider an open set ω . If H0 (ω ) = {0}, then mT0 = mT0 /mω by Theorem III.6.3. Hence mω = 1 and therefore no zero of the Blaschke product B lies in ω ∩ D, and µ (β ) = 0 for the open arcs β of C of which α = C ∩ ω is composed. But µ (β ) = 0 implies that mT0 is analytic on β . It follows from Theorem III.5.1 that ω is contained in the resolvent set of T0 . On the other hand, if H1 (ω ) = Hα = {0}, it follows from Theorem VII.5.2 that C\α is residual for T1 and hence ΘT1 (eit ) is isometric a.e. in (α ); by virtue of Proposition VI.4.4 the set α (composed of open arcs) is contained in the resolvent set of T1 . From these results it follows that if H(ω ) = {0}, then ω is contained in the resolvent sets of T0 and T1 , so its intersection with σ (T ) = σ (T0 ) ∪ σ (T1 ) is void. This implies the validity of (v) and thus concludes the proof of Theorem 3.1. 2. By virtue of this theorem the subspaces H(ω ) have properties analogous to those of the spectral subspaces for a normal operator. In particular, if σ (T ) has more than one point, this theorem establishes a nontrivial spectral decomposition of the space H, and, consequently, the existence of nontrivial hyperinvariant subspaces for T . Consider indeed two disjoint open sets, say ω1 and ω2 , both having nonempty intersections with σ (T ). The corresponding spaces H(ω1 ) and H(ω2 ) will then be different from {0}, and because H(ω1 ) ∩ H(ω2 ) = {0}, neither of them will equal H; thus H(ω1 ) and H(ω2 ) will be nontrivial disjoint subspaces of H, both hyperinvariant for T . Let us consider now a weak contraction T on H, whose spectrum consists of a single point τ . We distinguish two cases accordingly as |τ | < 1 or |τ | = 1. (1) If |τ | < 1, then we have, by the spectral radius theorem, kT n k1/n → |τ | < 1 (cf. [Func. Anal.] Sec. 149), hence kT n k → 0. Consequently, T has no unitary part and even no C11 part; thus T ∈ C0 . From Theorem III.5.1 it follows that the minimal function is of the form   λ −τ n mT (λ ) = (n a positive integer). (3.13) ¯ 1 − τλ Now for a weak contraction T , (3.13) implies dim H < ∞. In fact, denoting by Lτ the subspace formed by the solutions h of the equation T h = τ h, Lτ is invariant for T and Tτ = T |Lτ satisfies the equation Iτ − Tτ∗ Tτ = (1 − |τ |2)Iτ (Iτ : the identity operator on Lτ ). Since (1−|τ |2 )Iτ has to be of finite trace, we have dim Lτ = dτ < ∞. Now let L be an arbitrary finite dimensional subspace of H, say of dimension d.

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Because (3.13) implies (T − τ I)n = O, and thus T n = c0 I + c1 T + · · · + cn−1 T n−1 ,

(3.14)

the subspace M spanned by L, T L, . . . , T n−1 L is invariant for T . As M has finite dimension (≤ nd), we can choose in M a basis with respect to which the matrix of T |M has Jordan normal form. From (3.14) it follows that the orders of the Jordan cells in this matrix do not exceed n; hence the number ν of these cells is ≥ dim M/n. To every cell corresponds an eigenvector of T , and the eigenvectors so obtained are linearly independent, thus we must have ν ≤ dτ . Consequently, d ≤ dim M ≤ nν ≤ ndτ . This implies obviously that H itself has dimension ≤ ndτ . (2) For any contraction T such that σ (T ) = {τ }, |τ | = 1, a necessary and sufficient condition in order that T be c.n.u., is that τ not be an eigenvalue of T . This follows readily from the fact, implied by the spectral theorem, that every isolated point of the spectrum of a unitary operator is an eigenvalue. Let us assume therefore that T is a contraction on H with σ (T ) = {τ }, |τ | = 1, and that τ is not an eigenvalue of T . As T is then c.n.u., we deduce from Proposition II.6.7 that T ∈ C00 . If, moreover, the characteristic function of T has a scalar multiple—in particular, if T is a weak contraction on a separable space H—then T ∈ C00 implies T ∈ C0 . From Theorem III.5.1 we deduce that the minimal function of T is of the form ¯ ) (a > 0), mT (λ ) = ea (τλ (3.15) where ea (λ ) = exp(a(λ + 1)/(λ − 1)); see the proof of Proposition III.7.3. It is clear that this case can only occur if H is of infinite dimension. We come back to the study of such contractions in Chapter X.

4 Dissipative operators. Class (Ω0+ ) 1. By virtue of the relations between a one-parameter continuous semigroup of contractions and its cogenerator, and between a maximal accretive or dissipative operator and its Cayley transform, our results obtained from Chap. VI on, concerning functional models, invariant subspaces, spectral decompositions, and so on, carry over in a more or less immediate manner from individual contractions to oneparameter continuous semigroups of contractions, or to accretive and dissipative operators (as regards the models of semigroups see the Notes to Chap. VI). Let us consider, for example, operators on the space H of the form A = R + iQ,

(4.1)

where R and Q are self-adjoint operators, with O ≤ Q ≤ 2q · I.

(4.2)

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It is obvious that A is dissipative. The resolvent set of A contains in particular the points z = x + iy with y < 0 and y > 2q. In fact, we have A − zI = M + N,

where

M = (R − xI) + i(q − y)I,

N = i(Q − qI);

condition (4.2) implies that kNk ≤ q, and if y 6= q then M is boundedly invertible, with kM −1 k ≤ |q − y|−1. As a consequence we have A − zI = M(I + M −1 N),

where

kM −1 Nk ≤ q · |q − y|−1.

The right-hand side of the last inequality is < 1 if y < 0 or y > 2q; hence, in these cases, A − zI also is boundedly invertible. It follows in particular that the Cayley transform of A, that is, the operator T = (A − iI)(A + iI)−1 = I − 2i(A + iI)−1

(cf. (IV.4.12))

(4.3)

is defined everywhere on H, which implies that A is maximal dissipative. Moreover, because the homography z→λ =

z−i z+i

(4.4)

transforms the points z(6= −i) of the resolvent set of A to points of the resolvent set of T , it follows that the spectrum of T does not cover the unit disc |λ | < 1. Moreover, if A is bounded, the spectrum of T does not even cover the circle |λ | = 1. We obtain by a simple calculation: I − T ∗ T = J ∗ QJ, where

I − T T ∗ = JQJ ∗ ,

(4.5)

J = 2i(A + iI)−1 = I − T ;

thus it follows that the transformations

τ : DT h → Q1/2 Jh,

τ∗ : DT ∗ h → Q1/2 J ∗ h (h ∈ H)

(4.6)

are isometric. On account of the relation Q1/2 JH = Q1/2 JH = Q1/2 H = QH, and the analogous one with J ∗ in place of J, the transformations (4.6) extend by continuity to unitary ones:

τ : DT → Q,

τ∗ : DT ∗ → Q, where Q = QH.

(4.7)

It follows that the characteristic function {DT , DT ∗ , ΘT (λ )} of T coincides with {Q, Q, τ∗ΘT (λ )τ −1 }.

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Let us set

ΘA (z) = τ∗ΘT (λ )τ

−1

 1+λ z=i , 1−λ

|λ | < 1



(4.8)

and calculate ΘA (z) explicitly. On account of the relation

ΘT (λ )DT = DT ∗ (I − λ T ∗ )−1 (λ I − T ), see (VI.1.2), and the definition (4.6) of τ and τ∗ , we have for h ∈ H:

ΘA (z)Q1/2 Jh = τ∗ΘT (λ )DT h = Q1/2 J ∗ (I − λ T ∗ )−1 (λ I − T )h = −1  z−i ∗ (A + iI)(A∗ − iI)−1 = Q1/2 J ∗ I − z+i   z−i I − (A − iI)(A + iI)−1 h · z+i = − 2iQ1/2[(z + i)(A∗ − iI) − (z − i)(A∗ + iI)]−1 · [(z − i)(A + iI) − (z + i)(A − iI)](A + iI)−1h

= Q1/2 (A∗ − zI)−1 (A − zI)Jh = Q1/2 [I + 2i(A∗ − zI)−1 Q]Jh = [I + 2iQ1/2(A∗ − zI)−1 Q1/2 ]Q1/2 Jh;

hence we deduce that

ΘA (z) = [I + 2iQ1/2(A∗ − zI)−1 Q1/2 ]|Q.

(4.9)

When Q is compact, with spectral decomposition Qh = ∑ ωk (h, ϕk )ϕk ,

(4.10)

where {ϕk } is an orthonormal system of eigenvectors corresponding to the eigenvalues ωk > 0, this system is a basis of the subspace Q of H, and the operator ΘA (z) has, with respect to this basis, the (finite or infinite) matrix with the entries (ΘA (z)ϕ j , ϕk ) = δ jk + 2i(ω j ωk )1/2 ((A∗ − zI)−1 ϕ j , ϕk ).

(4.11)

2. We now show that if A is bounded, then A and T have the same invariant subspaces L, and TL = T |L is the Cayley transform of the dissipative operator AL = A|L. To this end, first consider a subspace L invariant for A. Choose a circle having σ (A) in its interior and the point −i in its exterior. If z0 is the center and r the radius of this circle, we obtain from the spectral radius theorem that r ≥ lim k(A − z0I)n k1/n . n→∞

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As a consequence we have for |z − z0 | > r the expansion, convergent in norm, ∞

(A − zI)−1 = [(A − z0 I) − (z − z0)I]−1 = − ∑(z − z0 )−n−1 (A − z0 I)n , 0

(4.12)

which implies that L is invariant for (A − zI)−1 also. As we can choose in particular z = −i, we obtain that L is invariant for (A + iI)−1 and, as a consequence of (4.3), of T also. Moreover, relation (4.12) implies for z = −i (A + iI)−1|L = (AL + iIL )−1 and hence TL = (A − iI)(A + iI)−1|L = (AL − iIL)(AL + iIL )−1 ; this proves that TL is the Cayley transform of AL . If we suppose, conversely, that L is invariant for T , then the expansion ∞

(T − µ I)−1 = − ∑ µ −n−1 T n 0

(|µ | > 1)

implies that L is invariant for (T − µ I)−1 also. Because T − I is boundedly invertible, with (T − I)−1 = (1/2i)(A + iI) (cf. (4.3)), (T − I)−1 is the limit (in norm) of (T − µ I)−1 as µ → 1; thus we conclude that L is invariant for (T − I)−1 and hence for A = i(I + T )(I − T )−1 as well. One of the consequences of the statement just proved is that if A is bounded, A is unicellular if and only if T is unicellular. 3. Suppose A is bounded, σ (A) consists of the single point 0, and 0 is not an eigenvalue of A. Then the spectrum of the contraction T ′ = −T consists of the single point 1, which is not an eigenvalue of T ′ . Moreover, T ′ is completely nonunitary (cf. the remark made at the end of Sec. 3.2) and hence A is completely nonselfadjoint. T ′ is the cogenerator of a continuous one-parameter semigroup of contractions {T ′ (s)}s≥0 , T ′ (s) = es (T ′ ) (s ≥ 0); (4.13) see Theorem III.8.1. From (4.3) we deduce that the generator A′ of this semigroup is given by A′ = (iA)−1 (A−1 exists and has dense domain, because −1 is not an eigenvalue of T , and hence not of T ∗ either). The functional calculus introduced in Sec. IV.4.4 justifies the notation T ′ (s) = exp(sA′ ).

(4.14)

Under the additional hypothesis that Q is of finite trace, it follows from (4.5) that T , and hence T ′ also, are weak contractions. By virtue of Theorem 1.1 the characteristic function of T ′ admits a scalar multiple. Because σ (T ′ ) = {1}, we have T ′ ∈ C00 (cf. Proposition II.6.7). It follows that T ′ ∈ C0 (cf. Theorem VI.5.1) and

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351

mT ′ (λ ) = ea (λ ), with a = aT ′ > 0. This means that ea (T ′ ) = O, whereas eb (T ′ ) 6= O for 0 ≤ b < a; using (4.13) and (4.14) we can express this in the alternative form: exp(aA′ ) = O

exp(bA′ ) 6= O for 0 ≤ b < a.

and

Let us introduce the notation (Ω0+ ) for the class of bounded dissipative operators A = R + iQ for which Q is of finite trace, σ (A) = {0}, and 0 is not an eigenvalue of A. The above results prove the following proposition (except the final statement (4.15)). Proposition 4.1. For A ∈ (Ω0+ ) the operator A′ = (iA)−1 is the generator of a continuous one-parameter semigroup of contractions {T ′ (s)}s≥0 , such that T ′ (s) = O for s large enough. The smallest of these values, denoted by sA , is equal to the value a = aT ′ occurring in the minimal function ea (λ ) of T ′ = (iI − A)(iI + A)−1 . The following relation also holds. sA = lim sup[|z| · logk(A + zI)−1k]. z→0

(4.15)

Proof. In order to prove (4.15) let us observe first that, setting z′ = (iz)−1 (z 6= 0), we have A′ + z′ I = (zI + A)(izA)−1 , and hence (A′ + z′ I)−1 = izA(A + zI)−1 .

(4.16)

We also have A′−1 = iA, thus every (finite) complex number belongs to the resolvent set of A′ . By virtue of relation (IV.4.17) we have (−A′ − z′ I)−1 =

Z sA 0

′

etz T ′ (t) dt

(4.17)

if Re z′ < 0; as both sides of (4.17) are entire functions of z′ , they must coincide for every complex z′ . An immediate consequence is the inequality k(A′ + z′ I)−1 k ≤ hence

Z sA 0



′

et|z | dt < exp(sA |z′ |) for |z′ | ≥ 1;

 1 ′ ′ −1 lim sup ′ · logk(A + z I) k ≤ sA . |z | z′ →∞

(4.18)

On the other hand, it follows from the well-known theorem of PALEY AND W IENER [1] that for every scalar function ϕ (t) ∈ L2 (−σ , σ ) we have   Z σ ′ 1 ′ lim sup ′ · log |ϕb(z )| = αϕ , where ϕb(z′ ) = etz ϕ (t) dt |z | −σ z′ →∞

and αϕ denotes the smallest of the numbers α ≥ 0 such that ϕ (t) vanishes a.e. on the set (−σ , σ )\(−α , α ). By the definition of sA we have T ′ (s) 6= O for 0 ≤ s < sA , thus there exist for every ε > 0 elements h, g ∈ H such that the function ϕ (t) defined

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by (T ′ (t)h, g) for 0 ≤ t < sA and by 0 for −∞ < t < 0, satisfies αϕ > sA − ε . In this case we have, by virtue of (4.17),

we obtain that 

ϕb(z′ ) = ((−A′ − z′ I)−1 h, g),

 1 ′ ′ −1 lim sup ′ · log |((A + z I) h, g)| > sA − ε , |z | z′ →∞ and this implies that in (4.18) the equality sign holds. Using (4.16) we can express this equation also in the form sA = lim sup[|z| · logkzA(A + zI)−1k]. z→0

(4.19)

THe operator A is bounded, thus (4.19) implies sA ≤ lim sup[|z| · logkz(A + zI)−1k]. z→0

(4.20)

Let {zn } be a sequence (zn → 0) for which this upper limit is approached. As sA > 0, we have necessarily kzn (A + zn I)−1 k → ∞, and hence, on account of the relation z(A + zI)−1 = I − A(A + zI)−1, we obtain lim[|zn | · logkzn (A + znI)−1 k] ≤ lim sup[|zn | · log(1 + kA(A + znI)−1 k)] n

n

= lim sup[|zn | · logkA(A + znI)−1 k] ≤ sA n

(cf. (4.19)). Therefore in (4.20) and hence in (4.15), the equality sign holds. 4. Let A = R + iQ ∈ (Ω0+ ) be such that Q is of finite rank N. It follows from (4.5) that the defect indices of T are equal to the rank of Q, that is, to N. The determinant dT (λ ) is an inner function, divisible by mT (λ ), and a divisor of mT (λ )N (cf. Theorem VI.5.2). Because mT (λ ) = mT ′ (−λ ) = ea (−λ ) = exp(aζ ), where

ζ=

i −λ + 1 =− , −λ − 1 z

it follows that dT (λ ) = κ · exp(cζ ),

where

|κ| = 1, a ≤ c ≤ Na.

(4.21)

(Observe that |λ | < 1 implies Re ζ < 0, and conversely.) The value of c can be calculated using (4.8) and (4.11). By virtue of these formulas, the matrix of ΘT (λ ) corresponding to the orthonormal basis {τ −1 ϕk } of DT and the orthonormal basis {τ∗−1 ϕ j } of DT ∗ has the entries

ϑk j (λ ) = (ΘA (−i/ζ )ϕ j , ϕk ) = δ jk − 2(ω j ωk )1/2 ζ ((I + iζ A∗ )−1 ϕ j , ϕk ).

(4.22)

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As dT (λ ) is the determinant of this matrix, (4.21) and (4.22) imply d det[ϑk j (λ )]|ζ =0 = κc. dζ

(4.23)

Now from (4.22) we get

ϑk j (λ )|ζ =0 = δk j

and

d ϑk j (λ )|ζ =0 = −2(ω j ωk )1/2 δ jk , dζ

which shows that the left-hand side of (4.23) equals −2 ∑ ω j . As ω j > 0, we conclude that c = 2∑ω j . (4.24) So we have proved the following theorem. Theorem 4.2. For A ∈ (Ω0+ ), with Q = Im A of finite rank, the value sA defined in the preceding proposition satisfies the inequality sA ≤ 2 tr Q.

(4.25)

In Chap. X we show that equality holds in (4.25) if and only if A is unicellular. 5. If the defect indices of T are equal to 1, then T is determined up to unitary equivalence by its minimal function ea (−λ ), where a = aT = sA (cf. the remark following Theorem VI.5.2), and hence by sA . On account of the obvious relation srA = r · sA (r > 0) we conclude that if A0 = R0 + iQ0 ∈ (Ω0+ ) with rank Q0 = 1 and sA0 = 1, then every operator A = R + iQ ∈ (Ω0+ ) with rank Q = 1 is unitarily equivalent to sA · A0 . Consider, for example, the operator A0 defined on L2 (0, 1) by A0 f (x) = i

Z x 0

f (t) dt.

(4.26)

As A0 is a Volterra-type integral operator, we have σ (A0 ) = {0}, and it is obvious that 0 is not an eigenvalue of A0 . For Q = Im A0 we have Q f (x) =

1 2

Z 1 0

f (t) dt;

hence Q ≥ O and rank Q = 1. Thus A0 ∈ (Ω0+ ). It is easy to show that A′0 = (iA0 )−1 is the generator of the continuous one-parameter semigroup of contractions defined by  f (x − s) for x ∈ [s, ∞] ∩ [0, 1] ′ (exp sA0 ) f (x) = 0 otherwise,

and hence sA0 = 1. Thus the following statement is proved.

Proposition 4.3. The operator A0 on L2 (0, 1), defined by (4.26), is unicellular. Every operator A ∈ (Ω0+ ) for which Im A is of rank 1, is unitarily equivalent to a positive multiple of A0 , namely to sA · A0 .

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5 Dissipative operators similar to self-adjoint ones 1. We now consider operators A = R + iQ on the (separable) space H, where R and Q are self-adjoint, and Q is bounded and positive; such an operator A is maximal dissipative; see Sec. 4.1. It is obvious that A is similar to a self-adjoint operator1 if its Cayley transform T is similar (by the same affinity S) to a unitary operator. As we know that σ (T ) does not cover the unit disc D, we infer from Theorem VI.4.1 that there exist points λ ∈ D at which ΘT (λ ) is boundedly invertible. From Theorem VI.4.5 we deduce therefore that T is similar to a unitary operator if and only if kΘT (λ )gk ≥ ckgk

(g ∈ DT , λ ∈ D)

(5.1)

for some constant c > 0. By virtue of relations (4.8) and (4.9), and because τ and τ∗ are unitary, we obtain, setting q = τ g and z = i

1+λ 1−λ

(g ∈ DT , |λ | < 1, Im z > 0),

that kΘT (λ )gk = kΘA(z)qk = kq + 2iQ1/2(A∗ − zI)−1 Q1/2 qk and kgk = kqk.

(5.2) (5.3)

If, in particular, rank Q = 1, that is, Qh = ω (h, q0 )q0

(ω > 0, kq0k = 1),

(5.4)

then Q = QH consists of the numerical multiples of q0 , thus q = ω −1/2 Q1/2 q for q ∈ Q. As, moreover, kQ1/2 hk = (Qh, h)1/2 = [ω (h, q0 )(q0 , h)]1/2 = |(q0 , ω 1/2 h)|, the right-hand side of (5.2) can also be written in the form |(q0 , q + 2iω (A∗ − zI)−1 q)|, and hence we have with

kΘA (z)qk = |ϑ (z)| · kqk (q ∈ Q),

(5.5)

ϑ (z) = 1 − 2iω ((A − z¯I)−1 q0 , q0 ) (Im z > 0).

(5.6)

|ϑ (z)| ≥ c(> 0) (Im z > 0).

(5.7)

By virtue of (5.2)–(5.5), condition (5.1) takes the form

1

For unbounded operators A1 , A2 similarity is defined as for bounded ones: existence of an affinity S such that A1 S = SA2 .

5. D ISSIPATIVE OPERATORS SIMILAR TO SELF- ADJOINT ONES

355

2. As an example consider the operator A on H = L2 (0, 1) defined by Ah(x) = a(x)h(x) + i

Z x 0

(5.8)

h(t) dt,

where a(x) is a real-valued, a.e. finite, measurable function. We have then A = R+iQ with Z x Z 1  Z i 1 1 1 Rh(x) = a(x)h(x) + h(t) dt, Qh(x) = − h(t) dt = (h, e0 )e0 2 0 2 0 2 x and e0 (x) ≡ 1; thus Q satisfies (5.4) with q0 = e0 and ω = 12 . To determine ϑ (z) let us set uζ = (A + ζ I)−1 e0 Then (a(x) + ζ )uζ (x) + i

Z x 0

(ζ = −¯z).

uζ (t) dt = 1

(0 ≤ x ≤ 1),

or, setting (a(x) + ζ )uζ (x) = vζ (x), vζ (x) + i

Z x 0

1 v (t) dt = 1. a(t) + ζ ζ

This equation has the unique solution  Z vζ (x) = exp −i

0

x

(5.9)

 1 dt . a(t) + ζ

By virtue of (5.6), (5.9), and (5.10) we have therefore  Z ϑ (z) = 1 − i(uζ , e0 ) = vζ (1) = exp −i

0

1

dt a(t) + ζ

(5.10)



,

or, if we introduce the distribution function of a(x), that is, the function σ (a) = meas{x : 0 ≤ x ≤ 1, a(x) ≤ a} (−∞ < a < ∞), then   Z∞ d σ (a) (ζ = −¯z), ϑ (z) = exp −i (5.11) −∞ a + ζ and hence  Z |ϑ (z)| = exp −

 β d σ (a) 2 2 −∞ (a − α ) + β ∞

(z = α + iβ , β > 0).

(5.12)

(−∞ < α < ∞, β > 0),

(5.13)

Thus, condition (5.7) takes the form F(α , β ) ≡

Z ∞

−∞

β d σ (a) ≤ M (a − α )2 + β 2

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C HAPTER VIII. W EAK C ONTRACTIONS

with M = log(1/c) < ∞. If (5.13) is satisfied, we have for every finite interval (α1 , α2 ) and for every η > 0:  Z α2 Z ∞ Z α2 β dα M(α2 − α1 ) ≥ d σ (a) F(α , β ) d α = 2 2 α1 α1 (a − α ) + β −∞  Z ∞ α2 − a α1 − a = arctan d σ (a). − arctan β β −∞ If α1 and α2 are points of continuity of σ (a), then we can let β → 0 and obtain, using Fatou’s lemma, M(α2 − α1 ) ≥ π

Z α2 α1

d σ (a) = π [σ (α2 ) − σ (α1 )];

(5.14)

thus σ (a) satisfies the Lipschitz condition with constant M/π . Conversely, for any σ (a) satisfying (5.14) we have F(α , β ) ≤

M π

  β da a − α a=+∞ M arctan = = M, 2 2 π β −∞ (a − α ) + β a=−∞

Z ∞

and thus (5.13) holds. We have proved part (a) of the following proposition. Proposition 5.1. Let A be the operator defined by (5.8). (a) A is similar to a self-adjoint operator if and only if σ (a), the distribution function of a(x), satisfies a Lipschitz condition. (b) If σ (a) satisfies a Lipschitz condition, then the completely nonself-adjoint part of A is similar to the self-adjoint operator A0 defined on L2 (Ω ) by A0 f (ξ ) = ξ · f (ξ ), where Ω = {a : σ ′ (a) > 0}. Proof. It remains to prove part (b). From (5.2), (5.3), and (5.5) we obtain kΘT (eit )gk = lim |ϑ (z)| · kgk (g ∈ DT ), z→ξ

(5.15)

where ξ denotes that point of the real axis which is the image of λ = eit under the homography λ → z = i(1 + λ )/(1 − λ ), that is, ξ = − cot(t/2), and where z tends to ξ from the upper half-plane, nontangentially to the real axis. Now if σ (a) satisfies a Lipschitz condition, we have for almost every ξ ∈ (−∞, ∞), lim

Z ∞

z→ξ −∞

β d σ (a) = πσ ′ (ξ ) (a − α )2 + β 2

(5.16)

as z = α + iβ tends to ξ nontangentially to the real axis (cf., e.g., H OFFMAN [11] p. 123). It follows from (5.12), (5.15), and (5.16) that kΘT (eit )gk = exp(−πσ ′ (ξ )) · kgk (g ∈ DT )

5. D ISSIPATIVE OPERATORS SIMILAR TO SELF- ADJOINT ONES

357

for almost every point eit of the unit circle, and hence (∆T (t)2 g, g) = [1 − exp(−2πσ ′ (ξ ))] · (g, g) (g ∈ DT ) for almost every point t of (0, 2π ). This implies, for these t,

∆T (t)g = [1 − exp(−2πσ ′ (ξ ))]1/2 g (g ∈ DT ). Because DT = τ −1 Q has dimension 1, the space ∆T L20 (DT ) can be identified with the space δ L20 (0, 2π ),2 where

δ (t) = η (− cot(t/2)),

η (ξ ) = [1 − exp(−2πσ ′(ξ ))]1/2 .

In this way we see that T (1) , the c.n.u. part of T , is similar to multiplication by the function eit on the space δ L20 (0, 2π ). As a consequence, the completely nonselfadjoint part of A (i.e. A(1) = i(I + T (1) )(I − T (1) )−1 ), is similar to multiplication by i(1 + eit )/(1 − eit ) = − cot(t/2) in the same space. Consider the transformation   1 ξ −i (0 ≤ t ≤ 2π ; −∞ < ξ < ∞). F f (t) ≡ F(eit ) → ξ +i [π (1 + ξ 2)]1/2 It maps L20 (0, 2π ) unitarily onto L2 (−∞, ∞) in such a way that the space δ L20 (0, 2π ) is mapped onto the space η L2 (−∞, ∞). Moreover, to multiplication by − cot(t/2) in the first subspace corresponds multiplication by ξ in the second subspace. Now the space η L2 (−∞, ∞) can be identified in a natural way with the space L2 (Ω ), where

Ω = {ξ : η (ξ ) 6= 0} = {ξ : σ ′ (ξ ) 6= 0}, and this concludes the proof of Proposition 5.1. 3. When the operator A is completely non-self-adjoint, it is itself similar to A0 (if σ (a) is Lipschitzian). This is the case if, for example,   0 for a ≤ 0, (5.17) a(x) ≡ x, and hence σ (a) = a for 0 ≤ a ≤ 1  1 for 1 ≤ a < ∞;

then we have Ω = (0, 1). In fact, suppose there exists a subspace H′ of L2 (0, 1), which reduces A to a self-adjoint operator A′ . Because A is bounded, we have for h ∈ H′ , 0 = (A′ − A′∗)h = (A − A∗)h = 2iQh = i(h, e0 )e0 2

We use in this section the subscript 0 in L20 to indicate that integration has been taken with respect to the measure dt/(2π ).

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C HAPTER VIII. W EAK C ONTRACTIONS

and hence (h, e0 ) = 0. As a consequence, we also have (h, An e0 ) = (A∗n h, e0 ) = 0

for h ∈ H′

and n = 1, 2, . . . .

Now, Ae0 = x + ix = (1 + i)x,       i i i x2 , . . . , An e0 = (1 + i) 1 + ... 1 + xn , . . . A2 e0 = (1 + i) 1 + 2 2 n so that h ⊥ xn (n = 0, 1, . . .), which implies h = 0. Hence H′ = {0}, which proves that A is completely non-self-adjoint. We obtain the following corollary. Corollary 5.2. The operator A on L2 (0, 1) defined by Ah(x) = x · h(x) + i

Z x 0

h(t) dt

is completely nonself-adjoint, but similar to the self-adjoint operator A0 on L2 (0, 1) defined by A0 g(x) = x · g(x).

6 Notes The results of Secs. 1–3 of this chapter were announced in S Z .-N.–F. [7], where the contractions in question were called “almost unitary.” The term “weak contraction” was proposed by M.G. K RE˘I N [1]. (Thus we avoid such paradoxical expressions as “completely nonunitary, almost unitary contraction.”) The operators T for which I − T ∗ T and I − T T ∗ are of finite rank have been called by M.S. Livˇsic and “quasiunitary” by others, see P OLJACKI˘I [1]–[3]. In their general context, our theorems seem to be new even in the case of finite defect indices. The first systematic study of operators in the class (Ω0+ ) was undertaken by M.S. Brodski˘ı and M.S. Livˇsic (cf. B RODSKI˘I [1]) as well as by their collaborators. Recall that for an operator A in the class (Ω0+ ), the Cayley transform T of A is a weak contraction (cf. Sec. 4). Thus the spectral decomposition of T generates an analogous spectral decomposition of A. In the particular case that A is bounded and ω is either the intersection of σ (A) with the real line, or the part of σ (A) in the interior of the upper half-plane, the corresponding subspace H(ω ) was constructed earlier by M.S. B RODSKI˘I [8], using another method. G INZBURG [3] arrived at results similar to our Theorems 2.1 and 3.1, making use of certain parts of the paper S Z .-N.–F. [IX], but applying the method of multiplicative integrals. Proposition 5.1 appeared first in S Z .-N.–F. [X]. Particular cases were considered earlier in B RODSKI˘I AND L IV Sˇ IC [1] (e.g., Corollary 5.2, which is due to S AHNOVI Cˇ ). G OHBERG AND K RE˘I N [5] generalized Proposition 5.1 to integral operators of a more general type.

6. N OTES

359

Consider two completely nonunitary contractions T1 , T2 with defect indices equal to one, and let Θ1 , Θ2 be the corresponding characteristic functions, which are simply scalar contractive analytic functions. K RIETE [3] shows that T1 and T2 are similar if and only if Θ1 /Θ2 , Θ2 /Θ1 are bounded and {z ∈ C : |Θ1 (z)| = 1} = {z ∈ C : |Θ2 (z)| = 1} up to sets of measure zero. For an extension of this result to contractions with finite defect indices see S Z .-N.–F. [22]. In connection with this chapter see also M.S. B RODSKI˘I [5]; B RODSKI˘I ET ˇ [1]–[10]; AL . [1]; F RIEDRICHS [1]; K ALISCH [1]; K ISILEVS′ KI˘I [1]; S AHNOVI C S ARASON [2]; and Sˇ MUL′ JAN [2].

Chapter IX

The Structure of C1·· -Contractions 1 Unitary and isometric asymptotes 1. We systematically exploit the operators intertwining a given contraction with an isometry or unitary operator. Given operators T on H and T ′ on H′ , we denote by I (T, T ′ ) the set of all intertwining operators; these are the bounded linear transformations X : H → H′ such that X T = T ′ X. We also use the notation {T }′ = I (T, T ) for the commutant of T . Fix a contraction T on H, an isometry (resp., unitary operator) V on K, and X ∈ I (T,V ) such that kX k ≤ 1. The pair (X,V ) is called an isometric (resp., unitary) asymptote of T if for every isometry (resp., unitary operator) V ′ , and every X ′ ∈ I (T,V ′ ) with kX ′ k ≤ 1, there exists a unique Y ∈ I (V,V ′ ) such that X ′ = Y X and kY k ≤ 1. Assume that (X ,V ) is an isometric or unitary asymptote of T . If the operator X is zero then we also have K = {0}. Indeed, we have IK X = OX, and hence IK = O by uniqueness. If X 6= O, we must have kX k = 1. Indeed, setting X ′ = X /kX k, we deduce the existence of Y ∈ {V }′ such that Y X = X ′ and kY k ≤ 1. The desired conclusion follows because clearly kY k ≥ 1/kX k. The following result demonstrates a uniqueness property of asymptotes. Lemma 1.1. For any two isometric (resp., unitary) asymptotes (X,V ), (X ′ ,V ′ ) of a contraction T , there exists a unique unitary transformation Y ∈ I (V,V ′ ) such that Y X = X ′. Proof. The existence of a contractive Y ∈ I (V,V ′ ) such that Y X = X ′ follows because (X,V ) is an asymptote. Similarly, because (X ′ ,V ′ ) is an asymptote, there is a contractive Y ′ ∈ I (V ′ ,V ) such that Y ′ X ′ = X. The relation (Y ′Y )X = Y ′ (Y X) = Y ′ X ′ = X and the fact that Y ′Y is a contraction imply that Y ′Y = I by the uniqueness clause in the definition of asymptotes. Similarly, YY ′ = I so that Y is the desired unitary transformation. Corollary 1.2. Let T be a contraction on H, and let (X ,V ) be a unitary asymptote of T , with V acting on K. The smallest reducing subspace for V containing XH is K. B.Sz.-Nagy et al., Harmonic Analysis of Operators on Hilbert Space, Universitext, DOI 10.1007/978-1-4419-6094-8_9, © Springer Science + Business Media, LLC 2010

361

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C HAPTER IX. T HE S TRUCTURE OF C1·· -C ONTRACTIONS

Proof. It suffices to consider the case X 6= O. Let K′ be the smallest reducing subspace for V containing X H, denote by Y1 the orthogonal projection of K onto K′ , and by Y2 the identity operator on K. We have Y1 X = Y2 X = X and kY1 k = kY2 k ≤ 1. The definition of unitary asymptotes implies Y1 = Y2 , and therefore K′ = K. The following result allows us to deal with intertwining operators without worrying about their norms. Lemma 1.3. Let T be a contraction on H, and let (X ,V ) be a unitary asymptote of T , with V acting on K. (1) The pair (X,V |K+ ) is an isometric asymptote for T , where K+ = XH. (2) Assume that V ′ is an isometry and X ′ ∈ I (T,V ′ ). There exists a unique Y ∈ I (V |K+ ,V ′ ) such that X ′ = Y X . This operator satisfies kY k = kX ′ k. (3) Assume that V ′ is a unitary operator and X ′ ∈ I (T,V ′ ). There exists a unique Y ∈ I (V,V ′ ) such that X ′ = Y X . This operator satisfies kY k = kY |K+ k = kX ′ k. Proof. It suffices to consider the case X 6= O. To prove (1), assume that V ′ is an isometry on K′ , and W is the minimal unitary extension of V ′ . Given X ′ ∈ I (T,V ′ ) with kX ′ k ≤ 1, we also have X ′ ∈ I (T,W ), and therefore there exists a unique Y ∈ I (V,W ) such that Y X = X ′ and kY k ≤ 1. Clearly Y K+ ⊂ Y XH = X ′ H ⊂ K′ , so that the operator Z = Y |K+ belongs to I (V |K+ ,V ′ ), ZX = X ′ , and kZk ≤ 1. In fact, Z is unique with these properties. Indeed, assume that Z1 ∈ I (V |K+ ,V ′ ), Z1 X = X ′ , and kZ1 k ≤ 1. The range of X is dense in K, therefore the equation Z1 X = X ′ = ZX yields Z1 = Z, and this concludes the proof of (1). Let now V ′ and X ′ be as in (2), and assume Y1 ,Y2 ∈ I (V |K+ ,V ′ ) satisfy Y1 X = Y2 X = X ′ . Then Y1 Xh = Y2 Xh = X ′ h, h ∈ H, and hence Y1 = Y2 because X H is dense in K+ . To prove the existence of Y , it suffices to consider the case X ′ 6= O. There exists then Y0 ∈ I (V |K+ ,V ′ ) such that Y0 X = X ′ /kX ′ k and kY0 k ≤ 1; in fact kY0 k = 1 because X ′ /kX ′k has norm one. It suffices to take Y = kX ′ kY0 . Finally, let V ′ and X ′ be as in (3), and assume Y1 ,Y2 ∈ I (V,V ′ ) satisfy Y1 X = Y2 X = X ′ . Then Y1V −n Xh = V ′−nY1 X h = V ′−n X ′ h = V ′−nY2 X h = Y2V −n X h for all h ∈ H and n ≥ 0. The equality Y1 = Y2 follows then from Corollary 1.2. The existence of Y is proved as in case (2). The equality kY k = kY |K+ k follows because kYV −n X hk = kY X hk (x ∈ H), and the vectors {V −n Xh : h ∈ H, n ≥ 0} are dense in K. Isometric and unitary asymptotes also have a useful commutant extension property. Lemma 1.4. Given isometric (resp., unitary) asymptotes (X ,V ) and (X ′ ,V ′ ) of T and T ′ , respectively, and A ∈ I (T, T ′ ), there exists a unique transformation B =

1. U NITARY AND ISOMETRIC ASYMPTOTES

363

γ (A) ∈ I (V,V ′ ) such that BX = X ′ A. This operator satisfies kBk ≤ kAk. When T = T ′ , X = X ′ , and V = V ′ , the map γ is a unital algebra homomorphism. It follows in particular that σ (γ (A)) ⊂ σ (A) for every A ∈ {T }′ . Proof. The existence of B follows immediately from an application of the definition of asymptotes to the operator X ′ A ∈ I (T,V ′ ). The uniqueness of B, and the fact that γ is a contractive homomorphism follow easily from the uniqueness properties in Lemma 1.3. For instance, given A, A′ ∈ {T }′ , the relation XAA′ = γ (A)X A′ = γ (A)γ (A′ )X implies γ (AA′ ) = γ (A)γ (A′ ). The spectral inclusion follows from the fact that γ is a unital homomorphism. Indeed, if A ∈ {T }′ is invertible then A−1 ∈ {T }′ and γ (A−1 )γ (A) = γ (A)γ (A−1 ) = γ (I) = I. We now provide a direct construction proving the existence of isometric and unitary asymptotes. Later we identify them with parts of the unitary dilation of a given operator. Fix a contraction T on H, and note that for every x ∈ H the sequence {kT n xk}∞ n=1 is decreasing, hence convergent. The polar identity (T x, Ty) =

1 4 k ∑ i kT (x + ik y)k2 4 k=1

shows that the sequence {(T n x, T n y)}∞ n=1 also converges for every x, y ∈ H. The form defined by wT (x, y) = lim (T n x, T n y) (x, y ∈ H) n→∞

is linear in x, conjugate linear in y, and 0 ≤ wT (x, x) ≤ kxk2 for x ∈ H. Therefore there exists a unique operator AT on H such that (AT x, y) = wT (x, y)

(x, y ∈ H)

and O ≤ AT ≤ I. The obvious identity wT (T x, Ty) = wT (x, y) implies T ∗ AT T = AT , 1/2 1/2 and in particular kAT T xk = kAT xk for x ∈ H. Consider now the space K+ T = 1/2

AT H. The preceding identity implies the existence of an isometry VT on K+ T such 1/2 1/2 + that VT AT x = AT T x for every x ∈ H. We also define an operator XT : H → K+ T 1/2 by setting XT+ x = AT x for x ∈ H. The operator VT has a minimal unitary extension + WT acting on a space KT ⊃ K+ T . Define XT ∈ I (T,WT ) by setting XT x = XT x for x ∈ H. Proposition 1.5. The pair (XT+ ,VT ) (resp., (XT ,WT )) is an isometric (resp., unitary) asymptote of T . Proof. Clearly kXT+ k ≤ 1 and XT+ ∈ I (T,VT ). Let V be an arbitrary isometry, and X ∈ I (T,V ). The inequality kXxk = kV n X xk = kX T n xk ≤ kX kkT n xk

364

implies that

C HAPTER IX. T HE S TRUCTURE OF C1·· -C ONTRACTIONS

kXxk ≤ kX k lim kT n xk = kXkkXT+xk n→∞

(x ∈ H).

It follows that the map Y0 : XT+ x 7→ X x is well defined, and it extends uniquely to a linear transformation Y satisfying kY k ≤ kX k and Y XT+ = X . The equations VY XT+ = V X = X T = Y XT+ T = YVT XT+ , and the fact that XT+ has dense range, imply Y ∈ I (VT ,V ). The uniqueness of such an operator Y is obvious. This proves the statement concerning (XT+ ,VT ). Assume now that U is an arbitrary unitary operator, and X ∈ I (T,U). It follows from the first part of the proof that there exists a unique Y ∈ I (VT ,U) such that Y XT+ = X . To conclude the proof it suffices to show that Y has a unique extension Y1 ∈ I (WT ,U) with the same norm. Note that Y1 must satisfy Y1 (WT−n x) = U −n (Y x) (x ∈ H, n = 1, 2, . . . ). This formula defines Y1 as a continuous linear transformation on the linear manifold S −n + n≥1 WT KT , and the desired conclusion follows from the density of this set in KT .

Assume now that the operator T on H has an invariant subspace H′ , and consider the triangulation   ′ T ∗ T= O T ′′ associated with the orthogonal decomposition H = H′ ⊕ H′′ . The subspace K′ = −n ′ ′ ′′ n=1 WT XT H is reducing for WT , say WT = W ⊕W relative to the decomposition ′ ′′ ′ ′ KT = K ⊕ K . Because XT H ⊂ K , we can define operators X ′ ∈ I (T ′ ,W ′ ) and X ′′ ∈ I (T ′′ ,W ′′ ) by setting W∞

X ′ x = XT x and

X ′′ x = PK′′ XT x

(x ∈ H′ ) (x ∈ H′′ ).

Theorem 1.6. With the above notation, we have

(1) The pair (X ′ ,W ′ ) is a unitary asymptote of T ′ . (2) The pair (X ′′ ,W ′′ ) is a unitary asymptote of T ′′ . (3) WT is unitarily equivalent to WT ′ ⊕ WT ′′ . Proof. The fact that (XT ′ ,WT ′ ) is a unitary asymptote for T ′ implies the existence of a contraction Y ∈ I (WT ′ ,W ′ ) such that Y XT ′ = X ′ . Note that kXT ′ xk = lim kT n xk = kXT xk = kX ′ xk (x ∈ H′ ) n→∞

and therefore Y is isometric on the range of XT ′ , and its range contains X ′ H′ . Moreover, ′−n YWT−n Y XT ′ = W ′−n X ′ (n = 1, 2, . . . ), ′ XT ′ = W

2. T HE SPECTRA OF C1·· - CONTRACTIONS

365

and this implies that Y is an isometry and its range is dense, and hence Y is unitary. This proves (1). To prove (2), consider an arbitrary unitary operator U, and X ∈ I (T ′′ ,U). Observe that XPH′′ ∈ I (T,U) and therefore we can find a unique Y0 ∈ I (WT ,U) such that kY0 k ≤ kX PH′′ k = kXk and Y0 XT = X PH′′ . The operator Y0 is zero on XT H′ , and the identity Y0WT−n = U −nY0

(n = 1, 2, . . . )

implies that Y0 is zero on K′ . Therefore we can write Y0 = Y PK′′ , with Y ∈ I (W ′′ ,U) and kY k = kY0 k ≤ kX k. Moreover, we clearly have Y X ′′ = X. The uniqueness of Y follows from the uniqueness of Y0 . The last assertion follows from the fact that WT = W ′ ⊕W ′′ , because W ′ and W ′′ are unitarily equivalent to WT ′ and WT ′′ , respectively. Note that VT need not be unitarily equivalent to VT ′ ⊕VT ′′ . Indeed, take T to be a bilateral shift of multiplicity one, and H′ a nonreducing invariant subspace. 2. We now relate the unitary asymptote to the minimal unitary dilation of an operator. Let T be a contraction on H, and let U on K ⊃ H be the minimal unitary dilation of T . Consider the ∗-residual part R∗ of U on the reducing space R∗ ⊂ K, and the operator X : H → R∗ defined by X x = PR∗ x

(x ∈ H).

By Proposition II.3.1 and (II.3.6), we have X ∈ I (T, R∗ ) and kXxk = lim kT n xk = kXT+ xk n→∞

(x ∈ H).

It follows from the construction of VT that the pair (X , R∗ |XH) is an isometric asymptote of T . Proposition 1.7. The pair (X, R∗ ) is a unitary asymptote of T . Proof. We only have to verify that R∗ is the minimal unitary extension of R∗ |X H. W −n 0 The space R0∗ = ∞ n=1 R∗ X H reduces R∗ , and thus R∗ ⊕ M(L) reduces U. This reducing space for U contains H, and therefore it must equal K by the minimality of U. We deduce that R0∗ = R∗ , as desired. Proposition 1.8. Let (X,W ) be a unitary asymptote of the contraction T . The transformation X is one-to-one if and only if T is of class C1·· . If T is of class C11 then X is a quasi-affinity, and T is quasi-similar to W . Proof. The first assertion is obvious. The second one follows from Proposition II.3.5.

2 The spectra of C1·· -contractions 1. Let T be a contraction on the Hilbert space H, and let (XT ,WT ) be its unitary asymptote. It follows from Lemma 1.4 that σ (WT ) ⊂ σ (T ). We show that, even

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C HAPTER IX. T HE S TRUCTURE OF C1·· -C ONTRACTIONS

if T is of class C11 or C10 , there are very few restrictions on the set σ (T ) beside this inclusion. We restrict ourselves to the case in which T is c.n.u. In this case, the spectral measure of the minimal unitary dilation U of T is absolutely continuous relative to normalized Lebesgue measure m on the unit circle C (cf. Theorem II.6.4), and therefore so is the spectral measure of the ∗-residual part R∗ , and that of WT which is unitarily equivalent to R∗ . In other words, there exists a Borel subset ωT ⊂ C such that the spectral measure E of WT is mutually absolutely continuous with the scalar measure χωT dm; here we use χω to denote the characteristic function of the set ω . We emphasize that ωT is only determined up to sets of measure zero. Theorem VI.2.3 and Proposition 1.7 imply that

ωT = {ζ ∈ C : ∆∗ (ζ ) 6= O}, where ∆∗ (ζ ) = (I − ΘT (ζ )ΘT (ζ )∗ )1/2 . This follows from the representation of T ∗ as a functional model associated with its characteristic function ΘT ∗ (λ ), and from the fact that ΘT ∗ (λ ) = Θ T˜ (λ ). Thus, R∗ is unitarily equivalent to multiplication by ζ on ∆∗ L2 (DT ∗ ). If T is of class C11 then, for a.e. ζ ∈ C, ΘT (ζ ) is isometric if and only if it is unitary (cf. Propositions VI.3.5 and V.2.4), and therefore ωT is equal to the smallest Borel set that is residual for T in the sense of Definition VII.5.3. We recall from Definition VII.5.2 that the essential support of a Borel set α ⊂ C is the complement of the largest open set ω ⊂ C such that m(ω ∩ α ) = 0. We use the short notation α − for the essential support of α , which should not be confused with the closure α . Observe that the equality α1− = α2− can occur even when the sets α1 and α2 differ by a set of positive measure. We say that α is essentially closed if α = α − . Clearly the set σ (WT ) = ωT− is essentially closed. We are ready to describe a further condition the sets σ (T ) and σ (WT ) must satisfy if T ∈ C1·· . We say that an essentially closed set α ⊂ C is neatly contained in a compact set σ ⊂ D if α ⊂ σ and each nonempty closed subset σ ′ ⊂ σ , such that σ \ σ ′ is also closed, satisfies m(σ ′ ∩ α ) > 0. Remark. It is useful to note that, given a Borel set α ⊂ C of positive measure and an open arc β ⊂ C, then m(α ∩ β ) = 0 if and only if m(α − ∩ β ) = 0.

Proposition 2.1. For every c.n.u. contraction T ∈ C1·· , the spectrum σ (WT ) is neatly contained in σ (T ).

Proof. Let σ ′ ⊂ σ (T ) be a nonempty closed set such that σ (T ) \ σ ′ is closed. The Riesz–Dunford functional calculus provides an invariant subspace H′ for T such that σ (T |H′ ) = σ ′ (cf. Sec. 148 in [Func. Anal.]). Because T ′ ∈ C1·· , the operator WT |H′ acts on a nonzero space, and hence the essentially closed set σ (WT |H′ ) has positive measure. By Theorem 1.6.(3), σ (WT |H′ ) ⊂ σ (WT ), and therefore m(σ ′ ∩ σ (WT )) ≥ m(σ (WT |H′ )) > 0, as claimed. Remark. Every contraction T can be written as T = T0 ⊕ T1 , with T0 c.n.u. and T1 unitary. We say that T is an absolutely continuous contraction if the spectral measure of T1 is absolutely continuous relative to m. If T is absolutely continuous, the

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unitary asymptote WT ≃ WT0 ⊕ T1 is absolutely continuous, and ωT can be defined as in the c.n.u. case. Proposition 2.1 easily extends to absolutely continuous contractions. 2. We now show that Proposition 2.1 provides the only constraint on the spectrum of an operator T of class C11 and that of its unitary asymptote, even if T is assumed W n cyclic (i.e., ∞ n=0 T h = H for some h ∈ H). The examples we construct must have infinite defect indices. Indeed, Theorem VII.6.3 indicates that σ (T ) ⊂ C if T ∈ C11 and T has finite defect indices. An example of T ∈ C11 with σ (T ) = D was presented in VI.4.2. Theorem 2.2. Assume that ω0 ⊂ C is a Borel set of positive measure, ω = ω0− , and ω is neatly contained in a compact set σ ⊂ D. Then there exists a cyclic c.n.u. contraction T ∈ C11 such that σ (T ) = σ and ωT = ω0 . In particular, ω = σ (WT ).

We need a few preliminaries. If β ⊂ C is a Borel set, recall that L2 (β ) can be identified with χβ L2 , where χβ denotes the characteristic function of β . We denote by Mβ the unitary operator of multiplication by ζ on L2 (β ). It is well known that Mβ has a cyclic vector. For instance, the function u(eit ) = χβ (eit )e−1/t (0 < t ≤ 2π ) is cyclic for Mβ . Lemma 2.3. For every Borel set α ⊂ C with positive measure, and for every c > 0, there exists a cyclic c.n.u. contraction T ∈ C11 such that σ (T ) = α − , kT −1 k > c, and WT is unitarily equivalent to Mα . Proof. Fix ε ∈ (0, 1), and an outer function ϑ ∈ H ∞ such that |ϑ (ζ )| = χC\α (ζ ) + ε χα (ζ ) for a.e. ζ ∈ C. Let T be a c.n.u. contraction whose characteristic function coincides with {E 1 , E 1 , ϑ (λ )}. Because ϑ is outer, we deduce that T ∈ C11 and kT −1 k = |ϑ (0)|−1 = ε −m(α ) > c if ε is sufficiently small (cf. (VI.4.11) and (III.1.14)). The operator WT is unitarily equivalent to R∗ , and this operator is unitarily equivalent to Mα because, up to sets of measure zero, α = {ζ ∈ C : |ϑ (ζ )| < 1}. Finally, because T ∈ C11 , it is quasi-similar to Mα , and hence it has a cyclic vector. Next is a spectral mapping theorem. Proposition 2.4. Let T be a contraction on H, and assume that the function u ∈ H ∞ can be extended continuously to D. Then

σ (u(T )) = u(σ (T )). Proof. Let A, B be two commuting operators on H. We claim that for every λ ∈ σ (A) (resp., µ ∈ σ (B)) there exists µ ∈ σ (B) (resp., λ ∈ σ (A)) such that |λ − µ | ≤ kA − Bk. Indeed, assume that a complex number λ is at a distance greater than kA − Bk from every µ ∈ σ (B). Then the operator (B − λ I)−1 has a spectral radius less than

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1/kA − Bk, and by commutativity it follows that (B − λ I)−1(A − B) has a spectral radius less than one. Indeed, k[(B − λ I)−1(A − B)]nk1/n = k(B − λ I)−n(A − B)nk1/n ≤ k(B − λ I)−nk1/n kA − Bk

for every n, and the desired conclusion follows by letting n → ∞. Therefore the operator A − λ I = (B − λ I)(I + (B − λ I)−1(A − B))

is invertible. Observe that the functions ur (λ ) = u(rλ ) converge uniformly to u, and therefore kur (T ) − u(T )k → 0 as r → 1 − 0. Because the operators ur (T ) commute with u(T ), the preceding observation implies that the compact sets σ (ur (T )) converge to σ (u(T )) in the Hausdorff metric. Also, the sets ur (σ (T )) = u(rσ (T )) converge to u(σ (T )). Now, the operators ur (T ) can also be calculated by the Riesz–Dunford functional calculus because ur is analytic in a neighborhood of σ (T ). Therefore the desired conclusion follows from the spectral mapping theorem for the Riesz– Dunford functional calculus: σ (ur (T )) = ur (σ (T )). The construction of C11 -contractions with complicated spectra relies essentially on the following lemma. Lemma 2.5. Let Ω ⊂ D be a simply connected open set bounded by a rectifiable Jordan curve Γ . Assume that Γ ∩C contains a nontrivial arc J. Let α ⊂ J be a Borel set of positive measure, µ0 ∈ Ω , and c > 0. There exists a c.n.u. contraction T ∈ C11 with the following properties. (1) (2) (3) (4)

σ (T ) = α − . WT is unitarily equivalent to Mα . k(T − µ0 I)−1 k ≥ c. k(T − µ I)−1 k ≤ 1/dist(µ , Ω ) for every µ in the complement of Ω .

Proof. Fix a homeomorphism u : D → Ω such that u|D is holomorphic and u(0) = µ0 . The existence of the conformal map u|D follows from the Riemann mapping theorem. The fact that u extends to D is due to Carath´eodory (cf. Theorem II.4 in G OLUZIN [1]) because Γ is a Jordan curve. By results of F. and M. Riesz, because Γ is rectifiable, a set ω ⊂ C has Lebesgue measure zero if and only if u(ω ) has arclength zero (cf. Sec. X.1, Theorem 2 in G OLUZIN [1]). Therefore the set β = u−1 (α ) ⊂ C has positive Lebesgue measure, and

β − = u−1 (α − ).

(2.1)

u(λ ) − µ0 = u(λ ) − u(0) = λ v(λ ) (λ ∈ D),

(2.2)

We can factor the difference

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with v ∈ H ∞ . Apply now Lemma 2.3 (with β and ckvk∞ in place of α and c) to produce a cyclic c.n.u. contraction T1 ∈ C11 such that

σ (T1 ) = β − ,

kT1−1 k ≥ ckvk∞ ,

(2.3)

and WT1 is unitarily equivalent to Mβ . The required operator T is defined as T = u(T1 ). Theorem III.2.1(e) implies that T is a c.n.u. contraction, and the equality

σ (T ) = u(σ (T1 )) = u(β − ) = α − follows from Proposition 2.4 and (2.1). Because T1 is quasi-similar to Mβ , the operator T = u(T1 ) is quasi-similar to u(Mβ ). We claim that u(Mβ ) is unitarily equivalent to Mα . Indeed, an explicit unitary equivalence Z : L2 (α ) → L2 (β ) is provided by the formula ( |u′ (ζ )|1/2 g(u(ζ )) if ζ ∈ β (Zg)(ζ ) = 0 if ζ ∈ C \ β , and this is well defined for g ∈ L2 (α ) due to the F. and M. Riesz theorem referred to above. We conclude that T is quasi-similar to Mα , so that T is cyclic, of class C11 , and ωT = α . To verify the other properties of T , we apply (2.2) to deduce that T − µ0 I = u(T1 ) − u(0)I = T1 v(T1 ) so that ckvk∞ ≤ kT1−1 k ≤ kv(T1 )kk(T − µ0 I)−1 k ≤ kvk∞ k(T − µ0 I)−1 k, where we use (2.3) and Theorem III.2.1(b) to estimate the norm of v(T1 ). The inequality (3) follows. For (4), note that the function vµ (λ ) = 1/(u(λ ) − µ ) be/ Ω , and kvµ k∞ = 1/dist(µ , Ω ). Moreover, for such µ we longs to H ∞ if µ ∈ have (T − µ I)−1 = vµ (T1 ), and (4) follows from another application of Theorem III.2.1(b). The lemma is proved. We are now ready to prove Theorem 2.2. Proof. Let { µn }∞ n=1 be a dense sequence in σ , such that each of its terms appears infinitely many times. We construct a sequence {αn }∞ n=1 of pairwise disjoint subsets of ω0 with m(αn ) > 0, and a sequence {Tn }∞ n=1 of cyclic c.n.u. C11 -contractions with the following properties. (1) (2) (3) (4)

σ (Tn ) = αn− . WTn is unitarily equivalent to Mαn . If (Tn − µnI)−1 exists then k(Tn − µn I)−1 k > n. k(Tn − µ I)−1 k ≤ 1/[dist(µ , σ ) − 1/n] if dist(µ , σ ) > 1/n. S

Once these operators are constructed, we set α0 = ω0 \ ( ∞ n=1 αn ), and construct by Lemma 2.3 a c.n.u. contraction T0 ∈ C11 such that σ (T0 ) = α0− and WT0 is unitarily equivalent to Mα0 ; if α0 has measure zero, we can take T0 to act on the trivial space L {0}. The desired operator is the c.n.u. contraction defined as T = ∞ n=0 Tn . Each Tn is quasi-similar to WTn , and therefore T is quasi-similar to Mω0 . In particular, T is of class C11 , it is cyclic, and ωT = ω0 . The spectrum of T is determined by observing

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C HAPTER IX. T HE S TRUCTURE OF C1·· -C ONTRACTIONS

that, given a complex number λ , the operator T − λ I is invertible if and only if Tn − λ I is invertible for all n, and the sequence {k(Tn − λ I)−1 k}∞ n=0 is bounded. Condition (3) then indicates that µi belongs to σ (T ) for all i, and thus σ ⊂ σ (T ) by the density of { µn }∞ n=1 . On the other hand, condition (4) shows that σ (T ) ⊂ σ . To conclude the proof we need to construct sets αn and operators Tn satisfying the above conditions. Let us set Gn = {λ : dist(λ , σ ) < 1/n} (n = 1, 2, . . . ), and denote by G′n the connected component of Gn which contains µn . The set G′n ∩ σ thus contains µn , it is closed, and σ \ (G′n ∩ σ ) is closed as well. Because ω0 is neatly contained in σ , we must have m(G′n ∩ ω0 ) > 0 (cf. the remark preceding Proposition 2.1). We can find inductively Borel subsets βn ⊂ G′n ∩ ω0 such that 1 0 < m(βn ) ≤ m(βn−1 ) (n ≥ 2). 3 S∞

The Borel set αn = βn \ (

k=1 βn+k )

has positive measure because   ∞ ∞ 1 m(αn ) ≥ m(βn ) − ∑ m(βn+k ) ≥ 1 − ∑ 3−k m(βn ) = m(βn ). 2 k=1 k=1

There is a closed arc Jn ⊂ G′n ∩ C such that m(αn ∩ Jn ) > 0. Replacing each αn by αn ∩ Jn we can also assume that αn ⊂ Jn . The sets {αn }∞ n=1 are pairwise disjoint. Next we construct the operators Tn . Fix n ≥ 1, and choose a point µn′ ∈ G′n ∩ D such that 1 |µn′ − µn | < ; 2n ′ we can take µn = µn if µn ∈ D. Assume that ζ1 and ζ2 are the two endpoints of the arc Jn . The set G′n ∩ D is connected, therefore we can find a simple rectifiable curve Γn ⊂ (G′n ∩ D) ∪ {ζ1 , ζ2 }, with endpoints ζ1 and ζ2 , such that the simply connected region Ωn bounded by Jn ∪ Γn is entirely contained in G′n ∩ D, and µn′ ∈ Ωn . We now apply Lemma 2.5 with Ωn , αn , µn′ , 3n in place of Ω , α , µ0 , c, respectively. We obtain a cyclic c.n.u. C11 -contraction Tn satisfying conditions (1) and (2) above, such that k(Tn − µn′ I)−1 k ≥ 3n and

k(Tn − µ I)−1 k ≤ 1/dist(µ , Ωn )

/ Ωn ). (µ ∈

The last condition implies (4) because Ωn ⊂ Gn , and therefore dist(µ , σ ) ≤ dist(µ , Gn ) +

1 1 ≤ dist(µ , Ωn ) + n n

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371

for all scalars µ . Finally choose a unit vector x such that k(Tn − µn′ I)xk ≤ 1/2n. We have 1 k(Tn − µ n I)xk ≤ k(Tn − µn′ I)xk + |µn′ − µn | < , n and this in turn implies condition (3). The proof is complete. 3. We consider now the case of a contraction T ∈ C10 , which is necessarily c.n.u. If T has a finite defect index, then the two defect indices must be different; indeed, ΘT is inner, but not ∗-inner. In this case it follows that σ (T ) = D. Surprisingly, however, the general form of the spectrum for the class C10 is the same as for the class C11 . Theorem 2.6. Assume that ω0 ⊂ C is a Borel set of positive measure, ω = ω0− , and ω is neatly contained in a compact set σ ⊂ D. Then there exists a cyclic contraction T ∈ C10 such that σ (T ) = σ and ωT = ω0 . In particular, ω = σ (WT ).

The proof of this result depends on an analogue of Lemma 2.3 for the class C10 . This is considerably more difficult because the contractions involved must have infinite defect indices. Our construction depends on identifying appropriate invariant subspaces of bilateral weighted shifts. Consider a sequence β = {β (n)}∞ n=−∞ of positive numbers such that β (n) ≥ β (n + 1) ≥ 1 for all n. We denote by L2β the Hilbert space consisting of those funcn tions f ∈ L2 whose Fourier series ∑∞ n=−∞ un ζ is such that ∞

∑ β (n)2 |un |2 < ∞,

n=−∞

where the norm of f is given by ∞

k f k2β = ∑ β (n)2 |un |2 . n=−∞

The functions en (ζ ) = ζ n form an orthonormal basis in L2 , and β (n)−1 en form an orthonormal basis in Lβ2 . Denote by U the bilateral shift on L2 , and observe that ULβ2 ⊂ Lβ2 . Moreover, the restriction Uβ of U to Lβ2 is a contraction satisfying Uβ (β (n)−1 en ) = β (n)−1 en+1 =

β (n + 1) (β (n + 1)−1en+1 ) β (n)

for all integers n. Thus Uβ is a weighted bilateral shift. The inclusion operator Xβ : L2β → L2 is obviously a contraction, and Xβ ∈ I (Uβ ,U). Lemma 2.7. Assume that limn→+∞ β (n) = 1. Then the pair (Xβ ,U) is an isometric asymptote for Uβ .

Proof. It suffices to observe that the range of Xβ is dense, U is unitary, and kXβ f k = limn→∞ kUβn f kβ whenever f is a finite linear combination of the vectors {ek }∞ k=−∞ .

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The following results show that restrictions of Uβ provide operators T such that WT = WT+ is unitarily equivalent to Mα for suitable choices of the sequence β . Lemma 2.8. Let β = {β (n)}∞ n=−∞ be a sequence of positive numbers such that β (n) ≥ β (n + 1) and limn→∞ β (n) = 1. Assume that α ⊂ C is a Borel set with the property that the functions ζ n χα (ζ ) belong to L2β for all integers n. Denote by H the closed linear subspace of L2β generated by {ζ n χα (ζ )}∞ n=−∞ , set T = Uβ |H, and define X ∈ I (T, Mα ) by X = Xβ |H. Then the pair (X , Mα ) is an isometric asymptote for T . Proof. It is clear that H is invariant for Uβ and Xβ H is dense in L2 (α ), so that the result follows from Theorem 1.6. The operator Xβ is one-to-one, therefore the operator T constructed in the preceding lemma is of class C1·· , and we have α − = σ (Mα ) ⊂ σ (T ) by Lemma 1.4. We show that sequences β can be found so that the hypothesis of Lemma 2.8 is satisfied, and in addition T ∈ C10 and σ (T ) = α − . The sequences we need consist entirely of powers of 2. More precisely, ( 1 for n ≥ 0 β (n) = 2 p for r p ≤ −n < r p+1 , where 1 = r1 < r2 < · · · are integers such that the sequence {r p+1 − r p }∞p=1 is increasing and unbounded. Such a sequence β is called a simple weight sequence. Lemma 2.9. If β is a simple weight sequence, then Uβ ∈ C10 and σ (Uβ ) = C. Moreover, kUβ−nk = β (−n) for n ≥ 0. Proof. Note that Uβ∗ (β (k)−1 ek ) = (β (k)/β (k − 1))(β (k − 1)−1ek−1 ), so that Uβ∗n (β (k)−1 ek ) =

β (k) (β (k − n)−1 ek−n ) β (k − n)

tends to zero because β (k − n) → ∞ as n → ∞, thus Uβ ∈ C10 . We already know that C = σ (U) ⊂ σ (Uβ ), and it is clear that Uβ is invertible with kUβ−1 k = 2. More generally, β (k) = β (−n) = 2 p kUβ−nk = sup k β (k + n) for r p ≤ n < r p+1 . Therefore kUβ−n k1/n = 2 p/n ≤ 2 p/r p , and it follows that the spectral radius of U −1 is equal to 1 because lim

p

p→∞ r p

= 0.

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373

It follows that σ (Uβ ) ⊂ C, and the lemma is proved.

2 Lemma 2.10. For any sequence { fk }∞ k=1 ⊂ L there exists a simple weight sequence ∞ 2 β such that { fk }k=1 ⊂ Lβ .

Proof. There is no loss of generality in assuming that ∑∞ k=1 k f k k < ∞. Consider the   (k) 2 1/2 (k) n ∞ ∞ . It suffices to find β fourier series ∑n=−∞ un ζ of fk , and set un = ∑k=1 |un |

n 2 such that the function f ∈ L2 with Fourier series ∑∞ n=−∞ un ζ belongs to Lβ . Choose ∞ 2 −p integers 1 = r1 < r2 < · · · such that ∑n=r p |u−n | ≤ 5 for p ≥ 2. Increasing r p if necessary, we may assume that the sequence r p+1 − r p tends increasingly to infinity. We have then ∞

r p+1 −1

r2 −1

∞

p=2

n=r p

n=r1

n=0

k f k2β = ∑ 4 p ∑ |u−n |2 + 4 ∑ |u−n |2 + ∑ |un |2  p 4 + 4k f k2 < ∞, ≤ ∑ 5 p=1 ∞

as desired. In order to control σ (T ) in Lemma 2.8 we need one more result. Lemma 2.11. If T is an invertible C1·· -contraction such that ∞

∑ n−pkT −n k < ∞

(2.4)

n=1

for some integer p, then σ (T ) = σ (WT ). Proof. Lemma 1.4 implies the inclusion σ (WT ) ⊂ σ (T ). Because the sequence −1 is at most 1; thus σ (T ) ⊂ C. {n−pkT −n k}∞ n=1 is bounded, the spectral radius of T To conclude the proof, we must show that T − ζ0 I is invertible for each ζ0 ∈ C \ σ (WT ). Fix such a scalar ζ0 , and construct an infinitely differentiable function g on C such that g(ζ ) = −1 in a neighborhood of ζ0 , and g(ζ ) = 0 in an open set containing σ (WT ). If we factor g(ζ ) − g(ζ0) = (ζ − ζ0 )h(ζ )

(ζ ∈ C),

n the function h is also infinitely differentiable. Let ∑∞ n=−∞ un ζ be the Fourier series ∞ k n of h, so that ∑n=−∞ (in) un ζ is the Fourier series of the kth derivative of h. The sequence {nk un }n must thus be bounded for every k ≥ 1, and (2.4) implies that n ′ ∑∞ n=−∞ |un |kT k < ∞. We can then define the operator Y ∈ {T } by setting ∞

Y = ∑ un T n . n=−∞

Note that I = g(WT ) − g(ζ0 )I = (WT − ζ0 I)h(WT ), and XT ∈ I (Y, h(WT )). We deduce that XT (T − ζ0 I)Y = (WT − ζ0 I)h(WT )XT = XT .

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Finally, because T ∈ C1·· , the operator XT is one-to-one, and the last identity implies that Y is the inverse of T − ζ0 I. We are now ready for an analogue of Lemma 2.3, but without cyclicity. Proposition 2.12. For every Borel set α ⊂ C with positive measure, and for every c > 0, there exists a C10 -contraction A such that σ (A) = α − , ωA = α , and kA−1 k ≥ c. Proof. Let β be a simple weight sequence provided by Lemma 2.10 applied to the functions fn (ζ ) = ζ n χα (ζ ) (n = 0, ±1, ±2, . . .). Lemma 2.8 produces an operator T ∈ C1·· such that WT is unitarily equivalent to Mα . The operator Uβ ∈ C10 is invertible and T has dense range. We conclude that T ∈ C10 is invertible as well, and kT −n k ≤ kUβ−n k for n ≥ 1. Therefore the equality σ (T ) = α − follows from Lemma −n −n −3 2.11 if ∑∞ n=1 n kUβ k < ∞. This condition is achieved if kUβ k = β (−n) ≤ 2n for n ≥ 1, and for this it suffices to take r p ≥ 2 p−1 in the definition of β . (Note that enlarging the numbers r p also enlarges the space L2β .) Observe, however, that kT −1 k ≤ 2. In order to construct the required operator, consider the characteristic function {DT , DT ∗ , ΘT (λ )}. The properties of T imply that this function is inner and ∗outer, ΘT (λ ) is invertible for every λ ∈ D, the set {ζ ∈ C : ΘT (ζ ) is not unitary} coincides with α up to sets of measure zero, and for every ζ ∈ C \ α − the function ΘT extends analytically to a neighborhood of ζ (cf. Theorem VI.4.1). Fix an arbitrary unit vector x0 ∈ DT , and choose a unitary transformation Z : DT ∗ → DT such that ZΘT (0)x0 = kΘT (0)x0 kx0 . Such a unitary transformation exists because the spectral conditions on T imply that dT = dT ∗ = ∞. The function {DT , DT , ZΘT (λ )} coincides with ΘT . For each natural number n, denote by An a c.n.u. contraction whose characteristic function coincides with {DT , DT , (ZΘT (λ ))n }. The function (ZΘT (λ ))n is also inner, ∗-outer, invertible for λ ∈ D, it extends analytically at points in C \ α − , and α = {ζ ∈ C : (ZΘT (ζ ))n is not unitary}. It follows that An is of class C10 , ωAn = α , and σ (An ) = α − . Now, −n −n −n kA−1 →∞ n k = k(ZΘT (0)) k ≥ k(ZΘT (0)) x0 k = kΘT (0)x0 k

as n → ∞ (cf. formula (VI.4.11)). Therefore the conclusion of the proposition is satisfied by An for sufficiently large n. The following result shows the application of functional calculus (as in the proof of Lemma 2.5) produces operators of class C10 . Then we show how to produce a cyclic operator. Lemma 2.13. Let T be a C10 -contraction such that σ (T ) does not contain D, and let u ∈ H ∞ be a nonconstant function which extends continuously to D such that kuk∞ = 1 and |u| = 1 on ωT . Then the contraction u(T ) also belongs to C10 and ωu(T ) = u(ωT ).

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375

Proof. First note that XT u(T ) = u(WT )XT , and u(WT ) is unitary. Because XT is oneto-one, we deduce that u(T ) ∈ C1·· . Setting µ = u(0) ∈ D, we know that u(T ) ∈ C· 0 if and only if the operator A = (u(T ) − µ )(I − µ u(T ))−1 belongs to C· 0 (cf. Theorem III.2.1, Sec. VI.1.3, and Proposition VI.3.5). Now, there exists a function v ∈ H ∞ such that u(λ ) − µ = λ v(λ ) (λ ∈ D). 1 − µ u(λ )

Considering the values of v on C we see that kvk∞ ≤ 1. We conclude that for every x ∈ H we have kA∗n xk = kv(T )∗n T ∗n xk ≤ kT ∗n xk → 0

as n → ∞, and it follows that A ∈ C· 0 . Thus u(T ) ∈ C10 , as claimed. To verify the last assertion, observe that the adjoint of WT+ is a quasi-affine transform of T ∗ . The requirement on σ (T ), and the Wold decomposition, imply that WT+ = WT , and therefore XT is a quasi-affinity. Because XT ∈ I (u(T ), u(WT )) and u(WT ) is unitary, there exists Y ∈ I (Wu(T ) , u(WT )) such that XT = Y Xu(T ) . The range of Y is necessarily dense and therefore u(WT ) is unitarily equivalent to a direct summand of Wu(T ) (cf. the proof of Proposition II.3.4). An application of Proposition 2.4 shows that Xu(T ) is also a quasi-affinity. By Lemma 1.4, there exists a contraction Te ∈ {Wu(T ) }′ satisfying Xu(T ) T = TeXu(T ) . The equalities Wu(T ) Xu(T ) = Xu(T ) u(T ) = u(Te)Xu(T )

yield then u(Te) = Wu(T ) . Considering the canonical decomposition of Te, we infer by Theorem III.2.1 that Te is in fact a unitary operator. Thus there exists an operator Z ∈ I (WT , Te) such that Xu(T ) = ZXT . Because Z ∈ I (u(WT ), u(Te)) has dense range, it follows that Wu(T ) is unitarily equivalent to a direct summand of u(WT ). We conclude that Wu(T ) and u(WT ) are unitarily equivalent, and therefore ωu(T ) = u(ωT ) (Cf. K ADISON AND S INGER [1] and the proof of Lemma 2.5). To see that the construction in the proof of Theorem 2.6 produces a cyclic operator, we need some auxiliary results. The first one is a general observation about direct sums. In the proof we use a well known theorem of Runge. In our context it simply says that for every proper compact subset A ⊂ C, every continuous function f on C, and every ε > 0, there exists a polynomial p such that | f − p| < ε in a neighborhood of A. Lemma 2.14. Let {Tn }∞ disn=1 be a sequence of cyclic contractions with pairwise L joint spectra contained in the unit circle C. Then the direct sum T = ∞ T n=1 n is cyclic as well. Proof. Assume that Tn acts on Hn . For n ≤ N, we denote by QN,n the orthogonal L projection of Nk=1 Hk onto Hn . The spectra are assumed disjoint, therefore Runge’s

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C HAPTER IX. T HE S TRUCTURE OF C1·· -C ONTRACTIONS

theorem mentioned above, and the Riesz–Dunford functional calculus imply the existence of a polynomial pN,n satisfying

  N

pN,n L Tk − QN,n < 1 .

N k=1

Choose positive numbers δN such that

N sup{kpN ′ ,n (T )k : 1 ≤ n ≤ N ′ ≤ N} < δN−1

(N = 1, 2, . . . ).

Choose for each n, a cyclic vector en ∈ Hn for Tn such that ken k = δn . We show that L the vector e = ∞ n=1 en is cyclic for T . For n ≤ N we have

     N N N

L L L

Tk ek ek − QN,n kpN,n (T )e − en k ≤ pN.n

k=1 k=1 k=1

   ∞ ∞

L L + Tk ek

pN.n

k=N+1

k=N+1

1/2 kek 2 ≤ + kpN,n (T )k ∑ δk N k=N+1   ∞ kek 1 1/2 kek 1 ≤ < + +√ . ∑ 2 N k N N k=N+1 

∞

Letting N → ∞, we see that en belongs to the cyclic space for T generated by e, and therefore e is a cyclic vector for T . For the following lemma, denote by ωT,x the set ωT |Hx , where Hx =

∞ _

T nx

n=0

is the cyclic space for T generated by x. Recall that the sets ωT,x and ωT are only determined up to sets of measure zero. Lemma 2.15. Let T be an a. c. contraction on the separable Hilbert space H. The set {x ∈ H : ωT,x = ωT } is a dense Gδ in H. that m(ωT ) > 0. Let Proof. The case in which m(ωT ) = 0 is trivial, so we assume L (X ,V ) be a unitary L asymptote of T . We can take V = ∞ M k=1 ωk with C ⊃ ω1 ⊃ ω2 ⊃ · · · , and Xx = ∞ ). The equality ωT,x = ωT holds X x, with X ∈ I (T, M ωk k k=1 k if the set {ζ ∈ ω1 : (X1 x)(ζ ) = 0} has measure zero (cf. Theorem 1.6). The complement of the set in the statement is ∞ S

{x ∈ H : m(ωT \ ωT,x ) ≥ 1/n}.

n=1

2. T HE SPECTRA OF C1·· - CONTRACTIONS

377

Thus it suffices to prove that the set Sδ = {x ∈ H : m(ωT \ ωT,x ) ≥ δ } is closed and nowhere dense for δ > 0. Let {xn }∞ n=1 ⊂ Sδ be a sequence with limit x, and set ωn = ωT \ ωT,xn . Passing to a subsequence, we may assume that the sequence ∞ 2 R{ χωn }n=1 converges weakly in L (ωT ) to a function f . Note that 0 ≤ f ≤ χωT and f dm = limn→∞ m(ωn ) ≥ δ , so that m({ζ : f (ζ ) > 0}) ≥ δ . We have Z

|(Xk x)(ζ )| f (ζ ) dm(ζ ) = lim

n→∞

Z

|(Xk xn )(ζ )|χωn (ζ ) dm(ζ ) = 0

for all k, so that m({ζ ∈ ω1 : (Xk x)(ζ ) = 0, k ≥ 1}) ≥ m({ζ : f (ζ ) > 0}) ≥ δ , and hence x ∈ Sδ . This proves that Sδ is closed. To show that Sδ has empty interior it suffices to produce a single vector x such that ωT,x = ωT . Indeed, if x ∈ H is such a vector, and y ∈ H is an arbitrary vector, the set Eε = {ζ ∈ ω1 : (X1 y)(ζ ) + ε (X1 x)(ζ ) = 0} has positive measure for at most countably many values of the scalar ε . It follows that ωT,y+ε x = ωT for values of ε arbitrarily close to zero, in particular y + ε x ∈ / Sδ . To prove the existence of such a vector x, choose a total sequence {xn }∞ ⊂ H such that kxn k < 2−n , and set fn = n=1 X1 xn . The inequalities k fn k1 ≤ k fn k2 ≤ kxn k < 2−n implySthat ∑∞ n=1 | f n (ζ )| < ∞ for a.e. ζ ∈ ω1 . Setting αn = {ζ ∈ ω1 : fn (ζ ) 6= 0}, we have ∞ α n=1 n = ω1 a.e. We can construct inductively nonzero scalars cn ∈ D such that, for every n ≥ 1, n

∑ ck fk (ζ ) 6= 0 for a.e. ζ ∈

k=1

and the set

αn′ m(αn′ )

n S

k=1

αk ,

  n−1 n−1 S −n αk : |cn fn (ζ )| ≥ 3 ∑ ck fk (ζ ) = ζ∈

2−n

k=1

k=1

α1′

satisfies < (here = ∅). Indeed, set c1 = 1, and assume that n > 1 and c j has been defined for j < n. Choosing δn ∈ (0, 1) sufficiently small, we have m(αn′ ) < 2−n if |cn | = δn . Choose next a complex number λn with absolute value 1 such that n−1

∑ ck fk (ζ ) + λn δn fn (ζ ) 6= 0 for a.e. ζ ∈

k=1

n S

k=1

αk ,

′ and set cn = λn δn . Because ∑∞ n=2 m(αn ) < ∞, we conclude that a.e. ζ ∈ ω1 is only contained in finitely many of the sets αn′ . It follows then that the function ∑∞ n=1 cn f n is different from zero a.e. on ω1 , and hence the vector x = ∑∞ n=1 cn xn satisfies the equality ωT,x = ω1 = ωT .

We have now the necessary ingredients to prove an analogue of Lemma 2.5 for the class C10 .

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C HAPTER IX. T HE S TRUCTURE OF C1·· -C ONTRACTIONS

Lemma 2.16. Let Ω ⊂ D be a simply connected open set bounded by a rectifiable Jordan curve Γ . Assume that that the set Γ ∩ C contains a nontrivial arc J with 0 < m(J) < 1. Let α ⊂ J be a Borel set of positive measure, µ0 ∈ Ω , and c > 0. There exists a cyclic contraction T ∈ C10 with the following properties. (1) (2) (3) (4)

σ (T ) = α − . WT is unitarily equivalent to Mα . k(T − µ0 I)−1 k ≥ c. k(T − µ I)−1 k ≤ 1/dist(µ , Ω ) for every µ in the complement of Ω .

Proof. Fix a homeomorphism u, a set β , and a factorization u(λ ) − µ0 = λ v(λ ) (λ ∈ D) as in the proof of Lemma 2.5. Proposition 2.12 provides an operator T1 ∈ C10 such that σ (T1 ) = β − , kT1−1 k > ckvk∞ ,

and ωT1 = β . (Actually, WT1 is unitarily equivalent to the orthogonal sum of some copies of Mβ by the proof of Proposition 2.12.) Proposition 2.4 and Lemma 2.13 now show that the operator T2 = u(T1 ) is of class C10 ,

σ (T2 ) = u(σ (T1 )) = u(β − ) = α − , and

ωT2 = α ,

k(T2 − µ0 I)−1 k > c.

The operator T2 also satisfies condition (4) by virtue of Theorem III.2.1(b). The required operator T is obtained as the restriction T = T2 |M, where M is the cyclic space generated by a vector x such that ωT2 ,x = ωT2 = α . Such vectors are dense, therefore T satisfies (3) for an appropriate choice of x, provided T − µ0 I is invertible. Property (2) is verified by virtue of the equality ωT,x = ωT = α . Observe now that the operators (T2 − λ I)−1 (λ ∈ / σ (T2 )) can be approximated in norm by polynomials in T2 ; this follows from Runge’s theorem via the Riesz–Dunford functional calculus. It follows that M is invariant for (T2 − λ I)−1 (λ ∈ / σ (T2 )), and this immediately implies that σ (T ) ⊂ σ (T2 ) = α − and condition (4) is satisfied. Finally, condition (1) is also satisfied because α − = σ (WT ) ⊂ σ (T ). The following result is a slight variation of Lemma 2.16. Lemma 2.17. Let β ⊂ C, ∅ 6= β 6= C, be an essentially closed set, and fix ε ∈ (0, 1). There exists a cyclic C10 -contraction T with the following properties. (1) σ (T ) = β . (2) WT is unitarily equivalent to Mβ . (3) k(T − µ I)−1 k ≤ 1/(dist(µ , β ) − ε ) whenever dist(µ , β ) > ε .

2. T HE SPECTRA OF C1·· - CONTRACTIONS

379

Proof. There exist a finite number of pairwise disjoint open arcs J1 , J2 , . . . , Jn ⊂ C such that

β⊂ The sets

n S

k=1

Jk ⊂ {µ : dist(µ , β ) < ε /2}.

Ωk = {rζ : ζ ∈ Jk , r ∈ (1 − (ε /2), 1)}

are contained in { µ : dist(µ , β ) < ε }. Lemma 2.16 provides then cyclic C10 contractions Tk such that σ (Tk ) = β ∩ Jk , WTk is unitarily equivalent to Mβ ∩Jk , and L k(Tk − µ I)−1k ≤ 1/dist(µ , Ωk ) for µ ∈ / Ωk . The operator T = nk=1 Tk satisfies all the requirements of the lemma. Its cyclicity is guaranteed by Lemma 2.14. We can now prove Theorem 2.6.

Proof. Let { µn }∞ n=1 be a dense sequence in σ , such that each of its terms appears infinitely many times. Using Lemma 2.16 in place of Lemma 2.5, and applying the regularity of Lebesgue measure, we construct a sequence {αn }∞ n=1 of pairwise disjoint essentially closed subsets of ω0 with m(αn ) > 0, and a sequence {Tn′ }∞ n=1 of cyclic C10 -contractions with the following properties. (1) (2) (3) (4)

σ (Tn′ ) = αn− = αn . WTn′ is unitarily equivalent to Mαn . if (Tn′ − µn I)−1 exists then k(Tn′ − µnI)−1 k > n. k(Tn′ − µ I)−1 k ≤ 1/[dist(µ , σ ) − 1/n] if dist(µ , σ ) > 1/n. L

S

∞ ′ We can then form the C10 -contraction T ′ = ∞ n=1 Tn that satisfies ωT ′ = n=1 αn ′ and σ (T ) = σ . The set ω1 = ω0 \ ωT ′ may have positive measure. If that is the case, regularity of Lebesgue measureSimplies the existence of pairwise disjoint essentially closed sets βn ⊂ ω1 such that ∞ n=1 βn = ω1 (up to sets of measure zero). By Lemma 2.17, there exist cyclic C10 -contractions Tn′′ such that:

(a) σ (Tn′′ ) = βn . (b) WTn′′ is unitarily equivalent to Mβn . (c) k(Tn′′ − µ I)−1k ≤ 1/[dist(µ , βn ) − 1/n] if dist(µ , βn ) > 1/n.

Condition (c) implies, of course,

(c′ ) k(Tn′′ − µ I)−1k ≤ 1/[dist(µ , σ ) − 1/n] if dist(µ , σ ) > 1/n.

These conditions imply that the C10 -contraction T ′′ =

∞ M

Tn′′

n=1

satisfies σ (T ′′ ) = ω1− and ωT ′′ = ω1 . Moreover, note that the family {αn , βn }∞ n=1 consists of pairwise disjoint closed sets. Therefore Lemma 2.14 implies that T = T ′ ⊕ T ′′ is cyclic. Thus T satisfies all the requirements of Theorem 2.6.

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C HAPTER IX. T HE S TRUCTURE OF C1·· -C ONTRACTIONS

3 Intertwining with unilateral shifts 1. In the preceding sections we studied intertwinings in I (T,W ), where W is an isometric or unitary operator. In this section we focus on I (W, T ), where T is an absolutely continuous contraction, and W is isometric. The results are particularly useful for contractions of class C1· , but they can be applied whenever T has a nontrivial ∗-residual part, which is why we chose to prove them in greater generality. Let W = W0 ⊕W1 on K0 ⊕ K1 be the Wold decomposition of W , with W1 unitary and W0 a unilateral shift. If T ∈ C· 0 and X ∈ I (W, T ), then obviously X|K1 = 0. Therefore we restrict ourselves to the case of a unilateral shift W . We start by explaining in rough outline the idea of this section. Fix a contraction T on a Hilbert space H, and let U be its minimal unitary dilation on K ⊃ H. If the ∗-residual space R∗ has nonzero intersection with H, then H ∩ R∗ is invariant for T . Indeed, for x ∈ H ∩ R∗ , kPR∗ (T x)k = kR∗ PR∗ xk = kR∗ xk = kxk ≥ kT xk, and therefore PR∗ (T x) = T x. If, in addition, T is c.n.u., then T |(H ∩ R∗ ) is a unilateral shift. We show that the space R∗ always contains vectors which are arbitrarily close to H in the L∞ norm, and this provides restrictions of T similar to unilateral shifts when ωT = C. The meaning of L∞ approximation is explicated by Lemma 3.1. We focus first on the c.n.u. case, so let us assume that {E, E∗ , Θ (λ )} is a purely contractive analytic function, and that T is the model operator associated with this function (as in Sec. VI.3) acting on H, whereas U and U+ are the operators of multiplication by ζ on the spaces K = L2 (E∗ ) ⊕ ∆ L2 (E),

K+ = H 2 (E∗ ) ⊕ ∆ L2 (E),

respectively, where ∆ (ζ ) = (I − Θ (ζ )∗Θ (ζ ))1/2 for a.e. ζ ∈ C. The spaces E and E∗ are assumed to be separable. The crucial approximation result is the following purely function theoretical result. Lemma 3.1. Fix an essentially bounded function u ∈ L2 (E∗ ) and a positive number ε . There exist u′ ∈ H 2 (E∗ ) and an inner function ϕ ∈ H ∞ such that ku′ (ζ ) − ϕ (ζ )u(ζ )k < ε for a.e. ζ ∈ C. Proof. We may assume that ε < 1 and ku(ζ )k ≤ 1 a.e. There exists a measurable S partition C = ∞ n=1 σn , and there are vectors xn ∈ E∗ such that kxn k ≤ 1, ku(ζ ) − xnk <

ε 2

(a.e. ζ ∈ σn , n ≥ 1).

This is easily seen by constructing a Borel partition of the unit ball of E∗ into sets of diameter less than ε /2, and defining σn to be preimages under u of elements of this partition. Construct outer functions ψn ∈ H ∞ such that |ψn | = χσn +

ε χ 10n C\σn

(a.e. on C).

3. I NTERTWINING WITH UNILATERAL SHIFTS

381

∞ The function ψ = ∑∞ n=1 ψn belongs to H , and

ε ε = n 9 n=1 10 ∞

||ψ (ζ )| − 1| ≤ ∑

(a.e. ζ ∈ C).

Factor now ψ = gϕ , with g outer and ϕ inner. Because

ε ||g(ζ )| − 1| = ||ψ (ζ )| − 1| ≤ , 9 the function g is invertible in H ∞ , and kg−1k∞ < 2. Finally, we define ∞

u′ = ∑ g−1 ψn xn . n=1

The series above converges a.e. and in L2 (E∗ ), and therefore its sum belongs to H 2 (E∗ ). For almost every ζ ∈ σn we have

ε + ku′(ζ ) − ϕ (ζ )xn k 2 ε ≤ + kg−1k∞ kg(ζ )u′ (ζ ) − g(ζ )ϕ (ζ )xn k 2 ε ≤ + 2|ψn (ζ ) − ψ (ζ )| + 2 ∑ kψk (ζ )xk k 2 k6=n ∞ ε ε ≤ + 4 ∑ k < ε. 2 k=1 10

ku′ (ζ ) − ϕ (ζ )u(ζ )k <

Thus the functions u′ and ϕ verify the conclusion of the lemma. In this functional representation we have R∗ = K ⊖ M(L) = K ⊖ {Θ u ⊕ ∆ u : u ∈ L2 (E)}, and the projection onto M(L) is the multiplication operator by the operator-valued function     ΘΘ ∗ Θ ∆ Θ [Θ ∗ , ∆ ] = . ∆ ∆Θ ∗ ∆ 2

It follows that PR∗ is the operator of multiplication by the projection-valued function   I − Θ (ζ )Θ (ζ )∗ −Θ (ζ )∆ (ζ ) P(ζ ) = , −∆ (ζ )Θ (ζ )∗ Θ (ζ )∗Θ (ζ ) where P(ζ ) must be viewed as an operator on the space E∗ ⊕ ∆ (ζ )E. The operator R∗ is determined, up to unitary equivalence, by the measurable spectral multiplicity function

µ (ζ ) = rank(P(ζ )) (ζ ∈ C),

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C HAPTER IX. T HE S TRUCTURE OF C1·· -C ONTRACTIONS

whose possible values are nonnegative integers or ℵ0 . More precisely, if we set

ωn = {ζ : µ (ζ ) ≥ n} (n = 1, 2, . . . ),

L

L

∞ then R∗ is unitarily equivalent to ∞ n=1 Mωn . Let X ∈ I ( n=1 Mωn , R∗ ) be a unitary operator, and consider the sequence {χωn }∞ n=1 , where χωn is viewed as an eleL 2 (ω ). The sequence {w }∞ ⊂ R defined by ment in the nth component of ∞ L n n n=1 ∗ n=1 wn = X χωn has the property that wn (ζ ) = 0 for n > µ (ζ ) and {wn (ζ )}0≤n−1<µ (ζ ) is an orthonormal basis in the range of P(ζ ) for a.e. ζ ∈ C. Such a sequence {wn }∞ n=1 ∞ is called a basic sequence for R∗ . Conversely, given a basic sequence {w } for n n=1 L R∗ , there exists a unitary operator X ∈ I ( ∞ , R ) such that X = w . This χ M n ωn n=1 ωn ∗ discussion, including the concept of a basic sequence, can be applied to any a.c. unitary operator on a separable Hilbert space. We refer to D UNFORD AND S CHWARTZ [2], where the more general case of normal operators is considered. Fix a separable, infinite-dimensional Hilbert space F with orthonormal basis ∞ {en }∞ n=1 , and a basic sequence {wn }n=1 for an a.c. unitary operator W . Denote by 2 SF the unilateral shift on H (F). We can then construct an operator J ∈ I (SF ,W ) satisfying Jen = wn ; here en is regarded as a constant function in H 2 (F). Such an operator is called a basic operator. The existence of J is seen most easily when L ∞ 2 W= ∞ n=1 Mωn and wn = χωn . If f = ∑n=1 f n en is an arbitrary element of H (F), 2 with fn ∈ H , the corresponding basic operator is defined by

Jf =

∞ L

n=1

χωn fn .

The following theorem is the main result in this section. Theorem 3.2. Let T be an a.c. contraction on a separable Hilbert space H, let U be its minimal unitary dilation on K ⊃ H, and let ε ∈ (0, 1). Denote by R∗ the ∗-residual part of U. There exist operators Y,Y ′ ∈ I (SF , T ) and Z, Z ′ ∈ I (T, R∗ ) such that

(i) The norms kY k, kY ′ k, kZk, kZ ′ k are less than 1 + ε . (ii) ZY and Z ′Y ′ are basic operators. (iii) Y H 2 (F) ∨Y ′ H 2 (F) = H.

Proof. Let T = T0 ⊕ T1 be the decomposition of T such that T0 is c.n.u. and T1 is an absolutely continuous unitary operator. We may assume that T0 is the model operator T associated with the purely contractive analytic function {E, E∗ , Θ (λ )}, and T1 = L∞ L∞ 2 n=1 L (αk ), k=1 Mαk , where α1 ⊃ α2 ⊃ · · · are Borel subsets of C. Setting H1 = the minimal unitary dilation of T acts on K = K ⊕ H1, and R∗ = R∗ ⊕ H1. Consider a basic sequence {wn }∞ for R∗ . Writing wn = un ⊕ vn ⊕ gn with un ∈ L2 (E∗ ), vn ∈ L∞n=1 2 2 ∆ L (E), and gn ∈ k=1 L (αk ), we deduce from Lemma 3.1 that there exist inner functions ϕn ∈ H ∞ , and functions u′n ∈ H 2 (E∗ ), such that ku′n (ζ ) − ϕn (ζ )un (ζ )k <

ε 10n

(a.e. ζ ∈ C, n ≥ 1).

3. I NTERTWINING WITH UNILATERAL SHIFTS

383

The functions w′n = u′n ⊕ ϕn vn ⊕ ϕn gn belong then to K+ , and kw′n (ζ ) − ϕn (ζ )wn (ζ )k <

ε 10n

(a.e. ζ ∈ C, n ≥ 1).

(3.1)

∞ Replacing {wn }∞ n=1 by the basic sequence {ϕn wn }n=1 we can assume that ϕn = 1 for every n. Let us also set

w′′n = PR∗ w′n

(n ≥ 1).

As in the c.n.u. case, PR∗ is a multiplication operator by a projection valued function Q(ζ ), and therefore by (3.1) and by the assumption that ϕn = 1, we also have kw′′n (ζ ) − wn (ζ )k <

ε 10n

(a.e. ζ ∈ C, n ≥ 1).

The construction of the desired operators proceeds as follows: ′

(a) Show that there is an operator A ∈ {R∗ } such that kAk < 1 + ε and Aw′′n = wn . (b) Define Z = APR∗ |H. (c) Show that there is an operator X ∈ I (SF ,U+ ) such that kX k < 1 + ε and X en = w′n for all n. (d) Define Y = PH X . (e) Verify that ZYen = wn , and therefore ZY is basic. We start with the construction of A. The vectors {wn (ζ )}0≤n−1


ε2 ∑ n n=1 100 ∞

1/2

ε < . 9

Here we use kDk2 to denote the Hilbert–Schmidt norm: kDk22 = ∑n kDwn (ζ )k2 for an operator on the range of Q(ζ ). The function B(ζ ) is strongly measurable, and therefore the operator B of pointwise multiplication by B(ζ ) belongs to {R∗ }′ and kB − Ik < ε /9. It follows that B is invertible, and the operator A = B−1 satisfies condition (a). Analogously, for a.e. ζ ∈ C there are operators X (ζ ) defined on F such that X(ζ )en = w′n (ζ ) and kX (ζ )k < 1 + ε . Indeed, X (ζ ) differs from the partially isometric operator X0 (ζ ) : en 7→ wn (ζ ) by a Hilbert–Schmidt operator with norm less than ε . The operator X is defined as pointwise multiplication by the function X (ζ ). To verify (e), note that K+ ⊖ H ⊂ M(L) = R⊥ ∗ , thus PR∗ PH |K+ = PR∗ |K+ . Therefore ZYen = ZPH w′n = APR∗ PH w′n = APR∗ w′n = Aw′′n = wn for n ≥ 1. Furthermore, because PH |K+ ∈ I (U+ , T ) and PR∗ |H ∈ I (T, R∗ ), the operators Z and Y defined in (b) and (d) satisfy the required commutation and norm requirements.

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C HAPTER IX. T HE S TRUCTURE OF C1·· -C ONTRACTIONS

To prove (iii), it suffices to construct a second operator X ′ such that XH 2 (F) ∨ X ′ H 2 (F) = K+ . Fix then a total sequence {gn }∞ n=1 in K+ consisting of essentially bounded functions. Replace the vector w′n in the above construction by w′n + δn gn , where δn 6= 0 is chosen so that the condition kwn (ζ ) − (w′n (ζ ) + δn gn (ζ ))k <

ε 9n

is satisfied a.e. The above construction of X shows that there exists a unique operator X ′ ∈ I (SF ,U+ ) satisfying X ′ en = w′n + δn gn (n ≥ 1) and kX ′ k < 1 + ε Then δn gn = X ′ en − Xen, so that the ranges of X and X ′ span the whole space K+ . The theorem is proved. 2. Fix an a.c. contraction T on H, and let U on K ⊃ H be the minimal unitary dilation of T . As noted before, there exist Borel sets C ⊃ ω1 ⊃ ω2 ⊃ · · · such that L R∗ is unitarily equivalent to ∞ equivalent to n=1 Mωn . The operator R∗ is unitarily T WT , thus it follows that ω1 = ωT . We also use the notation ωℵ0 = ∞ n=1 ωn . Given (n) a Borel set α ⊂ C, and a cardinal number n ≤ ℵ0 , we denote by Mα the direct sum of n copies of Mα . Fix also a separable Hilbert space F with orthonormal basis {ek }∞ k=1 , and denote by Fn the closed linear space generated by {ek : k − 1 < n}; note that Fℵ0 = F. Proposition 3.3. With the notation above, let α ⊂ C be a Borel set. The following are equivalent for every n: (1) m(α \ ωn ) = 0. (n) (2) There exist operators Y ∈ I (SFn , T ) and Z ∈ I (T, Mα ) such that ZY is a basic operator. If these conditions are satisfied and ε > 0, the operators Y and Z can be chosen such that kY k < 1 + ε and kZk < 1 + ε .

Proof. Theorem 3.2 provides operators Y0 ∈ I (SF , T ) and Z0 ∈ I (T, R∗ ) such that kY0 k < 1 + ε , kZ0 k < 1 + ε , and Z0Y0 is a basic operator. Denote by {wk = Z0Y0 ek }∞ k=1 the corresponding basic sequence for R∗ . Assume first that (1) is true. The reducing space N for R∗ generated by the functions {χα wk : k − 1 < n} is such (n) that R∗ |N is unitarily equivalent to Mα . The required operators Y, Z are obtained (n) simply as Y = Y0 |H 2 (Fn ) and Z = APN Z0 , where A ∈ I (R∗ |N, Mα ) is unitary. The operators Y, Z thus constructed also satisfy kY k, kZk < 1 + ε . Conversely, assume that (2) holds, and observe that the smallest reducing sub(n) space for Mα which contains the range of Z is the entire space. This follows from the fact that ZH contains the range of the basic operator ZY . Because (PR∗ |H, R∗ ) (n) is a unitary asymptote of T , there exists A ∈ I (R∗ , Mα ) such that Z = APR∗ |H. (n)−1 We also have A ∈ I (R−1 ), and this implies that AR∗ is a reducing sub∗ , Mα (n) space for Mα , and therefore A has dense range because AR∗ ⊃ ZH. Equivalently,

3. I NTERTWINING WITH UNILATERAL SHIFTS

385

(n)

A∗ ∈ I (Mα , R∗ ) is one-to-one, and we see as above that the closure of the range of A∗ is reducing R∗ . The restriction of R∗ to this reducing subspace is unitarily equiv(n) alent to Mα . Spectral multiplicity decreases when passing to a reducing subspace, thus (1) follows. When n = ℵ0 , the argument of Theorem 3.2 shows that the preceding construction actually produce two operators Y,Y ′ such that Y H 2 (Fℵ0 ) ∨Y ′ H 2 (Fℵ0 ) = H where, of course, Fℵ0 = F. For finite n we have the following result. Proposition 3.4. With the notation of the preceding proposition, assume that m(α \ ωn ) = 0. Then, for every ε > 0, there exist sequences {Yk }∞ k=1 ⊂ I (SFn , T ) and (n) ∞ {Zk }k=1 ⊂ I (T, Mα ) such that (1) ZkYk is a basic operator for k ≥ 1. (2) W kYk k, kZk k < 1 + ε for k ≥ 1. 2 (3) ∞ k=1 Yk H (Fn ) = H.

Proof. We adapt the last part of the argument of Theorem 3.2. Using the notation in that proof, we construct operators Xk ∈ I (SFn ,U+ ) such that Xk e j = w′j for j = 2, 3, . . . , n, and Xk e1 = w′1 + δk gk , where δk > 0 is sufficiently small. It is clear that the ranges of the resulting operators X, X1 , X2 , . . . generate a dense subspace of K+ . The proposition follows easily from this observation. 3. The preceding results are most useful when the set α can be chosen to be equal to C. In this case, basic operators are isometric. Thus, if Y, Z satisfy the conclusion of Proposition 3.3, then (1 + ε )−1khk ≤ kY hk ≤ (1 + ε )khk (h ∈ H 2 (Fn )).

(3.2)

In particular, Y H 2 (Fn ) is closed and invariant for T , and T |(Y H 2 (Fn )) is similar to SFn . We have proved one of the main results of this chapter. Theorem 3.5. Assume that T is an a.c. contraction and ωn = C for some n, 1 ≤ n ≤ ℵ0 . Then there exists an invariant subspace M for T such that T |M is similar to SFn . This result can be strengthened by using Proposition 3.4. For this purpose, we denote by Lat(T, n, ε ) the collection of those invariant subspaces M for T for which there is an invertible operator Y ∈ I (SFn , T |M) satisfying (3.2). Proposition 3.4 yields then the following result. The sets ωn in the statement are such that R∗ or WT L is unitarily equivalent to ∞ n=1 Mωn .

Theorem 3.6. Let T be an a.c. contraction on a separable Hilbert space H, and let n be a cardinal number such that ωn = C. For every ε > 0, there exists a sequence W∞ {Mk }∞ k=1 ⊂ Lat(T, n, ε ) such that k=1 Mk = H. If n = ℵ0 , there exist subspaces M1 , M2 ∈ Lat(T, n, ε ) such that M1 ∨ M2 = H. Conversely, if Lat(T, n, ε ) 6= ∅, it follows that ωn = C.

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Proof. We only need to verify the last statement. Assume therefore that M is an invariant subspace for T , and X ∈ I (T |M, SFn ) is an invertible operator. Because WT |M is a direct summand of WT , it suffices to prove the result for T |M. Assume therefore that M = H. Inasmuch as H 2 (Fn ) ⊂ L2 (Fn ), X can also be viewed as an operator in I (T,W ), where W denotes multiplication by ζ in L2 (Fn ). Therefore there exists A ∈ I (WT ,W ) such that AXT = X . The range of A contains H 2 (Fn ), and therefore it must be dense in L2 (Fn ). The conclusion m(C \ ωn ) = 0 is reached as in the proof of Proposition 3.3. When the set ωT 6= C, we can still obtain a result about invariant subspaces for powers of T . We use the notation T ≺ S to indicate that T is a quasi-affine transform of S. Theorem 3.7. Let T ∈ C1·· be an a.c. contraction on a separable Hilbert space H, and let n > 2π /m(ωT ) be an integer. Then T n has no cyclic vectors. Proof. Let (X ,VT ) be the isometric asymptote of T . The operator X is a quasiaffinity because T ∈ C1·· , so that T ≺ VT . Without loss of generality, we may assume that T has a cyclic vector. In this case VT must have a cyclic vector so that either VT is a unilateral shift of multiplicity one, or VT is unitarily equivalent to MωT . Because T n ≺ VTn , it suffices to show that VTn has no cyclic vectors for n > 2π /m(ωT ). If VT is a shift of multiplicity one then ωT = C, and VTn does not have cyclic vectors because it is a shift of multiplicity n ≥ 2. Assume then that VT is unitarily equivalent to MωT L and n > 2π /m(ωT ). The operator Mωn T is unitarily equivalent to nk=1 Mαk , where    2(k − 1)π 2kπ , αk = eint : eit ∈ ωT ,t ∈ n n

(k = 1, 2, . . . , n).

Because ∑nk=1 m(αk ) = nm(ωT ) > 2π , these sets cannot be disjoint. Thus Mωn T has spectral multiplicity greater than one, and hence it has no cyclic vectors. 4. We conclude this section with a reflexivity result. Given an operator T on a Hilbert space H, we denote by Lat(T ) the collection of all invariant subspaces of T . Given a collection L of closed subspaces of H, we denote by Alg(L ) the algebra consisting of all operators T such that T M ⊂ M for every M ∈ L . It is clear that the algebra AlgLat(T )=Alg(Lat(T )) contains T and is closed in the weak operator topology. The operator T is said to be reflexive if AlgLat(T ) is the smallest algebra containing T and I which is closed in the weak operator topology. If T is an a.c. contraction and u ∈ H ∞ , then u(T ) is a strong limit of polynomials in T . Therefore the following result implies the reflexivity of T provided that ωT = C. Theorem 3.8. Let T be an a.c. contraction on a separable space, and assume that ωT = C. Then AlgLat(T ) = {u(T ) : u ∈ H ∞ }.

Proof. We prove the result first when T is the unilateral shift S on H 2 . The functions kµ ∈ H 2 defined for |µ | < 1 by ∞

kµ (λ ) = (1 − µλ )−1 = ∑ µ k λ k k=0

(λ ∈ D)

4. H YPERINVARIANT SUBSPACES OF C11 - CONTRACTIONS

387

satisfy S∗ kµ = µ kµ and ( f , kµ ) = f (µ ) for f ∈ H 2 . In particular, the set {kµ : µ ∈ D} is total in H 2 . Consider now an operator A ∈ AlgLat(S), and note that A∗ ∈ AlgLat(S∗ ). In particular, A∗ kµ = αµ kµ for some scalar αµ . We have then A∗ S∗ kµ = S∗ A∗ kµ and, because the vectors kµ form a total set, A commutes with S. An application of Lemma V.3.2 (with U+ = U+′ = S) implies that A = u(S) for some u ∈ H ∞ . Thus the theorem is true for T = S, and it therefore be true for any a.c. contraction that is similar to S. Consider now the general case, and let A ∈ AlgLat(T ). Theorem 3.6 implies the existence of operators {Yn }∞ n=1 ⊂ I (S, T ) such that 1 khk ≤ kYn hk ≤ 2khk (n ≥ 1, h ∈ H 2 ), 2 and

W∞

n=1 Yn H

2

= H. The operator T |(Yn H 2 ) is then similar to S, and A|(Yn H 2 ) ∈ AlgLat(T |(Yn H 2 )).

The first part of the proof implies the existence of functions un ∈ H ∞ such that Ax = un (T )x for n ≥ 1 and x ∈ Yn H 2 . Replacing A by A − u1(T ) ∈ AlgLat(T ), we may assume that u1 = 0. It suffices to prove that un = 0 for n ≥ 2. Assume to the contrary that un is not zero for some n ≥ 2. In this case the operator A|(Yn H 2 ) = un (T )|(Yn H 2 ) is similar to un (S), and is therefore one-to-one; hence (Yn H 2 ) ∩ (Y1 H 2 ) = {0}. The operator Y = Y1 + (1/8)Yn ∈ I (S, T ) satisfies the inequalities 9 1 khk ≤ kY hk ≤ khk (h ∈ H 2 ), 4 4 and therefore T |(Y H 2 ) is similar to S. There is thus u ∈ H ∞ such that Ax = u(T )x for x ∈ Y H 2 . Note then that for f ∈ H 2 \ {0} 1 1 1 u(T )Y1 f + u(T )Yn f = u(T )Y f = AY f = AY1 f + AYn f = un (T )Yn f , 8 8 8 so Y1 (u f ) = u(T )Y1 f = (1/8)(un (T ) − u(T ))Yn f ∈ (Yn H 2 ) ∩ (Y1 H 2 ) = {0}. We deduce that u = 0, and hence un (T )Yn = O as well. This implies the desired conclusion that un = 0.

4 Hyperinvariant subspaces of C11 -contractions 1. In this section we use the unitary asymptotes of T and T ∗ to carry out a thorough investigation of the lattice of hyperinvariant subspaces of a contraction T ∈ C11 . A different approach was used in Sec. VII.5. Given a unitary asymptote (X,W ) for T ∗ , the pair (W ∗ , X ∗ ) is called a unitary ∗-asymptote for T . We record for further use the properties of a ∗-asymptote; these follow directly from the definition of unitary asymptotes, and from their concrete construction.

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Lemma 4.1. Let T on H be a contraction, and let (V,Y ) be a unitary ∗-asymptote for T . (1) For every unitary operator U, and every X ∈ I (U, T ), there exists a unique Z ∈ I (U,V ) such that X = Y Z. Moreover, we have kZk = kX k. (2) There is a contractive homomorphism γ∗ : {T }′ → {V }′ such that AY = Y γ∗ (A) for A ∈ {T }′ . (3) We have kY ∗ xk = limn→∞ kT ∗n xk for x ∈ H. (4) If T ∈ C11 then T is quasi-similar to V .

Assume now that T ∈ C1·· on H. As seen in Sec. II.4, there exists a largest invariant subspace H1 ∈ Lat(T ) such that the restriction T |H1 is in C11 (cf. Proposition II.4.2). We have H ⊖ H1 = {x ∈ H : lim kT ∗n xk = 0} = kerY ∗ , n→∞

where (V,Y ) is a unitary ∗-asymptote for T . Therefore H1 is precisely the closure of the range of Y . More generally, assume that M ∈ Lat(T ). Then the restriction T |M is also in C1·· , and therefore there exists a largest space M1 ∈ Lat(T ) such that M1 ⊂ M and T |M1 is in C11 . We use the notation M1 = Ψ11 (M). Thus Ψ11 is a map from Lat(T ) to the collection Lat1 (T ) of those invariant subspaces N such that T |N ∈ C11 . When T ∈ C11 , the elements of Lat1 (T ) are also called quasi-reducing subspaces for T . The quasi-reducing subspaces of a unitary operator are obviously the same as the reducing subspaces, but this is generally not true for general C11 contractions. It is easy to check that Lat1 (T ) is a lattice. More precisely, if M1 , M2 ∈ Lat1 (T ), then M1 ∨ M2 ∈ Lat1 (T ) is the least upper bound of the two spaces, and Ψ11 (M1 ∩ M2 ) is their greatest lower bound. The examples at the end of this section show that Ψ11 (M1 ∩ M2 ) may be different from M1 ∩ M2 . We use the more general notation Lat S for the collection of all closed subspaces of H that are invariant for every element A in a family S of operators on H. Thus, Lat{T }′ represents the collection of hyperinvariant subspaces of T . Denote T by {T }′′ = A∈{T }′ {A}′ the double commutant of T . Proposition 4.2. Let T on H be a C11 -contraction.

(1) For every M ∈ Lat1 (T ) there exists A ∈ {T }′ such that M = AH = AM. (2) Lat1 (T ) ⊂ Lat{T }′′ .

Proof. Let (X,W ) be a unitary asymptote of T , and let (V,Y ) be a unitary ∗asymptote of T |M. Note that the range of Y is dense in M and W |XM is unitary because T |M ∈ C11 . We know that T |M is quasi-similar to its unitary asymptote and ∗-asymptote, and the unitary asymptote of T |M is unitarily equivalent to the direct summand W |X M of W . It follows that there exists a map B ∈ I (W,V ) such that BX|M has dense range. The desired operator is then defined as A = Y BX. Part (2) obviously follows from (1). We later show that the opposite inclusion to (2) is not true in general. Let us set Lat1 {T }′ = Lat1 (T ) ∩ Lat{T }′ .

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Proposition 4.3. Let T ∈ C11 be a cyclic operator. Then {T }′ = {T }′′ and hence Lat1 (T ) ⊂ Lat{T }′ . Proof. Let (X ,W ) be a unitary asymptote of T , and let γ : {T }′ → {W }′ be the homomorphism satisfying γ (A)X = X A for A ∈ {T }′ . Because W is quasi-similar to T , it is also cyclic, and therefore {W }′ is commutative (cf. Theorems IX.3.4 and IX.6.6 in C ONWAY [1]). The fact that X is one-to-one implies then that γ is one-toone, and therefore {T }′ is commutative. Thus {T }′ = {T }′′ , and the final inclusion follows from the preceding corollary. Quasi-reducing hyperinvariant subspaces have a basic maximality property. We write T ∼ S to indicate that T and S are quasi-similar.

Proposition 4.4. Let T ∈ C11 , M ∈ Lat1 {T }′ , and N ∈ Lat1 (T ). If T |M ∼ T |N then M ⊃ N. If in addition N ∈ Lat1 {T }′ , we must have M = N. Proof. It is enough to prove the first assertion. Fix an operator A ∈ {T }′ such that M = AM = AH, and a quasi-affinity Q ∈ I (T |M, T |N). The operator QA commutes with T and therefore it leaves M invariant. Therefore N = QM = QAM ⊂ M, as claimed. 2. We now relate the set Lat1 {T }′ to the corresponding set for the unitary asymptote of T . First, a preliminary result. Lemma 4.5. Let T and T ′ be two contractions such that T ≺ T ′ . If T ∈ C· 1 then T ′ ∈ C· 1 as well. If T and T ′ are in C11 then T ∼ T ′ . Proof. Let A ∈ I (T, T ′ ) be a quasi-affinity. We have A∗ T ′∗n x = T ∗n A∗ x for n ≥ 1. If limn→∞ kT ′∗n xk = 0 we deduce that limn→∞ T ∗n A∗ x = 0. If T ∈ C· 1 this is possible only for x = 0, and therefore T ′ ∈ C· 1 . Finally, if both T and T ′ are of class C11 , they are quasi-similar to their unitary asymptotes WT ,WT ′ , and therefore WT ≺ WT ′ . It follows that WT and WT ′ are unitarily equivalent, and the desired conclusion T ∼ T ′ follows. Theorem 4.6. Let T be a contraction of class C11 , let (X,W ) be a unitary asymptote of T , and let (W,Y ) be a unitary ∗-asymptote of T . There exists a bijection ϕ : Lat1 {T }′ → Lat{W }′ such that T |M ∼ W |ϕ (M) for every M ∈ Lat1 {T }′ . We have ϕ (M) = X M for M ∈ Lat1 {T }′ and ϕ −1 (N) = Y N for N ∈ Lat{W }′ . Proof. Assume that T acts on H and W acts on H′ . We know from Proposition 4.4 applied to W (resp., T ) that for every M ∈ Lat1 {T }′ (resp., N ∈ Lat{W }′ ) there exists at most one subspace N ∈ Lat{W }′ (resp., M ∈ Lat1 {T }′ ) such that T |M ∼

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W |N. It suffices therefore to show that such spaces do exist. We start with M ∈ Lat1 {T }′ , and construct the space N ∈ Lat{W }′ by setting W

N = {AXM : A ∈ {W }′ }. Note that YAX ∈ {T }′ for every A ∈ {W }′ , and because M is hyperinvariant we have W XY N = {XYAX M : A ∈ {W }′ } ⊂ X M. On the other hand, XY ∈ {W }′ , PN ∈ {W }′′ , and XY has dense range. Thus XY N = XY PN H′ = PN XY H′ = PN H′ = N, and the last two relations imply XM = N. Note that X |M ∈ I (T |M,W |XM), hence T |M ≺ W |X M. Because W |XM is unitary, the preceding lemma shows that T |M ∼ W |X M. Finally, fix a space N ∈ Lat{W }′ , and set M = Y N. As in the preceding argument, we have W |N ≺ T |M, and W |N is unitary. The preceding lemma implies then that T |M ∈ C· 1 , and therefore T |M ∈ C11 . The same lemma implies now that W |N ∼ T |M. To conclude the proof, it suffices to show that M is hyperinvariant. Consider then an arbitrary operator A ∈ {T }′ , and denote by B ∈ {W }′ the unique operator satisfying AY = Y B provided by Lemma 4.1(2). We have AM ⊂ AY N = Y BN ⊂ Y N = M, where we used the fact that N is hyperinvariant. The theorem is proved. The preceding result shows, in particular, that Lat 1 {T }′ is a lattice, isomorphic to Lat{W }′ . Indeed, the map ϕ and its inverse preserve inclusions. As noted already, quasi-similar C11 -operators have unitarily equivalent unitary asymptotes. The preceding theorem implies they also have isomorphic lattices of quasi-reducing hyperinvariant subspaces. The following result relates this isomorphism to the construction in the proof of Proposition II.5.1. Recall that Ψ11 (M) denotes the C11 part of an invariant subspace M for a C1·· operator. Corollary 4.7. Let T1 ∼ T2 be two C11 -contractions, and fix quasi-affinities X ∈ I (T1 , T2 ),Y ∈ I (T2 , T1 ). There exists a unique bijection ϕ : Lat1 {T1 }′ → Lat1 {T2 }′ such that T1 |M1 ∼ T2 |ϕ (M2 ) for every M1 ∈ Lat1 {T1 }′ , namely W

ϕ (M1 ) = {AX M1 : A ∈ {T2 }′ } = Ψ11 (Y −1 M1 ) (M1 ∈ Lat1 {T1 }′ ). Its inverse is given by W

ϕ −1 (M2 ) = {BY M2 : B ∈ {T1 }′ } = Ψ11 (X −1 M2 ) (M2 ∈ Lat1 {T2 }′ ).

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391

Proof. The existence and uniqueness of ϕ follow from the preceding theorem. To establish the formula for ϕ , fix M1 ∈ Lat1 {T1 }′ , and set M2 = ϕ (M1 ). Note first that T2 |M2 ≺ T1 |Y M2 . An application of Lemma 4.5 shows that T1 |Y M2 is of class C11 , and T1 |Y M2 ∼ T2 |M2 ∼ T1 |M1 .

Proposition 4.4 shows then that Y M2 ⊂ M1 , and therefore M2 ⊂ Y −1 M1 . Set N = Ψ11 (Y −1 M1 ). Because N is the largest subspace of Y −1 M1 where the restriction of T2 is C11 , we have M2 ⊂ N.

Obviously, Y N ⊂ M1 . By symmetry, we also have X M1 ⊂ M2 , and therefore N′ = XY N is contained in M2 . Because XY is a quasi-affinity, Lemma 4.6 yields T2 |N′ ∼ T2 |N, and thus there exists a quasi-affinity Z ∈ I (T2 |N′ , T2 |N). By Proposition 4.2, there exists B ∈ {T2 }′ such that N′ = BN′ = BH2 . Because ZB ∈ {T2 }′ and M2 is hyperinvariant for T2 , we conclude that N = ZBN′ ⊂ ZBM2 ⊂ M2 . The opposite inclusion was verified earlier. W Finally, the space q(M1 ) = {AXM1 : A ∈ {T2 }′ } is clearly in Lat{T2 }′ , and XM1 ⊂ M2 ∈ Lat{T2 }′

implies that q(M) ⊂ M2 . Because T2 |X M1 ∼ T1 |M1 ∼ T2 |M2 , there exists a quasi-affinity Q ∈ I (T2 |XM1 , T2 |M2 ). Proposition 4.2 yields an operator D ∈ {T2 }′ such that X M1 = DXM1 = DH2 . It follows that QD ∈ {T2 }′ , and M2 = QDXM1 ⊂ QDq(M1 ) ⊂ q(M1 ). The formula for ϕ −1 is obtained by interchanging the roles of T1 and T2 . The corollary is proved. If W is a unitary operator on a separable space, the projections in {W }′′ are of the form χα (W ), and therefore the spaces in Lat{W }′ are precisely the ranges of these operators (cf. Sec. IX.8 in C ONWAY [1]). Assume that W is absolutely continuous, L and is therefore unitarily equivalent to an operator of the form ∞ M n=1 ωn for some Borel sets C ⊃ ω1 ⊃ ω2 ⊃ · · · . Then χα1 (W ) = χα2 (W ) if and only if α1 ∩ ω1 = α2 ∩ ω1 . If T is a C11 -contraction quasi-similar to W , and α ⊂ C is a Borel set, we denote by HT,α the space that corresponds to the range of χα (W ) under the isomorphism between Lat{W }′ and Lat1 {T }′ . These are easily seen to be precisely the spaces considered in Theorem VII.5.2. We can give a new description of the spaces HT,α when T = T is a model operator. This description does not use regular factorizations, and depends explicitly on the characteristic function and on the set α . Thus, assume that E and E∗

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are separable Hilbert spaces, {E, E∗ , Θ (λ )} is a purely contractive analytic function, and the spaces K, K+ , H are constructed as usual. Let us also set ∆∗ (ζ ) = (I − Θ (ζ )Θ (ζ )∗ )1/2 for a.e. ζ ∈ C.

Corollary 4.8. Assume that T is of class C11 and α ⊂ C is a Borel set. We have then HT,α = Ψ11 ({u ⊕ v ∈ H : −∆∗ (ζ )u(ζ ) + Θ (ζ )v(ζ ) = 0 for a.e. ζ ∈ C \ α }). Proof. The pair (PR∗ |H, R∗ ) is a unitary asymptote for T, and the preceding corollary implies that HT,α = Ψ11 ((PR∗ |H)−1 (χα R∗ )). Now, χα R∗ = {w ∈ R∗ : w(ζ ) = 0 for a.e. ζ ∈ C \ α }, and PR∗ is the operator of pointwise multiplication by the projection-valued function   I − Θ (ζ )Θ (ζ )∗ −Θ (ζ )∆ (ζ ) P(ζ ) = . −∆ (ζ )Θ (ζ )∗ Θ (ζ )∗Θ (ζ ) The corollary follows because P(ζ ) can also be written as P(ζ ) =



   ∆∗ (ζ )2 −∆∗ (ζ )Θ (ζ ) −∆∗ (ζ ) [−∆∗ (ζ ), Θ (ζ )], = Θ (ζ )∗ −Θ (ζ )∗ ∆∗ (ζ ) Θ (ζ )∗Θ (ζ )

and

is an isometry for a.e. ζ ∈ C.



−∆∗ (ζ ) Θ (ζ )∗



3. We conclude with a few observations about the lattice structure of Lat(T ) and Lat1 (T ). For an arbitrary set S of operators on H, any family {M j } j∈J of subspaces in Lat S has a least upper bound and a greatest lower bound, namely W T j∈J M j and j∈J M j . If T ∈ C11 and {M j } j∈J ⊂ Lat1 (T ), the family {M j } j∈J still has a least upper bound and a greatest lower bound in Lat1 (T ), namely the spaces W T M = j∈J M j and Ψ11 ( j∈J M j ), respectively. We show that the latter space can in T fact be different from j∈J M j . Example 1. There exist a cyclic absolutely continuous contraction T ∈ C11 , and two subspaces M1 , M2 ∈ Lat1 {T }′ such that M = M1 ∩ M2 ∈ / Lat1 (T ). In particular, M ∈ Lat{T }′ \ Lat1 (T ). In order to facilitate the construction, observe that an operator of the form   ADT T

is always a contraction if T and A are contractions, where A could even act between different spaces. Similarly, [T, DT ∗ A] is a contraction if T and A are contractions. We start with c.n.u. contractions T j ∈ C11 on H j ( j = 0, 1, 2) that are not boundedly invertible; the existence of such operators was proved in Sec. VI.4.2. Fix vec-

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393

tors x j ∈ H j with kx j k = 1/2 ( j = 0, 1, 2), and denote by E 1 the one-dimensional space of complex scalars. Define an operator T on H = H0 ⊕ E 1 ⊕ H1 ⊕ H2 by setting   T0 X0 O O  O O Y1 Y2   T =  O O T1 O  , O O O T2

where X0 : E 1 → H0 is defined by X0 λ = λ DT0∗ x0 , λ ∈ E 1 , and Y j : H j → E 1 is defined by Y j v = (DTj v, x j ) for v ∈ H j and j = 1, 2. In order to see that T is a contraction, note that T maps H0 ⊕ E 1 ⊕ {0} ⊕ {0} and {0} ⊕ {0} ⊕ H1 ⊕ H2 into orthogonal subspaces. Thus it is enough to verify contractivity separately on these two spaces. The restrictions of T to these two spaces are contractions of the form just discussed above. For instance,     Y1 Y2  T1 O  = ADT1 ⊕T2 , T1 ⊕ T2 O T2

where A : H1 ⊕ H2 → E 1 is a contraction defined by A(h1 ⊕ h2 ) = (h1 , x1 ) + (h2 , x2 ). Obviously the only vectors x ∈ H satisfying limn→∞ kT n xk = 0 belong to H0 ⊕ 1 E ⊕ {0} ⊕ {0}, and these vectors x are in the kernel of T . Choosing x0 such that DT0∗ x0 does not belong to T0 H0 will then ensure that T ∈ C1·· . Analogously, if DTj x j is not in T j∗ H j for j = 1, 2, then that T ∈ C· 1 . Such choices are in fact possible. For instance, the operator T0 is not onto, and therefore we can choose x0 = DT0∗ y0 for / T0 H0 . Making analogous some y0 ∈ H0 \ T0 H0 . Therefore DT0∗ x0 = y0 − T0 T0∗ y0 ∈ choices for x1 and x2 , we have constructed an operator of class C11 . Next, repeated application of Theorem 1.6 shows that WT is unitarily equivalent to WT0 ⊕ WT1 ⊕ WT2 . By Theorem 2.2 we can choose WTj to be cyclic unitary operators with disjoint spectra. With this choice, WT is also a cyclic a.c. operator, and therefore so is T ∼ WT . Observe now that the spaces M1 = H0 ⊕ E 1 ⊕ H1 ⊕ {0} and M2 = H0 ⊕ E 1 ⊕ {0} ⊕ H2 are invariant for T , and the above arguments show that T |M j ∈ C11 for j = 1, 2. Proposition 4.4 shows that in fact M j ∈ Lat1 {T }′ for j = 1, 2. Finally, M = M1 ∩ M2 = H0 ⊕ E 1 ⊕ {0} ⊕ {0}, and obviously (T |M)∗ has nonzero kernel. Thus M ∈ / Lat1 (T ). The interested reader be able to verify that T is in fact c.n.u. A more elaborate construction yields the following example. Example 2. There exist a cyclicTa.c. contraction T ∈ C11 , and a sequence M1 ⊃ M2 ⊃ · · · in Lat1 {T }′ such that ∞ / Lat1 (T ). n=1 Mn ∈

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C HAPTER IX. T HE S TRUCTURE OF C1·· -C ONTRACTIONS

We only sketch the construction briefly. Start with a sequence of c.n.u. C11 contractions T0 , T1 , . . . that are not boundedly invertible, and construct   T0 X0 O O O · · ·  O O Y1 Y2 Y3 · · ·     O O T1 O O · · ·    T =  O O O T2 O · · ·     O O O O T3 · · ·    .. .. .. .. .. . . . . . . . .

on H0 ⊕ E 1 ⊕ H1 ⊕ H2 ⊕ · · · , where the operators X0 and Y j are defined as in the preceding example. With proper choices, this is a cyclic c.n.u. C11 -contraction. We L then set Mn = H0 ⊕ E 1 ⊕ ∞j=1 K j , with K j = H j for j > n and K j = {0} for j ≤ n. The intersection of these spaces is H0 ⊕ E 1 ⊕ {0} ⊕ {0} ⊕ · · ·, and it does not belong to Lat1 (T ) as seen in the preceding example. Remark. The situation discussed in Example 1 shows that part (v) of Theorem VII.6.2 is not necessarily true if the characteristic function is not assumed to have a scalar multiple. Example 2 shows that, in the absence of scalar multiples, Theorem VII.6.2(v) may fail even if the sequence {αn } is decreasing. In terms of characteristic functions, the subspace M in Example 1 shows that there exists a regular factorization Θ (λ ) = Θ2 (λ )Θ1 (λ ) of a function Θ (λ ), outer from both sides, such that Θ1 (λ ) is not outer, although Θ (ζ ) is isometric on a set of positive measure (cf. Theorem VII.1.1 and Propositions VII.2.1 and VI.3.5). Remark. It is interesting to note that for a subspace M ∈ Lat{T }′ , the C11 part Ψ11 (M) is also hyperinvariant. Consider indeed an operator X ∈ {T }′ , and consider the matrices     T T X X T |M = 11 12 , X |M = 11 12 O T22 X21 X22 relative to the decomposition M = Ψ11 (M) ⊕ (M ⊖ Ψ11 (M)). These operators commute, thus X21 ∈ I (T11 , T22 ). Because T11 ∈ C· 1 and T22 ∈ C· 0 , it follows immediately that X21 = O. Thus Ψ11 (M) is invariant for an arbitrary X ∈ {T }′ . Consider now a family {M j } j∈J ⊂ Lat1 {T }′ . It follows that the greatest lower bound T Ψ11 ( j∈J M j ) of this family in Lat1 (T ) actually belongs to Lat1 {T }′ . On the other hand, Lat1 {T }′ is obviously closed under the usual operation of taking the closed linear span of a family of subspaces.

5 Notes As seen in Sec. II.5, power-bounded operators of class C11 are quasi-similar to unitary operators. This idea was developed in K E´ RCHY [8], where isometric and unitary asymptotes were constructed for arbitrary power-bounded operators. Section 1 has been organized so as to suggest this more general development. Theorem 1.6 is proved in K E´ RCHY [8] in this general setting. In the context of contractions, part

5. N OTES

395

(3) of this theorem also appears in B ERCOVICI AND K E´ RCHY [1] where the tool is regular factorization. The identification of the unitary asymptote with the ∗-residual part of the minimal unitary dilation, and the explicit identification of the corresponding intertwiners is from K E´ RCHY [5]. Isometric and unitary asymptotes were introduced for some operators that are not even power-bounded. For instance, if {kT n k}∞ n=1 is a regular sequence (or more generally, for certain semigroups with regular norm behavior) this was done in K E´ RCHY [10],[12] and in K E´ RCHY AND L E´ KA [1]. The regularity property was characterized ¨ in K E´ RCHY [11] and K E´ RCHY AND M ULLER [1]. A study of C1·· -contractions based on the unitary asymptote can also be found in B EAUZAMY [2, Chapter XII]. The first example of a C11 -contraction such that D 6= σ (T ) 6⊂ C is given in E CKSTEIN [1]. An example of a cyclic C11 -contraction with σ (T ) = D appears in B ERCOVICI AND K E´ RCHY [1]; a noncyclic example is in VI.4.2. Theorem 2.2 is from B ERCOVICI AND K E´ RCHY [2]. Proposition 2.4 is a special form of a result of F OIAS¸ AND M LAK [1]. G ILFEATHER [1] presented a weighted bilateral shift T ∈ C10 such that σ (T ) = C. B EAUZAMY [1] constructed a contraction T ∈ C10 whose spectrum contains a nontrivial arc of C disjoint from σ (WT ). The complete description of the spectra of C10 -contractions in Theorem 2.6 is from B ERCOVICI ´ RCHY [3], which extends results in K E´ RCHY [4]. Lemma 2.11 also folAND K E lows from the fact that T is a generalized scalar operator; see C OLOJOAR A˘ AND F OIAS¸ [1, Theorem 5.1.4]. A spectral mapping theorem generalizing the relation ωu(T ) = u(ωT ) (cf. Lemma 2.13) can be found in K E´ RCHY [15]. The proof of Lemma 2.14 is inspired by N IKOLSKI˘I [1]. The material of Sec. 3 is from K E´ RCHY [9],[16]. However, the approximation Lemma 3.1 is based on ideas from K E´ RCHY [6],[7], and Theorem 3.7 is due to S Z .-N AND F. [31]. The concept of reflexivity was introduced by S ARASON [4] who proved that normal operators and unilateral shifts are reflexive. An overview of this area is in H ADWIN [1]. W U [3],[5] proved that C11 -contractions with finite defect indices are reflexive. Theorem 3.8 was proved in TAKAHASHI [4]. The case when T ∈ C11 was done earlier by K E´ RCHY [6]. The reflexivity of such operators also follows from B ROWN AND C HEVREAU [1], where it was shown that an a.c. contraction T is reflexive if ku(T )k = kuk∞ for every u ∈ H ∞ . The isomorphism of Lat1 {T }′ to Lat{WT }′ was proved in K E´ RCHY [8] for power-bounded operators; this is an extension of Theorem 4.6. Theorem 4.6 can also be derived, at least in the c.n.u. case, from Theorem VII.5.2 and a result of T EODORESCU [3] on regular factorizations. Examples 1, 2 in Sec. 4, and the subsequent remarks are from K E´ RCHY [1]. The examples also use some ideas from B ERCOVICI AND K E´ RCHY [1]. The isomorphisms of various invariant subspace lattices, and their implementation, are studied in K E´ RCHY [3]. In particular, Corollary 4.8 is from that paper. Proposition 4.3, along with a more detailed discussion of cyclic C11 -contractions, is in K E´ RCHY [2]. The classification of lattices of invariant subspaces of isometries is discussed in C ONWAY AND G ILLESPIE [1] and K E´ RCHY [14]. The existence of hyperinvariant subspaces for C1·· -contractions T was discussed in B EAUZAMY

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[2, Theorem XII.8.1]. It is shown there that an invertible T ∈ C1·· has a nontrivial hyperinvariant subspace if it is not a scalar multiple of the identity, and −2 −n xk < ∞ for some x 6= 0. This result was extended by K E´ RCHY ∑∞ n=1 n log kT [13] to C1·· -operators with regular norm sequences, and it was shown that there is in fact an infinite family of completely disjoint hyperinvariant subspaces. An interesting connection between unitary asymptotes and the existence of disjoint invariant subspaces is discussed in TAKAHASHI [6]. TAKAHASHI [2] shows that a contraction T of class C1· , whose defect operators are Hilbert–Schmidt, is completely injection-similar to an isometry. The relation of injection-similarity was introduced in S Z .-N.–F. [24].

Chapter X

The Structure of Operators of Class C0 1 Local maximal functions and maximal vectors 1. Let T be a c.n.u. contraction on the Hilbert space H, and h ∈ H. Denote by Mh the cyclic space for T generated by h. Observe that for a function u ∈ H ∞ , we have u(T )h = 0 if and only if u(T |Mh ) = O.

Definition. The operator T is said to be locally of class C0 if, for every h ∈ H, there exists a function u ∈ H ∞ (depending, generally, on h) such that u(T )h = 0. The minimal function of T |Mh is denoted mh if T is locally of class C0 .

The purpose of this section is to prove that operators T which are locally of class C0 are actually of class C0 ; that is, the function u in the definition above can be chosen independently of h (cf. Sec. III.4). The proof follows from the existence of maximal vectors, defined below. Definition. Assume that T is locally of class C0 , and h ∈ H. The vector h is said to be T -maximal (or simply maximal when no confusion may arise) if mg divides mh for every g ∈ H.

Observe that, provided that T has a maximal vector h, then T is of class C0 , and m T = mh . 2. For the purposes of this chapter, we need to extend some of the concepts in Sec. III.1 as follows. Consider functions ϕ , ψ ∈ H 2 , not both identically zero. We denote by ϕ ∧ ψ the greatest common inner divisor of the functions ϕ and ψ . More V generally, i ϕi denotes the greatest common inner divisor of a family {ϕi } of functions, not all identically zero. Analogously, ϕ ∨ ψ stands for the least common inner multiple of ϕ and ψ , with a corresponding notation for inner multiples of families of functions. In this chapter, an equality u = v between two inner functions is understood to hold only up to a constant factor of absolute value one. Proposition 1.1. Let {ϕi : i ∈ I} be a family of nonconstant inner divisors of the inner function ϕ ∈ H ∞ . If ϕi ∧ ϕ j = 1 for i 6= j, then the set I is at most countable. B.Sz.-Nagy et al., Harmonic Analysis of Operators on Hilbert Space, Universitext, DOI 10.1007/978-1-4419-6094-8_10, © Springer Science + Business Media, LLC 2010

397

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Proof. Choose λ ∈ D such that ϕ (λ ) 6= 0. If i1 , i2 , . . . , in ∈ I are distinct, then

ϕi1 ϕi2 · · · ϕin = ϕi1 ∨ ϕi2 ∨ · · · ∨ ϕin divides ϕ and, in particular, n

∑ − log|ϕik (λ )| ≤ − log|ϕ (λ )|.

k=1

We conclude that ∑i∈I − log|ϕi (λ )| < ∞, and hence the set I1 = {i ∈ I : |ϕi (λ )| 6= 0} is at most countable. By assumption, I = I1 and the proposition follows. 3. Assume that T is locally of class C0 , and K ⊂ H is a subspace of dimension two with basis {h1 , h2 }. Let us set mK = mh 1 ∨ m h 2 , and note that mK does not depend on the particular basis. Indeed, mK is the greatest common inner divisor of all functions u ∈ H ∞ satisfying u(T )K = {0}.

Lemma 1.2. Let T be locally of class C0 , and let K ⊂ H be a subspace of dimension two. Then the set {h ∈ K : mh 6= mK } is the union of an at most countable family of subspaces of dimension one.

Proof. Denote by A the set in the statement, and observe that 0 ∈ A, and mλ h = mh whenever λ is a nonzero scalar. We conclude that A is the union of a family of subspaces of dimension one: S A = E 1 hi , i∈I

where E 1 denotes, as usual, the complex numbers, and hi , h j are linearly independent for i 6= j. Define ϕi = mK /mhi , and note that ϕi is not constant because hi ∈ A for i ∈ I. If i 6= j, the vectors hi and h j form a basis of K, and therefore

ϕi ∧ ϕ j = mK /(mhi ∨ mh j ) = mK /mK = 1. The lemma follows now immediately from Proposition 1.1. Lemma 1.3. Let T be locally of class C0 . For each λ0 ∈ D and every α > 0, the set

σ = {h ∈ H : |mh (λ0 )| ≥ α } is closed in H. Proof. Let {hn } ⊂ σ be a sequence converging to h. An application of the Vitali and Montel theorem allows us to assume, upon dropping to a subsequence, that the sequence {mhn } converges uniformly on the compact subsets of D to a function

1. L OCAL MAXIMAL FUNCTIONS AND MAXIMAL VECTORS

399

u ∈ H ∞ . We certainly have |u(λ )| ≤ 1 for λ ∈ D, and |u(λ0 )| ≥ α . By Theorem III.2.1, {mhn (T )} converges weakly to u(T ), and therefore for k ∈ H we have |(u(T )h, k)| ≤ |((u(T ) − mhn (T ))h, k)| + |(mhn (T )(h − hn), k)| ≤ |((u(T ) − mhn (T ))h, k)| + kh − hnkkkk → 0, as n → ∞; here we made use of the relation mhn (T )hn = 0. Because k is arbitrary, we conclude that u(T )h = 0, and therefore mh |u. We can thus write u = mh ϕ with ϕ ∈ H ∞ , and |ϕ (eit )| = |u(eit )| a.e. It follows that |ϕ (λ0 )| ≤ 1, and

α ≤ |u(λ0 )| ≤ |mh (λ0 )| so that h ∈ σ , as desired. The next result follows from an application of the Baire category theorem. Lemma 1.4. Assume that T is locally of class C0 . The set {k ∈ H : |mk (λ0 )| = inf |mh (λ0 )|} h∈H

is a dense Gδ set in H for each λ0 ∈ D. Proof. Fix λ0 ∈ D, and set

α = inf |mh (λ0 )|. h∈H

The complement of the set in the statement can be written as   1 . σ j = h ∈ H : |mh (λ0 )| ≥ α + j

S∞

j=1 σ j ,

where

The preceding lemma implies that each σ j is a closed set, and to finish the proof it suffices to show that each σ j has empty interior. Suppose to the contrary that σ j contains the open ball B = {h : kh − h0k < ε }. Because σ j 6= H, we can consider a linear space K generated by h0 and some vector k ∈ / σ j , k 6= 0. Lemma 1.2 implies the existence of f ∈ K ∩ B such that m f = mK ; in particular mk |m f , from which we infer |m f (λ0 )| ≤ |mk (λ0 )| < α + (1/ j) because k ∈ / σ j . On the other hand, f ∈ B ⊂ σ j , a contradiction. The lemma follows. Theorem 1.5. Assume that T is locally of class C0 on H. Then there exist T -maximal vectors, and the set of T -maximal vectors is a dense Gδ in H. In particular, T is of class C0 and mT = mh for each T -maximal vector h. Proof. The intersection of countably many Gδ sets is still a dense Gδ , and therefore the set M = {h ∈ H : |mh (λn )| = inf |mk (λn )|, n ≥ 0} k∈H

is a dense Gδ for any choice of sequence {λn } ⊂ D. Choose this sequence to be dense in D. If h ∈ M and k ∈ H, we have |mh (λn )| ≤ |mk (λn )| for all n, and by

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continuity

|mh (λ )| ≤ |mk (λ )|,

λ ∈ D.

We conclude that mk |mh , and thus every element of M is T -maximal. The other assertions of the theorem are now obvious. 4. We need the following variation of Theorem 1.5 on the existence of maximal vectors. Theorem 1.6. Let T be an operator of class C0 on H, B a Banach space, and X : B → H a bounded linear operator. If H=

W

T n XB,

n≥0

then the set is a dense Gδ in B.

{k ∈ B : mXk = mT }

Proof. The proof closely imitates that of Theorem 1.5. We provide the relevant details. Fix λ0 ∈ D, and set α = inf |mXk (λ0 )|. k∈B

The sets     1 1 −1 =X h ∈ H : |mh (λ0 )| ≥ α + σ j = k ∈ B : |mXk (λ0 )| ≥ α + j j are closed by Lemma 1.3. We then proceed as in the proof of Lemma 1.4 to show that each σ j has empty interior. It follows that the set {k ∈ B : |mXk (λ0 )| = α } is a dense Gδ in B. Then the argument of Theorem 1.5 shows that the set M = {k ∈ B : |mXk (λ )| = inf |mXh (λ )|, λ ∈ D} h∈B

is a dense Gδ . For k ∈ M it follows that mXk is a multiple of mXh for all h ∈ B, and hence mXk (T )(X B) = {0}. This last relation implies   W n mXk (T ) T XB = {0}, n≥0

and hence mXk (T ) = O. Therefore, for such k we have mXk = mT . The theorem follows.

2 Jordan blocks 1. As usual, we denote by S the unilateral shift of multiplicity one acting on H 2 .

2. J ORDAN BLOCKS

401

Definition. For each inner function ϕ ∈ H ∞ , the Jordan block S(ϕ ) is the operator defined on H(ϕ ) = H 2 ⊖ ϕ H 2 by S(ϕ ) = PH(ϕ ) S|H(ϕ ) or, equivalently, S(ϕ )∗ = S∗ |H(ϕ ). We have already seen in Proposition III.4.4 that the Jordan block S(ϕ ) is an operator of class C0 with minimal function ϕ . These operators can be viewed as the basic building blocks of arbitrary operators of class C0 , and it is worthwhile to study their properties in more detail.

Lemma 2.1. If ϕ is a nonconstant inner function, then S is the minimal isometric dilation of S(ϕ ). Proof. This follows immediately from Theorem VI.3.1. Indeed, if ϕ is not constant, then {E 1 , E 1 , ϕ (λ )} is a purely contractive analytic function. Corollary 2.2. We have dS(ϕ ) = dS(ϕ )∗ = 1 for every nonconstant inner function ϕ . Proof. By Theorem VI.3.1, {E 1 , E 1 , ϕ } coincides with the characteristic function of S(ϕ ). The corollary follows at once. Proposition 2.3. Let T be a contraction of class C· 0 on H such that dT ∗ = 1. Then one of the following mutually exlusive possibilities holds. (1) T is unitarily equivalent to S. (2) T is unitarily equivalent to S(ϕ ) for some nonconstant inner function ϕ . Proof. The minimal isometric dilation of T is a unilateral shift of multiplicity one; in other words, it is unitarily equivalent to S. Thus we may assume that H ⊂ H 2 is invariant for S∗ , and T ∗ = S∗ |H. The proposition clearly follows from the classification of invariant subspaces of S. Indeed, either H = H 2 , or H 2 ⊖ H = ϕ H 2 for some inner function ϕ . In this last case, H = H(ϕ ) and T = S(ϕ ). The function ϕ cannot be constant because dim(H) ≥ dim(DT ∗ ) = 1. Recall that the adjoint of a function ϕ is defined by ϕ ˜(λ ) = ϕ (λ¯ ), λ ∈ D.

Corollary 2.4. Let ϕ be an inner function in H ∞ . The adjoint S(ϕ )∗ is unitarily equivalent to S(ϕ ˜). Proof. As noted above, {E 1 , E 1 , ϕ (λ )} coincides with the characteristic function of S(ϕ ). Therefore the characteristic function of S(ϕ )∗ coincides with {E 1 , E 1 , ϕ ˜(λ )}, and this yields the desired unitary equivalence by virtue of Proposition VI.1.1. 2. We study next the invariant subspaces and maximal vectors of S(ϕ ). It is convenient to denote by ran X the range of an operator X . Proposition 2.5. Let ϕ be a nonconstant inner function. (1) For every h ∈ H(ϕ ) we have mh = ϕ /(h ∧ ϕ ).

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(2) Every invariant subspace of S(ϕ ) has the form ψ H 2 ⊖ ϕ H 2 for some inner divisor ψ of ϕ . We have

ψ H 2 ⊖ ϕ H 2 = ker(ϕ /ψ )(S(ϕ )) = ran ψ (S(ϕ )) for each inner divisor ψ of ϕ . (3) If M = ψ H 2 ⊖ ϕ H 2 is invariant for S(ϕ ) then S(ϕ )|M is unitarily equivalent to S(ϕ /ψ ), and the compression of S(ϕ ) to H(ϕ ) ⊖ M = H(ψ ) is precisely S(ψ ). (4) A vector h ∈ H(ϕ ) is cyclic for S(ϕ ) if and only if ϕ ∧ h = 1, that is, if and only if h is maximal. The set of cyclic vectors for S(ϕ ) is a dense Gδ in H(ϕ ). Proof. (1) Set u = mh and v = ϕ /(h ∧ ϕ ). We have

 v(S(ϕ ))h = PH(ϕ ) v(S)h = PH(ϕ ) (vh) = PH(ϕ ) ϕ

 h = 0, h∧ϕ

and consequently u|v. Conversely, we know that u(S(ϕ ))h = 0, so that uh = ϕ g for some g ∈ H 2 . Because u divides ϕ , it follows that h = (ϕ /u)g, and hence (ϕ /u)|h. Obviously (ϕ /u)|ϕ , thus (ϕ /u)|(h ∧ ϕ ) or, equivalently, v|u. We deduce that v = u, as desired. (2) The description of the invariant subspaces of S(ϕ ) is part b) of Proposition III.4.3. Let ψ be an inner divisor of ϕ . We have (ϕ /ψ )(S(ϕ ))h = 0 if and only if mh |(ϕ /ψ ) or, equivalently by (1), ψ |h. Because {h ∈ H(ϕ ) : ψ |h} = ψ H 2 ⊖ ϕ H 2 , we proved that ker(ϕ /ψ )(S(ϕ )) = ψ H 2 ⊖ ϕ H 2 . For the second equality we note that

ψ (S(ϕ ))H(ϕ ) = PH(ϕ ) ψ (S)H(ϕ ) = PH(ϕ ) ψ (S)H 2 = PH(ϕ ) ψ H 2 = ψ H 2 ⊖ ϕ H 2 . Part (3) follows from Theorem VI.1.1. (4) If h is cyclic, we must have mh = mS(ϕ ) , so that h ∧ ϕ = 1 by (1). Conversely, if h ∧ ϕ = 1, (2) shows that h does not belong to any proper invariant subspace of S(ϕ ), and hence h is a cyclic vector. The last statement follows from Theorem 1.5. The proposition is proved. Corollary 2.6. Every invariant subspace of S(ϕ ) is hyperinvariant. Proof. This follows from the equality

ψ H 2 ⊖ ϕ H 2 = ran (ψ (S(ϕ ))) if ψ is an inner divisor of ϕ .

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Corollary 2.7. Assume that ϕ , u ∈ H ∞ , and ϕ is inner. Then keru(S(ϕ )) = (ϕ /(u ∧ ϕ ))H 2 ⊖ ϕ H 2, (ran u(S(ϕ )))− = (u ∧ ϕ )H 2 ⊖ ϕ H 2 , S(ϕ )| keru(S(ϕ )) is unitarily equivalent to S(u ∧ ϕ ), and S(ϕ )|(ran u(S(ϕ )))− is unitarily equivalent to S(ϕ /(u ∧ ϕ )). Proof. Observe that u(S(ϕ ))h = 0 if and only if mh |u. Because mh always divides ϕ , we see that mh |u if and only if mh |(u ∧ ϕ ). In other words, we have ker u(S(ϕ )) = ker(u ∧ ϕ )(S(ϕ )). Now Proposition 2.5(2) proves the first equality in the statement. Analogously, H(ϕ ) ⊖ (ran u(S(ϕ )))− = ker u˜(S(ϕ )∗ )

= ker(u ∧ ϕ )˜(S(ϕ )∗ ) = H(ϕ ) ⊖ (ran (u ∧ ϕ )(S(ϕ ))),

so that

(ran u(S(ϕ )))− = ran (u ∧ ϕ )(S(ϕ )) = (u ∧ ϕ )H 2 ⊖ ϕ H 2 .

The last two assertions of the corollary follow from Part (3) of Proposition 2.5. Corollary 2.8. The set of cyclic vectors for S(ϕ ) is a dense Gδ in H(ϕ ). Proof. Proposition 2.5 implies that h ∈ H(ϕ ) is cyclic if and only if mh = ϕ . The corollary follows now from Theorem 1.5. 3. Quite interestingly, Theorem 1.6 has the following consequence of intrinsic interest for the arithmetic of Hardy spaces. This is used in Sec. 6. We denote by ℓ1 the Banach space of absolutely summable sequences of complex scalars. Theorem 2.9. Let { f j : j ≥ 0} be a bounded sequence of functions in H 2 ,and let ϕ be an inner function. The set of those sequences α = {α j } in ℓ1 satisfying the relation ! ! ∞

∑ αj fj

j=0

is a dense Gδ in ℓ1 .

∧ϕ =

∞ V

j=0

fj

∧ϕ

Proof. We may assume without loss of generality that ! ∞ V

j=0

f j ∧ ϕ = 1.


V Indeed, we can replace f j by f j /ψ and ϕ by ϕ /ψ , with ψ = ∞j=0 f j ∧ ϕ . Under this additional assumption, the invariant subspace for S(ϕ ) generated by the vectors {PH(ϕ ) f j : j ≥ 0} is H(ϕ ). Indeed, if this invariant subspace is ψ H 2 ⊖ ϕ H 2 , it fol-

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V lows that ψ | ∞j=0 f j ∧ ϕ , and hence ψ = 1. We can then apply Theorem 1.6 to the space B = ℓ1 and the linear operator X : B → H(ϕ ) defined by ! X α = PH(ϕ )

∞

∑ αj fj

j=0

(α = {α j } ∈ ℓ1 ).

We deduce that the set of those sequences α ∈ ℓ1 for which mX α = ϕ is a dense Gδ in ℓ1 . Finally, the  condition mX α = ϕ is equivalent to (X α ) ∧ ϕ = 1, and this is in turn equivalent to ∑∞j=0 α j f j ∧ ϕ = 1.

4. We conclude this section with a few facts about operators that intertwine Jordan blocks.

Theorem 2.10. Let ϕ be an inner function. For every operator X ∈ {S(ϕ )}′ there exists a function u ∈ H ∞ such that X = u(S(ϕ )) and kuk = kX k.

Proof. We may assume that ϕ is not constant, in which case S is the minimal isometric dilation of S(ϕ ). Given X ∈ {S(ϕ )}′ , Theorem II.2.3 implies the existence of an operator Y ∈ {S}′ such that kY k = kX k and X = PH(ϕ )Y |H(ϕ ). Apply now Lemma V.3.2 to deduce that Y = u(S) for some u ∈ H ∞ . This function satisfies the conclusion of the theorem. A more general form of Theorem 2.10 is useful in applications. Theorem 2.11. Let ϕ , ϕ ′ be inner functions, and let X : H(ϕ ) → H(ϕ ′ ) satisfy the intertwining relation XS(ϕ ) = S(ϕ ′ )X . There exists a function u ∈ H ∞ such that ϕ ′ |uϕ , kuk = kXk, and X = PH(ϕ ′ ) u(S)|H(ϕ ). Conversely, if u ∈ H ∞ is such that ϕ ′ |uϕ , then the above formula defines an operator such that XS(ϕ ) = S(ϕ ′ )X, and X = O if and only if ϕ ′ |u.

Proof. As in the preceding argument, we may assume that ϕ and ϕ ′ are not constant, and then the commutant lifting theorem yields Y ∈ {S}′ such that Y (ϕ H 2 ) ⊂ ϕ ′ H 2 and X = PH(ϕ ′ )Y |H(ϕ ). If we write Y = u(S), we see that the above inclusion is equivalent to ϕ ′ |uϕ . The remaining assertions are easily verified.

3 Quasi-affine transforms and multiplicity 1. Let T be an operator on the complex Hilbert space H. Definition. The cyclic multiplicity µT of T is the smallest cardinality of a subset M ⊂ H with the property that the set {T n h : h ∈ M, n ≥ 0} generates H. The operator T is said to be multiplicity-free if µT = 1. Note that T is multiplicity-free if and only if it has a cyclic vector. Proposition 3.1. Let V be a unilateral shift with wandering space F. We have µV = dim(F).

3. Q UASI - AFFINE TRANSFORMS AND MULTIPLICITY

405

W

n Proof. If M is an orthonormal basis in F, then ∞ n=0 V M = H, and therefore µV ≤ dim(F). To prove the opposite inequality, let M be an arbitrary set such that µV = W n card(M) and ∞ n=0 V M = H. Then     ∞ ∞ W W F = H ⊖VH = V nM ⊖ V nM , n=0

n=1

and it follows that F is spanned as a closed space by the set PF M. Consequently, dim(F) ≤ card(PF M) ≤ card(M) = µV .

Lemma 3.2. Let T and T ′ act on H and H′ , respectively, and let X : H′ → H be a bounded linear transformation such that X T ′ = T X . If X has dense range then µT ≤ µT ′ . Proof. Choose M ′ ⊂ H′ with card(M ′ ) = µT ′ and range then ∞ W

T n (XM ′ ) =

n=0

and therefore

∞ W

W∞

n=0 T

′n M ′

= H′ . If X has dense

X T ′n M ′ = (XH′ )− = H,

n=0

µT ≤ card(X M ′ ) ≤ card(M ′ ) = µT ′ .

Corollary 3.3. If T is a contraction of class C· 0 then µT ≤ dT ∗ .

Proof. Let T act on H, and let U+ on K+ be the minimal isometric dilation of T . By Proposition II.3.1, we have R = {0}, and hence U+ is a unilateral shift of multiplicity dT ∗ . Because PH K+ = H and T PH = PHU+ , we deduce

µT ≤ µU+ = dT ∗ from the preceding results. 2. Contractions of class C·0 with small defect dT ∗ , particularly with dT ∗ = 1, are relatively easy to understand (see Proposition 2.3). It is natural to reduce problems related to a contraction of class C· 0 to operators T with a small defect index dT ∗ . This is achieved in the following two results. Lemma 3.4. Let T be an operator of class C·0 on H. There exist a unilateral shift U on H1 , and a bounded linear transformation X : H1 → H such that X has dense range, XU = T X, and µU = µT . Proof. Let U+ on K+ be the minimal isometric dilation of T ; asWnoted above, U+ is n a unilateral shift. Fix a set M ⊂ H with card(M) = µT such that ∞ n=0 T M = H. We define the space H1 and the operators U, X as follows: H1 =

∞ W

U+n M,

n=0

U = U+ |H1 ,

X = PH |H1 .

The relation T X = XU follows because T PH = PHU+ . Next we see that (X H1 )− =

∞ W

XU+n M =

n=0

∞ W

T nX M =

n=0

∞ W

T n M = H,

n=0

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and therefore X has dense range. Finally, U is a unilateral shift,

µU ≤ card(M) = µT , and the opposite inequality µU ≥ µT follows from Lemma 3.2. Recall that T1 ≺ T2 indicates that T1 is a quasi-affine transform of T2 , and T1 ∼ T2 indicates that the T1 and T2 are quasi-similar. Theorem 3.5. For every contraction T of class C· 0 there exists a contraction T ′ of class C· 0 such that T ′ ≺ T, and

µT ′ = dT ′∗ = µT . Proof. Let H, H1 ,U, and X be as in the preceding lemma, and set H′ = H1 ⊖ kerX,

Y = X|H′ ,

T ′ = PH′ U|H′ .

Because T X = XU, ker X is an invariant subspace for U, and therefore T ′ is of class C· 0 as T ′∗ = U ∗ |H′ . Clearly Y is one-to-one, and Y H′ = X H1 , so that Y has dense range and is therefore a quasi-affinity. For every vector x′ ∈ H′ we have TY x′ = T X x′ = XUx′ = X (Ux′ − PkerX Ux′ ) = X PH′ Ux′ = X T ′ x′ = Y T ′ x′ . Thus TY = Y T ′ , and this proves the relation T ′ ≺ T . The inequalities

µT ≤ µT ′ ≤ dT ′∗ are obvious from Lemma 3.2 and its corollary. Finally, the wandering space F of U has dimension µT by Lemma 3.4, and IH′ − T ′ T ′∗ = IH′ − T ′U ∗ |H′

= PH′ (I − UU ∗)|H′ = PH′ PF |H′ .

We conclude that dT ′∗ = rank(I − T ′ T ′∗ ) ≤ rank(PF ) = dim(F) = µT , and this completes the proof of the theorem.

4 Multiplicity-free operators and splitting 1. The adjoint of a multiplicity-free operator is not generally multiplicity-free; for example, the adjoint of a unilateral shift of countably infinite multiplicity has a

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407

cyclic vector. An easy way to see this is to show that U ∗ ≺ (S∗ )(ℵ0 ) , where U is the bilateral shift S on L2 and S is the unilateral shift on H 2 . For this purpose, consider a 2(ℵ0 ) partition C = ∞ → n=1 αn into Borel sets of positive measure, and define X : H ∞ ∞ ∈ H 2(ℵ0 ) . Clearly X is a quasi-affinity L2 by X({ fn }∞ f χ for { f } ) = ∑n=1 n αn n n=1 n=1 in I (S(ℵ0 ) ,U) so that S(ℵ0 ) ≺ U, or equivalently U ∗ ≺ (S∗ )(ℵ0 ) , as claimed. Thus µ(S∗ )(ℵ0 ) ≤ µU ∗ = 1 by Lemma 3.2 and the remark preceding Lemma IX.2.3. We show that for operators T of class C0 we have in fact µT = 1 if and only if µT ∗ = 1. First we prove an auxiliary result. Proposition 4.1. Let T be an operator of class C0 . If T is multiplicity-free, then S(mT ) ≺ T . If T ∗ is multiplicity-free, then T ≺ S(mT ). Proof. Assume first that µT = 1. It follows from Theorem 3.5 that there exists an operator T ′ of class C· 0 such that T ′ ≺ T and dT ′∗ = µT = 1. The operator T ′ is of class C0 by Proposition III.4.6, and therefore it cannot be unitarily equivalent to S. Then Proposition 2.3 shows that T ′ is unitarily equivalent to S(ϕ ) for some inner function ϕ ∈ H ∞ . Thus we have S(ϕ ) ≺ T , and because ϕ = mS(ϕ ) = mT , we conclude that S(mT ) ≺ T , as desired. If µT ∗ = 1, the preceding proof shows that S(mT˜ ) = S(mT ∗ ) ≺ T ∗ , and hence T ≺ S(mT˜ )∗ . The proposition follows thus from Corollary 2.4. Theorem 4.2. Let T be an operator of class C0 . The following conditions are equivalent. (1) T is multiplicity-free. (2) T ∗ is multiplicity-free. (3) T is quasi-similar to S(mT ). Proof. It suffices to prove that (2) implies (1). Indeed, it follows then by symmetry that (1) implies (2). Furthermore, if (1) and (2) are satisfied, then T ∼ S(mT ) by the preceding proposition. Conversely, if S(mT ) ≺ T , then µT ≤ µS(mT ) = 1, and (1) follows. Assume therefore that T acts on H and T ∗ is multiplicity-free. By Proposition 4.1 we can choose a quasi-affinity X such that X T = S(mT )X . Theorem 1.5 allows us to W choose a T -maximal vector h ∈ H. Denote by K the cyclic space {T n h : n ≥ 0} generated by h. Thus we have T K ⊂ K and mT |K = mT . The operator T |K is multiplicityfree, therefore a second application of Proposition 4.1 yields an injective operator Y : H(mT ) → H such that Y H(mT ) is dense in K and Y S(mT ) = TY . We have then XY S(mT ) = X TY = S(mT )XY, so that XY ∈ {S(mT )}′ and, of course, XY is injective. By Theorem 2.10, we have XY = u(S(mT )) for some u ∈ H ∞ , and u ∧ mT = 1 because ker u(S(mT )) = {0}

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(cf. Corollary 2.7). Observe now that X(Y X − u(T )) = XY X − X u(T ) = XY X − u(S(mT ))X

= (XY − u(S(mT )))X = O,

and hence Y X = u(T ) because X is injective. Now, the relation u ∧ mT = 1 implies via Proposition III.4.7 that u(T ) is a quasi-affinity. In particular, H = (u(T )H)− ⊂ (Y H(mT ))− ⊂ K, so that K = H and h is a cyclic vector for T . The theorem is proved. The preceding argument also yields the following result. Corollary 4.3. Let T be a multiplicity-free operator of class C0 acting on H. A vector h ∈ H is cyclic for T if and only if it is T -maximal. The set of cyclic vectors for T is a dense Gδ in H. Corollary 4.4. Every restriction of a multiplicity-free operator of class C0 to an invariant subspace is multiplicity-free. Proof. Let T be a multiplicity-free operator of class C0 , and let K be an invariant subspace for T . If h is cyclic for T ∗ then PK h is cyclic for (T |K)∗ . Thus (T |K)∗ is multiplicity-free, and therefore so is T |K. 2. Some of the results concerning operators intertwining Jordan blocks can be transferred to general multiplicity-free operators of class C0 . Proposition 4.5. Let T and T ′ be two multiplicity-free operators of class C0 , and let A satisfy the equation AT = T ′ A. If mT = mT ′ then A is one-to-one if and only if it has dense range. Proof. Set ϕ = mT = mT ′ , so that T and T ′ are quasi-similar to S(ϕ ) by Theorem 4.2. Choose quasi-affinities X ,Y satisfying X S(ϕ ) = T ′ X and Y T = T S(ϕ ). The product XAY is easily seen to commute with S(ϕ ), and hence X AY = u(S(ϕ )) for some u ∈ H ∞ by Theorem 2.10. If A is either one-to-one or has dense range, then XAY has the same property, and hence u ∧ ϕ = 1 in either case (cf. Corollary 2.7). Next we note that X(AY X − u(T ′ )) = X AY X − X u(T )

= X AY X − u(S(ϕ ))X

= (X AY − u(S(ϕ )))X = O,

and hence u(T ′ ) = AY X because X is one-to-one. If u ∧ ϕ = 1, it follows that u(T ′ ) is a quasi-affinity, and consequently A has dense range. Analogously, one can show that Y XA = u(T ), and hence A is one-to-one if u ∧ ϕ = 1. The proposition follows easily from these observations.

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Proposition 4.6. Let T be a multiplicity-free operator of class C0 . There exists a function v ∈ KT∞ such that every operator A ∈ {T }′ can be written as A = (u/v)(T ) for some u ∈ H ∞ .

Proof. Let ϕ , X, and Y be as in the proof of the preceding proposition, with T ′ = T . If A = I, that proof implies the existence of v ∈ H ∞ such that v ∧ ϕ = 1 and Y X = v(T ); note that v ∈ KT∞ by Proposition III.4.7. Now, if A is arbitrary in {T }′ , we deduce the existence of u ∈ H ∞ such that Y X A = u(T ), so that v(T )A = u(T ). This means precisely that A = (u/v)(T ), as desired. 3. We prove next a result about invariant subspaces that justifies in particular the terminology “multiplicity-free”.

Theorem 4.7. Let T be an operator of class C0 . The following assertions are equivalent. (1) T is multiplicity-free. (2) For every inner divisor ϕ of mT , there exists a unique invariant subspace K for T satisfying the relation mT |K = ϕ . (3) If K and K′ are invariant for T , and T |K ≺ T |K′ , then K = K′ . (4) There are no proper invariant subspaces K for T such that mT |K = mT .

Moreover, if T is multiplicity-free, the unique invariant subspace considered in (2) is given by K = ker ϕ (T ) = [ran (mT /ϕ )(T )]− . Proof. Assume that T acts on H, it is multiplicity-free, K is invariant for T , and ϕ = mT |K . The operators T ′ = T |K and T ′′ = T | ker ϕ (T ) are multiplicity-free by Corollary 4.4, and they satisfy the relation JT ′ = T ′′ J, where J : K → ker ϕ (T ) is the inclusion operator. Because T ′ and T ′′ both have minimal function ϕ , Proposition 4.5 implies that J must have dense range, and therefore K = J ker ϕ (T ) = ker ϕ (T ). Thus (1) implies (2). It is obvious that (2) implies (4). Assume next that (4) holds, W and h is a T -maximal vector. If we define K = {T n h : n ≥ 0}, we have mT |K = mT , and hence K = H by (4). Thus h is a cyclic vector, and we conclude that (4) implies (1). It remains to show that (3) is equivalent to the other three conditions. The fact that (2) implies (3) is obvious because T |K ≺ T |K′ implies, in particular, the equality mT |K = mT |K′ . Conversely, assume that (3) holds and h, h′ are T -maximal vectors. Denote by K, K′ the cyclic spaces generated by h, h′ , respectively, and note that T |K ∼ T |K′ by Theorem 4.2 and the transitivity of of quasi-similarity. In particular T |K ≺ T |K′ , so that K = K′ and therefore h′ ∈ K. We conclude that K contains all maximal vectors, so that K = H because the set of maximal vectors is dense. Thus (3) implies (1). The last assertion of the theorem follows because both T |[ran (mT /ϕ )(T )]− and T | ker ϕ (T ) have minimal function ϕ . The last assertion of the preceding theorem yields the following result. Corollary 4.8. Every invariant subspace of a multiplicity-free operator of class C0 is hyperinvariant.

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We can now complete the characterization of unicellular operators of class C0 . Corollary 4.9. A contraction T of class C0 is unicellular if and only if it is multiplicity-free and its spectrum consists of a single point. Proof. Assume first that T is unicellular. We already know from the corollary to Proposition III.7.3 that σ (T ) is a singleton. If T is not cyclic, Theorem 4.7.(4) implies the existence of a maximal vector h such that Mh is not the whole space. Then Theorem 1.5 implies the existence of a maximal vector k 6∈ Mh . Neither of the spaces Mh and Mk is contained in the other, contradicting unicellularity. Thus T must be multiplicity-free. Conversely, assume that T is multiplicity-free, and σ (T ) is a singleton. In this case we have T ∼ S(mT ), and Theorem 4.7.(2) shows that T is unicellular if and only if the divisors of mT are totally ordered by divisibility. Thus mT is either a Blaschke product with a single zero, or a singular inner function determined by a measure supported by a single point. By Theorem III.5.1, this happens precisely when σ (T ) is a singleton. The corollary follows. 4. We now show how multiplicity-free operators can be used in the study of operators with larger multiplicity. Theorem 4.10. Let T be an operator of class C0 on H, h ∈ H a T -maximal vector, W and K = {T n h : n ≥ 0}. There exists an invariant subspace M for T such that K ∨ M = H and K ∩ M = {0}. Proof. The operator T1 = T |K is multiplicity-free, and by Theorem 4.2 there exists a vector k ∈ K cyclic for T1∗ . We now set K′ =

∞ W

T ∗n k,

n=0

M = H ⊖ K′ ,

and define T2 on K′ by T2∗ = T ∗ |K′ . Because K′ is invariant for T ∗ , we have PK′ T = T2 PK′ , and therefore the operator X : K → K′ defined by X = PK′ |K satisfies the intertwining relation XT1 = T2 X . Observe that K ∩ M = ker X, and H ⊖ (K ∨ M) = (H ⊖ K) ∩ K′ = K′ ∩ ker PK = ker X ∗ . To conclude the proof, it suffices to show that X is a quasi-affinity. To do this we first note that (ran X ∗ )− =

∞ W

X ∗ T2∗n k =

n=0

∞ W

T1∗n X ∗ k =

n=0

and thus X ∗ has dense range. If ϕ = mT2 , we have

∞ W

T1∗n k = K,

n=0

(ϕ (T1 ))∗ X ∗ = X ∗ (ϕ (T2 ))∗ = O, so that (ϕ (T1 ))∗ vanishes on a dense set. We conclude that mT = mT1 must divide ϕ , so that in fact ϕ = mT . The fact that X is a quasi-affinity follows now from Proposition 4.5 because X ∗ has dense range, and the operators T1 , T2 are multiplicity-free and have the same minimal function.

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Corollary 4.11. Let T be an operator of class C0 . There exist operators T ′ and T ′′ of class C0 such that S(mT ) ⊕ T ′ ≺ T ≺ S(mT ) ⊕ T ′′ . Proof. Let K and M be as in Theorem 4.10. Then we have (T |K) ⊕ (T |M) ≺ T with the intertwining quasi-affinity X : K ⊕ M → H defined by X (u ⊕ v) = u + v. Thus S(mT ) ⊕ (T |M) ≺ T because S(mT ) ∼ T |K. It follows that T ′ = T |M satisfies the required relation. The existence of T ′′ is deduced similarly replacing T by T ∗ .

Theorem 4.10 can be used to prove a converse to Proposition 4.6, hence yet another characterization of multiplicity-free operators. Theorem 4.12. The following assertions are equivalent for an operator of class C0 . (1) T is multiplicity-free. (2) {T }′ is commutative. (3) {T }′ consists of the bounded operators of the form f (T ) with f ∈ NT . Proof. We already know from Proposition 4.6 that (1) implies (3), and (3) trivially implies (2). It remains to show that the commutant of T is not commutative if µT ≥ 2. Let K, M, and H be as in Theorem 4.10; if µT ≥ 2 we must have K 6= H, and hence M 6= {0}. Define now K′ , T1 and T2 by K′ = H ⊖ M,

T1 = T |K,

T2∗ = T ∗ |K′ .

The operator X = PK′ |K is a quasi-affinity, and X T1 = T2 X. Both T1 and T2 are multiplicity-free, thus T1 and T2 are quasi-similar; indeed, both are quasi-similar to S(mT ). Let Y be a quasi-affinity satisfying Y T2 = T1Y , and define the operator A ∈ {T }′ by A = Y PK′ . We clearly have ker A = ker PK′ = M, and (AH)− = K. Assume that we can find a nonzero operator Z : K′ → M such that ZT2 = (T |M)Z. Then the operator B ∈ {T }′ defined by B = ZPK′ is such that AB = O and (BAH)− = (ZPK′ Y K′ )− = (ZPK′ K)− = (ZK′ )− 6= {0},

so that A and B do not commute. Thus, to conclude the proof, it suffices to produce such an operator Z. Because M 6= {0}, T |M has a nonzero cyclic subspace M1 , and it would suffice to find a nonzero operator Z : K′ → M1 such that ZT2 = (T |M)Z. Set now ϕ = mT = mT2 , and ϕ ′ = mT |M1 , so that T |M1 ∼ S(ϕ ′ ) and T2 ∼ S(ϕ ). The operator R = PH(ϕ ′ ) |H(ϕ ) is not zero, and RS(ϕ ) = S(ϕ ′ )R. The desired operator Z can now be constructed by composing R with the appropriate quasi-affinities.

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5 Jordan models 1. We have seen in the preceding section that multiplicity-free operators of class C0 are uniquely determined, up to quasi-similarity, by their minimal functions. We have thus a complete classification of multiplicity-free operators of class C0 up to quasi-similarity. In this section we extend this classification to all operators of class C0 acting on a separable space. Definition. Let Φ = {ϕ j : j ≥ 0} ⊂ H ∞ be a sequence of inner functions such that ϕ j+1 |ϕ j for all j ≥ 0. The operator S(Φ ) =

∞ L

S(ϕ j )

j=0

is called a Jordan operator. Note that some of the functions ϕ j in the above definition may be constant. If L this happens, the Jordan operator S(Φ ) is unitarily equivalent to k−1 j=0 S(ϕ j ), where k is the first integer such that ϕk is a constant function. Clearly, a Jordan operator is of class C0 , and mS(Φ ) = ϕ0 . Proposition 5.1. For every operator T of class C0 acting on a separable Hilbert space, there exists a Jordan operator S(Φ ) such that S(Φ ) ≺ T . Proof. Assume that T acts on the separable space H, choose a dense sequence {hn : n ≥ 0} in H, and let {kn : n ≥ 0} be a sequence in which each hi is repeated infinitely many times. We inductively construct vectors f0 , f1 , f2 , . . . in H, and invariant subspaces M−1 , M0 , M1 , . . . for T with the following properties: (1) (2) (3) (4)

M−1 = H; f j ∈ M j−1 , m f j = mT |M j−1 ; W K j ∨ M j = M j−1 , K j ∩ M j = {0}, where K j = {T n f j : j ≥ 0}; kk j − PK0∨K1 ∨···∨K j k j k ≤ 2− j

for j = 0, 1, 2 . . . . Assume, indeed that f j and M j have already been defined for j < n, and let us construct fn and Mn . (Note that if n = 0, only M−1 needs to be constructed, and there is no f−1 .) A repeated application of (3) yields H = M−1 = K0 ∨ M0 = K0 ∨ K1 ∨ M1 = · · · = K0 ∨ K1 ∨ · · · ∨ Kn−1 ∨ Mn−1 , so that we can find vectors un ∈ K0 ∨ K1 ∨ · · · ∨ Kn−1 and vn ∈ Mn−1 such that kkn − un − vn k ≤ 2−n−1. By Theorem 1.5, we can then find a vector fn ∈ Mn−1 such that m fn = mT |Mn−1 (in other words, fn is a T |Mn−1 -maximal vector), and kvn − fn k ≤ 2−n−1 .

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An application of Theorem 4.10 to the operator T |Mn−1 proves the existence of an invariant subspace Mn satisfying (3) for j = n. It remains to verify (4) for j = n, and this follows because kkn − PK0 ∨K1 ∨···∨Kn kn k ≤ kkn − un − fn k

≤ kkn − un − vn k + kvn − fn k ≤ 2−n .

Thus the existence of the vectors f j and of the spaces M j is established by induction. A useful consequence of (4) is that H=

∞ W

Kj.

j=0

Indeed, this follows from the equality lim dist kn ,

n→∞

∞ W

Kj

j=0

!

=0

and the fact that each hi is repeated infinitely many times among the kn , so that W hi ∈ ∞j=0 K j for all i. We define now Φ = {ϕ j : j ≥ 0} by setting ϕ j = m f j . Relation (4), and the fact that M j+1 ⊂ M j , easily imply that ϕ j+1 |ϕ j for all j, and hence S(Φ ) is a Jordan operator. We now prove that S(Φ ) ≺ T . The operator T |K j is multiplicityfree with minimal function ϕ j . Therefore Proposition 4.1 implies the existence of a quasi-affinity X j such that XS(ϕ j ) = (T |K j )X j . We can then define an operator X satisfying XS(Φ ) = T X by the formula ! ∞ ∞ ∞ ∞ 2− j L L L X j g j for X gj ∈ H(ϕ j ). gj = ∑ kX k j j=0 j=0 j=0 j=1 The reader will verify without difficulty that X is bounded. The range of X j is dense in K j , and the spaces K j span H, thus X has dense range. To prove that X is one-toone, suppose that g=

∞ L

g j ∈ ker X, g 6= 0,

j=0

and n is the first integer such that gn 6= 0. By the definition of X , we have ∞ 2− j Xn gn = − ∑ Xn+ j gn+ j . kXn k j=1 kXn+ j k

W

Thus Xn gn , a nonzero element of Kn , belongs to ∞j=1 Kn+ j ⊂ Mn . By (3), we must have Xn gn = 0, and this contradiction implies that X is one-to-one. We thus determined a quasi-affinity X such that T X = X S(Φ ), and this concludes the proof. Corollary 5.2. For every operator of class C0 acting on a separable Hilbert space, there exists a Jordan operator S(Φ ′ ) such that T ≺ S(Φ ′ ).

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Proof. Proposition 5.1, applied to T ∗ , shows the existence of a Jordan operator S(Ψ ), Ψ = {ψ j : j ≥ 0}, such that S(Ψ ) ≺ T ∗ . We have then T ≺ S(Ψ )∗ , and S(Ψ )∗ is unitarily equivalent to the Jordan operator S(Φ ′ ), where Φ ′ = {ψ ˜j : j ≥ 0}. 2. In order to complete the classification theorem, we prove that the operators S(Φ ) and S(Φ ′ ) constructed above are necessarily identical. If T is an operator acting on H, and n is a natural number, we denote by T (n) the orthogonal sum of n copies of T acting on the orthogonal sum H(n) of n copies of H. Lemma 5.3. Let n and k be natural numbers, and ϕ a nonconstant inner function. If there exists an injective operator X : H(ϕ )(k) → H(ϕ )(n) such that X S(ϕ )(k) = S(ϕ )(n) X, then k ≤ n. Proof. The operator X is represented by a matrix [Xi j ]1≤i≤n,1≤ j≤k in the sense that ! ! X

k L

j=1

hj

=

n L

i=1

k

∑ Xi j h j

j=1

for

k L

h j ∈ H(ϕ )(k) .

j=1

The condition XS(ϕ )(k) = S(ϕ )(n) X implies that the operators Xi j commute with S(ϕ ). By Theorem 2.10, we have Xi j h = PH(ϕ ) (ai j h),

h ∈ H(ϕ ),

where ai j ∈ H ∞ for all i, j. Now, the operator X is one-to-one and, in particular, it is not zero. Therefore ϕ cannot divide all the functions ai j . There exists then a minor of maximal rank of the matrix [ai j ]i, j that is not divisible by ϕ , and there is no loss of generality in assuming that this minor is |ai j |1≤i, j≤r , with r ≤ min{k, n}. Assuming now that k > n, consider the determinant   a11 a12 · · · a1r a1,r+1  a21 a22 · · · a2r a2,r+1     ..  = r+1x u . det  ... ... . . . ... ∑ j j .     ar1 ar2 · · · arr ar,r+1  j=1 x1 x2 · · · xr xr+1

The sum ∑r+1 j=1 ai j u j is zero if 1 ≤ i ≤ r, and it equals a minor of order r + 1 if i > r; therefore all of these sums are divisible by ϕ . We deduce that the vector L h = kj=1 h j ∈ H(ϕ )(k) defined by hj =

(

PH(ϕ ) u j 0

for 1 ≤ j ≤ r + 1 for j > r + 1,

satisfies the relation Xh = 0. The injectivity of X implies that h = 0. In particular, PH(ϕ ) ur+1 = 0, or ur+1 ∈ ϕ H 2 . However, the function ur+1 = det[ai j ]1≤i, j≤r was chosen not to be divisible by ϕ . This contradiction shows that necessarily k ≤ n, thus concluding the proof.

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Corollary 5.4. Let n and k be natural numbers, and ϕ a nonconstant inner function. If there exists an operator X : H(ϕ )(k) → H(ϕ )(n) with dense range such that XS(ϕ )(k) = S(ϕ )(n) X, then k ≥ n. Proof. The operator X ∗ is one-to-one, and X ∗ S(ϕ )∗(n) = S(ϕ )∗(k) X ∗ . Because S(ϕ )∗ is unitarily equivalent to S(ϕ ˜), the corollary follows immediately from Lemma 5.3. We recall that, given an integer N ≥ 1, an operator T of class C0 belongs to C0 (N) if dT ∗ = N. Lemma 5.5. Let T be an operator of class C0 (N) on H with minimal function ϕ . There exists a surjective operator X : H(ϕ )(N) → H such that X S(ϕ )(N) = T X . Proof. The minimal isometric dilation of T is a unilateral shift of multiplicity N. We may assume without loss of generality that H ⊂ (H 2 )(N) and T ∗ = S∗(N) |H. We have O = ϕ ˜(T ∗ ) = ϕ ˜(S∗(N) )|H, so that

H ⊂ ker ϕ ˜(S∗(N) ) = H(ϕ )(N) .

We simply define then X = PH |H(ϕ )(N) .

nonconstant comProposition 5.6. Let ϕ0 , ϕ1 , . . . , ϕn−1 be inner functions with aL mon inner divisor ϕ . The cyclic multiplicity of the operator T = n−1 j=0 S(ϕ j ) equals n. Proof. Each S(ϕ j ) has a cyclic vector, and hence the cyclic multiplicity of their direct sum is at most n. On the other hand, by Proposition 2.5(3) we have PH(ϕ )(n) T = S(ϕ )(n) PH(ϕ )(n) , L

and this implies that the multiplicity of n−1 j=0 S(ϕ j ) is at least equal to the multiplicity of S(ϕ )(n) . Thus, it suffices to prove that this last operator has multiplicity ≥ n. Set N = µS(ϕ )(n) , and use Theorem 3.5 to find an operator T of class C0 (N) such

that T ≺ S(ϕ )(n) . Fix a quasi-affinity Y satisfying Y T = S(ϕ )(n)Y . Next observe that mT = ϕ , and Lemma 5.5 provides a surjective operator X such that X S(ϕ )(N) = T X . We now have (Y X)S(ϕ )(N) = S(ϕ )(n) (Y X), and Y X has dense range. The inequality N ≥ n follows from Corollary 5.4. We are now ready for the classification theorem. Theorem 5.7. Let T be an operator of class C0 acting on a separable Hilbert space. There exists a Jordan operator S(Φ ) such that T ∼ S(Φ ). Moreover, S(Φ ) is uniquely determined by either S(Φ ) ≺ T or T ≺ S(Φ ). The operator S(Φ ) is called the Jordan model of T .

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Proof. By Proposition 5.1 and Corollary 5.2, there exist Jordan operators S(Φ ) and S(Φ ′ ) such that S(Φ ) ≺ T ≺ S(Φ ′ ); in particular, S(Φ ) ≺ S(Φ ′ ). It suffices then to prove that the relation S(Φ ) ≺ S(Φ ′ ) between two Jordan operators implies S(Φ ) = S(Φ ′ ). Assume therefore that S(Φ ′ )X = X S(Φ ) for some quasi-affinity X. If u is an arbitrary function in H ∞ , then clearly [X ran u(S(Φ ))]− = [ran u(S(Φ ′ ))X ]− = [ran u(S(Φ ′ ))]− , so that

X|[ran u(S(Φ ))]− : [ran u(S(Φ ))]− → [ran u(S(Φ ′ ))]−

is a quasi-affinity intertwining the restrictions of the two Jordan operators to these invariant subspaces. By Corollary 2.7, the operators S(Φ )|[ran u(S(Φ ))]− and S(Φ ′ )|[ran u(S(Φ ′ ))]− are unitarily equivalent to 

ϕj A= S u ∧ ϕj j=0 ∞ L



′

and A =

∞ L

j=0

S

ϕ ′j u ∧ ϕ ′j

!

,

respectively. We conclude that A ≺ A′ , and hence µA′ ≤ µA . When u = ϕn , we have A=

n−1 L j=0

S(ϕ j /ϕn ),

and therefore µA′ ≤ n. Proposition 5.6 implies in particular that the nth summand in A′ must be trivial. We deduce that ϕn′ = ϕn′ ∧ ϕn and hence ϕn′ |ϕn for all n. To conclude the proof, it suffices to show that ϕn also divides ϕn′ . But we have S(Φ ′ )∗ ≺ S(Φ )∗ , and S(Φ ′ )∗ , S(Φ )∗ are unitarily equivalent to the Jordan operators L∞ L∞ ′ ′ j=0 S(ϕ j ˜), j=0 S(ϕ ˜j ). By the first part of the argument we deduce that ϕ n˜ |ϕn ˜ for all n, and this is equivalent to ϕn |ϕn′ .

6 The quasi-equivalence of matrices over H ∞ 1. It is well known that the classical theorem of Jordan, concerning the classification of linear transformations on a finite-dimensional space, can be obtained as a consequence of a diagonalization theorem for polynomial matrices. One may ask whether the classification theorem for operators of class C0 can be proved in a similar fashion. We show that this is indeed the case for operators of class C0 with finite defect indices, and this follows from a diagonalization theorem for matrices over H ∞ . Let F be a separable Hilbert space, and {F, F, Θ (λ )} a bounded analytic function. Let {e j : 0 ≤ j < dim(F)} be an orthonormal basis in F. With respect to this basis, Θ (λ ) is represented by a matrix [θi j (λ )]0≤i, j
6. T HE QUASI - EQUIVALENCE OF MATRICES OVER H ∞

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order k, 1 ≤ k < dim(F), of Θ is a determinant of the form   θi1 j1 θi1 j2 · · · θi1 jk  θi j θi j · · · θi j  2 k   21 22 det  . ..  , .. . .  .. . .  .

θik j1 θik j2 · · · θik jk

where 0 ≤ i1 < i2 < · · · < ik < dim(F) and 0 ≤ j1 < j2 < · · · < jk < dim(F).

Definition. Assume that not all minors of order k of Θ are identically zero. then Dk (Θ ) is defined as the greatest common inner divisor of all minors of order k of Θ . If all minors of order k are equal to zero, we set Dk (Θ ) = 0. Observe that each minor of order k + 1 (with k + 1 ≤ dim(F)) is a linear combination, with coefficients in H ∞ , of minors of order k. Thus, if Dk+1 (Θ ) 6= 0, then Dk (Θ ) 6= 0, and Dk (Θ ) divides Dk+1 (Θ ). Definition. The invariant factors of Θ   D1 (Θ ) Ek (Θ ) = Dk (Θ )/Dk−1 (Θ )   0

are defined as follows:

if k = 1, if 2 ≤ k ≤ dim(F) and Dk (Θ ) 6= 0, if 2 ≤ k ≤ dim(F) and Dk (Θ ) = 0.

Next, we introduce the equivalence relation between matrices over H ∞ which allows us to prove a diagonalization theorem. Let us recall that a bounded analytic function {F, F, Φ (λ )} has a scalar multiple ϕ ∈ H ∞ if there exists a bounded analytic function {F, F, Φ ′ (λ )} satisfying the relations

Φ ′ (λ )Φ (λ ) = Φ (λ )Φ ′ (λ ) = ϕ (λ )IF

(λ ∈ D).

Definition. Let {F, F, Θ1 (λ )} and {F, F, Θ2 (λ )} be bounded analytic functions, and ω ∈ H ∞ an inner function. Then Θ1 and Θ2 are said to be ω -equivalent if there exist bounded analytic functions {F, F, Φ (λ )}, {F, F, Ψ (λ )} with scalar multiples ϕ , ψ , respectively, such that ϕ ∧ ω = ψ ∧ ω = 1, and

Φ (λ )Θ1 (λ ) = Θ2 (λ )Ψ (λ ) (λ ∈ D).

The functions {F, F, Θ1 (λ )} and {F, F, Θ2 (λ )} are said to be quasi-equivalent if they are ω -equivalent for every inner function ω ∈ H ∞ .

The fact that ω -equivalence (and hence quasi-equivalence) is reflexive and transitive is obvious. Symmetry is proved as follows. Let Φ , Ψ , ϕ , ψ be as in the above definition, and let Φ ′ , Ψ ′ satisfy

ΦΦ ′ = Φ ′ Φ = ϕ I,

ΨΨ ′ = Ψ ′Ψ = ψ I.

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The functions {F, F, ϕ (λ )Ψ ′ (λ )} and {F, F, ψ (λ )Φ ′ (λ )} both have the scalar multiple ϕψ relatively prime to ω , and (ψΦ ′ )Θ2 = Φ ′Θ2 ψ = Φ ′Θ2ΨΨ ′ = Φ ′ ΦΘ1Ψ ′ = Θ1 (ϕΨ ′ ). This shows that {F, F, Θ2 (λ )} is ω -equivalent to {F, F, Θ1 (λ )}. Thus both ω equivalence and quasi-equivalence are indeed equivalence relations. Lemma 6.1. Let the functions Θ1 , Θ2 , Φ , Φ ′Ψ , Ψ ′ , ϕ , and ψ satisfy the relations

ΦΘ1 = Θ2Ψ ,

Φ ′ Φ = ΦΦ ′ = ϕ I, and Ψ ′Ψ = ΨΨ ′ = ψ I.

For every integer k, 1 ≤ k ≤ dim(F), we have Dk (Θ1 )|ψ0k Dk (Θ2 ),

Dk (Θ2 )|ϕ0k Dk (Θ1 ),

where ϕ0 , ψ0 denote the inner factors of ϕ , ψ , respectively. Proof. Observe that

ϕΘ1 = Φ ′ ΦΘ1 = Φ ′Θ2Ψ ,

and clearly Dk (ϕΘ1 ) = ϕ0k Dk (Θ1 ). If we show that Dk (Θ2 )|Dk (Φ ′Θ2Ψ ),

(6.1)

then we obtain the relation Dk (Θ2 )|ϕ0k Dk (Θ1 ). Now, to prove (6.1) it suffices to prove the general fact that Dk (A)|Dk (AB) whenever {F, F, A(λ )} and {F, F, B(λ )} are bounded analytic functions. When dim(F) is finite, this divisibility is obvious. Indeed, each minor of order k of AB is a finite sum of terms, each term being the product of a minor of order k of A with a minor of order k of B. If F is infinitedimensional, let us denote by Pn the orthogonal projection of F onto the space generated by {e1 , e2 , . . . , en }. Then clearly Dk (A)|Dk (Pn APn )|Dk (Pn APn BPn ) (n ≥ 1), by the finite-dimensional case. Now, each minor of AB is the pointwise limit of the corresponding minors in Pn APn BPn as n → ∞. It follows easily that Dk (A) divides each minor of order k of AB, and therefore Dk (A)|Dk (AB), as desired. The relation Dk (Θ1 )|ψ0k Dk (Θ2 ) is proved in an analogous manner. Corollary 6.2. If Θ1 and Θ2 are quasi-equivalent, then Dk (Θ1 ) = Dk (Θ2 ) for every integer k, 1 ≤ k ≤ dim(F). Proof. Choose ω = Dk (Θ1 )Dk (Θ2 ) and apply Lemma 6.1 to the functions provided by ω -equivalence. 2. The following result shows the relationship between quasi-equivalence and quasi-similarity. We are using the fact that dT = dT ∗ if T is of class C00 , and thus ΘT coincides with a function of the form {F, F, Θ (λ )}.

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Proposition 6.3. Let T1 and T2 be operators of class C0 , and assume that the characteristic function of Ti coincides with {F, F, Θi (λ )} for i = 1, 2. If Θ1 and Θ2 are quasi-equivalent then T1 and T2 are quasi-similar. Proof. We assume without loss of generality that Ti = U+∗ |(H 2 (F) ⊖ Θi H 2 (F))

(i = 1, 2),

where U+ denotes the unilateral shift on H 2 (F). Let θi denote the minimal function of Ti , and set ω = θ1 θ2 . By hypothesis, we can find bounded operator-valued analytic functions Φ , Φ ′ , Ψ , Ψ ′ , and scalar functions ϕ , ψ ∈ H ∞ , such that

ΦΦ ′ = Φ ′ Φ = ϕ I,

ΨΨ ′ = Ψ ′Ψ = ψ I,

ΦΘ1 = Θ2Ψ ,

and ϕ ∧ ω = ψ ∧ ω = 1. We can then define a bounded linear transformation X : H 2 (F) ⊖ Θ1H 2 (F) → H 2 (F) ⊖ Θ2H 2 (F) by

Xu = P2 Φ u

for u ∈ H 2 (F) ⊖ Θ1H 2 (F),

where P2 denotes the orthogonal projection onto H 2 (F) ⊖ Θ2H 2 (F). Because

ΦΘ1 H 2 (F) = Θ2Ψ H 2 (F) ⊂ Θ2 H 2 (F), we see that X satisfies the relation X T1 = T2 X. In an analogous manner, the operator Y : H 2 (F) ⊖ Θ2H 2 (F) → H 2 (F) ⊖ Θ1H 2 (F) defined by

Y v = P1 (ψΦ ′ v),

satisfies Y T2 = T1Y because

v ∈ H 2 (F) ⊖ Θ2H 2 (F),

ψΦ ′Θ2 H 2 (F) = = = =

Φ ′Θ2 ψ H 2 (F) Φ ′Θ2ΨΨ ′ H 2 (F) Φ ′ ΦΘ1Ψ ′ H 2 (F) ϕΘ1Ψ ′ H 2 (F) ⊂ Θ1 H 2 (F).

Moreover, for v ∈ H 2 (F) ⊖ Θ2H 2 (F) and u ∈ H 2 (F) ⊖ Θ1H 2 (F) we have XY v = P2 (ΦψΦ ′ v) = P2 (ϕψ v) = (ϕψ )(T2 )v and

Y Xu = P1 (ψΦ ′ Φ u) = P1 (ϕψ u) = (ϕψ )(T1 )u.

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C HAPTER X. T HE S TRUCTURE OF O PERATORS OF C LASS C0

Thus XY = (ϕψ )(T2 ) and Y X = (ϕψ )(T1 ), and both of these operators are quasiaffinities because (ϕψ ) ∧ θ1 = (ϕψ ) ∧ θ2 = 1. We conclude that X and Y are also quasi-affinities, thus establishing the quasi-similarity of T1 and T2 . 3. Proposition 6.3 allows us to calculate the Jordan model of a given C0 contraction directly from its characteristic function. This is achieved by proving a diagonalization theorem relative to quasi-equivalence. For this purpose, we first introduce some notation. Given an integer k, 1 ≤ k < dim(F), we denote by Fk the subspace of F generated by {e j : j ≥ k}. Thus dim(F ⊖ Fk ) = k. If θ1 , θ2 , . . . , θk are functions in H ∞ , and {Fk , Fk , Θ0 (λ )} is a bounded analytic function, then we can define a bounded analytic function {F, F, Θ (λ )} such that ( θ j+1 (λ )e j , 0 ≤ j ≤ k − 1, Θ (λ )e j = Θ0 (λ )e j , j ≥ k. We use the notation Θ = Diag(θ1 , θ2 , . . . , θk , Θ0 ) for this function. The key step in the diagonalization process is as follows. Lemma 6.4. Let {F, F, Θ (λ )} be a bounded analytic function, and let ω ∈ H ∞ be an inner function. There exists a bounded analytic function {F1 , F1 , Θ1 (λ )} such that Θ is ω -equivalent to Diag(D1 (Θ ), Θ1 ), and D1 (Θ )|D1 (Θ1 ). We have Θ1 = O if and only if D2 (Θ ) = 0. Proof. The lemma is trivial when Θ = O, so we assume that Θ 6= O. It is also clear that, provided that ω ′ is an inner multiple of ω , ω ′ -equivalence implies ω equivalence. Therefore there is no loss of generality in assuming that D1 (Θ )2 |ω ; simply replace ω by D1 (Θ )2 ω . Let [θi j ]1≤i, j
and ∑

0≤ j
!

α j θi j ∧ ω =

V

0≤ j
!

θi j ∧ ω

(6.3)

for 0 ≤ i < dim(F). In order to realize simultaneously the conditions in (6.3), we must use the fact that a countable intersection of dense Gδ sets is also dense (Baire’s theorem). If we set θi = α j θi j , ∑ 0≤ j
6. T HE QUASI - EQUIVALENCE OF MATRICES OVER H ∞

421

we have V

0≤i
!

θi ∧ ω =

V

0≤i
V

=

0≤i
(θi ∧ ω ) "

V

=

0≤i, j
V

0≤ j
!

!

θi j ∧ ω

#

θi j ∧ ω

= D1 (Θ ) ∧ ω = D1 (Θ ). because of the assumption that D1 (Θ )2 |ω . One further application of Theorem 2.9 provides a sequence {β j : 0 ≤ j < dim(F)} such that

β0 6= 0, and ∑

|βi | < ∞,

∑

0≤i
0≤i
!

βi θi ∧ ω = D1 (θ ).

(6.4)

(6.5)

We now construct boundedly invertible operators A, B on F with matrices given by ( ( α j if i = 0 βi if j = 0 bi j = (0 ≤ i, j < dim(F)), ai j = δi j if i ≥ 1 δi j if j ≥ 1 where, as usual, δii = 1 and δi j = 0 for i 6= j. Thus the function Θ is trivially quasiequivalent to Θ ′ = BΘ A = [θi′j ], and we have ′ θ00 =

∑

0≤i
βi θi ,

′ ∧ ω = D (Θ ) = D (Θ ′ ) by (6.5). We have then so that θ00 1 1 ′ (θ00 /D1 (Θ )) ∧ (ω /D1 (Θ )) = 1,

(6.6)

and because D1 (Θ )|(ω /D1 (Θ )), we also have ′ (θ00 /D1 (Θ )) ∧ D1 (Θ ) = 1.

(6.7)

′ /D (Θ )) ∧ ω = 1. We can therefore write Relations (6.6) and (6.7) imply that (θ00 1 ′ θ00 = ϕ D1 (Θ ),

ϕ ∧ ω = 1.

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′ . Let us set We perform next an ω -equivalence to remove the factor ϕ from θ00 ′′ ′′ Θ = [θi j ]0≤i, j
  D1 (Θ ) if i = j = 0, θi′′j = θi′j if i j = 0 but i 6= j,   ′ ϕθi j if i j 6= 0.

We also set Φ = Diag(1, ϕ IF1 ) and Ψ = Diag(ϕ , IF1 ). Both Φ and Ψ have the scalar multiple ϕ , and clearly ΦΘ ′ = Θ ′′Ψ , so that Θ and Θ ′′ are ω -equivalent. Note that D1 (Θ ) continues to divide all the entries of Θ ′′ . The final step of the proof eliminates the entries θi0′′ and θ0i′′ for i 6= 0. We define two boundedly invertible matrices Φ1 = [ϕi j ]0≤i, j 0 or i = j = 0, and

ψi j =

(

−θi′′j /D1 (Θ ) if i = 0 and j ≥ 1, δi j if i > 0 or i = j = 0.

The boundedness of Ψ is verified as follows. If f = ∑0≤ j
for almost every ζ ∈ C, and hence kΨ (ζ )k ≤ 1 + ess sup{kΘ ′′ (ζ )k : ζ ∈ C} (a.e. ζ ∈ C). The boundedness of Φ is proved analogously, using its transpose matrix. It is also clear that Φ and Ψ are invertible; for instance, Φ −1 = 2I − Φ . Therefore Θ ′′ is quasi-equivalent to Θ ′′′ = ΦΘ ′′Ψ so that D1 (Θ ′′′ ) = D1 (Θ ′′ ) = D1 (Θ ). Clearly Θ ′′′ has the form Diag(D1 (Θ ), Θ1 ), and D1 (Θ ) divides all the entries of Θ1 . The last assertion of the lemma follows because D2 (Θ ) = 0 if and only if D2 (Θ ′′′ ) = 0, and D2 (Θ ′′′ ) = D1 (Θ )D1 (Θ1 ). Theorem 6.5. Assume that {F, F, Θ (λ )} is a bounded analytic function, and ω ∈ H ∞ is inner. (1) We have E j (Θ )|E j+1 (Θ ) for 0 ≤ j < dim(F).

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423

(2) If k is an integersuch that 1 ≤ k ≤ dim(F), there exists a bounded analytic function {Fk , Fk , Θk (λ )} such that Ek (Θ )|D1 (Θk ), and Θ is ω -equivalent to Diag(E1 (Θ ), E2 (Θ ), . . . , Ek (Θ ), Θk ). In order that Θk = O it is necessary and sufficient that Dk+1 (Θ ) = 0. Proof. We prove (2) first. As in the proof of Lemma 6.4, there is no loss of generality in assuming that D j (Θ )|ω whenever j ≤ k and D j (Θ ) 6= 0. We set Θ0 = Θ . An inductive application of Lemma 6.4 shows the existence of bounded analytic functions {F j , F j , Θ j (λ )}, i ≤ j ≤ k, with the following properties: Θ j is ω -equivalent to Diag(D1 (Θ j ), Θ j+1 ) and D1 (Θ j )|D1 (Θ j+1 ) for 0 ≤ j < k. Define functions δ j ∈ H ∞ by δ j = D1 (Θ j−1 ) for 1 ≤ j ≤ k. Then Θ is ω -equivalent to Θ ′ = Diag(δ1 , δ2 , . . . , δk , Θk ), we have δ j |δ j+1 for 1 ≤ j < k, and δk |D1 (Θk ). We now want to relate the functions δ j to D j (Θ ) and E j (Θ ). To do this, we note that by Lemma 6.1 there exist inner functions ϕ0 , ψ0 ∈ H ∞ satisfying the relations

ϕ0 ∧ ω = ψ0 ∧ ω = 1, and

D j (Θ )|ψ0j D j (Θ ′ ),

for 1 ≤ j ≤ dim(F). Now, it is clear that (6.9) can be rewritten as

D j (Θ ′ )|ϕ0j D j (Θ )

D j (Θ ′ ) = δ1 δ2 · · · δ j

D j (Θ )|ψ0j δ1 δ2 · · · δ j ,

(6.8) (6.9)

for 1 ≤ j ≤ k, and thus

δ1 δ2 · · · δ j |ϕ0j D j (Θ ).

(6.10)

These relations show, in particular, that the first j for which δ j = 0 coincides with the first j for which D j (Θ ) = 0. Assume that j is such that these two functions are not zero. Because D j (Θ )|ω , (6.8) implies D j (Θ ) ∧ ψ0j = D j (Θ ) ∧ ϕ0j = 1. An application of the second relation in (6.10) (with j − 1 in place of j) shows that D j (Θ )|ϕ0j−1 D j−1 (Θ )δ j . Hence D j (Θ )|D j−1 (Θ ) or, equivalently, E j (Θ )|δ j so that we can write (6.11) δ j = E j (Θ )η j (η j ∈ H ∞ , j = 1, 2, . . . , k).

Observe that

Dk (Θ ) = (E1 (Θ )η1 )(E2 (Θ )η2 ) · · · (Ek (Θ )ηk ) = δ1 δ2 · · · δk |ϕ0k Dk (Θ ),

so that η1 η2 · · · ηk |ϕ0k , and therefore η j ∧ ω = 1 for all j. Relation (6.11) can be extended to all j ≤ k by setting η j = 1 if D j (Θ ) = δ j = 0. We apply one further ω -equivalence to eliminate the factors η j . Define Ψ = Diag(η1 , η2 , . . . , ηk , IFk ), and note that

Θ ′ = Diag(E1 (Θ ), E2 (Θ ), . . . , Ek (Θ ), Θk )Ψ .

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Moreover, Ψ has the scalar multiple η1 η2 · · · ηk which is relatively prime with ω . Thus the ω -equivalence in (2) is proved. The relation Ek (Θ )|D1 (Θk ) follows from (6.11) and δk |D1 (Θk ). Finally, Dk+1 (Diag(E1 (Θ ), E2 (Θ ), . . . , Ek (Θ ), Θk )) = 0 if and only if Dk+1 (Θ ) = 0, and this happens if and only if Θk = O. In order to prove (1) for some integer j < dim(F), choose an integer k such that j < k ≤ dim(F), and perform the above construction. In (6.11) we have E j (Θ ) ∧ η j+1 = 1 because E j (Θ )|ω if E j (Θ ) 6= 0. Thus the relations E j (Θ )|δ j |δ j+1 and δ j+1 = E j+1 (Θ )η j+1 imply that E j (Θ )|E j+1 (Θ ), as desired. The preceding theorem yields actual quasi-equivalence if Dk (Θ ) = 0, or if F is finite-dimensional and k = dim(F). In particular, we obtain the following result. Theorem 6.6. Assume that T is an operator of class C0 (N), and its characteristic function coincides with {E N , E N , Θ (λ )}. Then T∼

N L

S(E j (Θ )).

j=1

Proof. This follows immediately from Theorem 6.5 and Proposition 6.3. We note that, under the conditions of the preceding theorem, the Jordan model of L T is ∞j=0 S(ϕ j ), where ϕ j = En− j−1 (Θ ) for j < N, and θ j = 1 for j ≥ N. Theorem 6.6 actually yields a new proof of the existence of Jordan models for the class C0 (N). This method for calculating the Jordan model generally fails if T has infinite defect numbers. Consider for instance a Jordan operator S(Φ ), where the sequence Φ = V {ϕ j : j ≥ 0} satisfies j ϕ j = 1. The characteristic function of S(Φ ) coincides with Θ = diag(ϕ j )∞j=0 , and it is easy to show that E j (Θ ) = D j (Θ ) = 1 for all j.

7 Scalar multiples and Jordan models 1. The natural context for studying determinants and minors is that of exterior powers, which we discuss now in the case of complex Hilbert spaces. Given Hilbert spaces F1 and F2 , we denote by F1 ⊗ F2 their Hilbert space tensor product. This is a Hilbert space generated by tensors x1 ⊗ x2 with x1 ∈ F1 , x2 ∈ F2 whose scalar product satisfies (x1 ⊗ x2 , y1 ⊗ y2 ) = (x1 , y1 )(x2 , y2 ),

x1 , y1 ∈ F 1 ,

x2 , y2 ∈ F 2 .

This tensor product can be realized concretely as the Hilbert space of Hilbert and Schmidt operators from the dual F∗1 of F1 to F2 ; F∗1 is simply the Hilbert space consisting of all linear functionals ϕ : F1 → E 1 . The tensor x1 ⊗ x2 corresponds then to the rank-one operator

ϕ → ϕ (x1 )x2

(ϕ ∈ F∗1 ).

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425

Given bounded linear operators T j on F j , j = 1, 2, there exists a unique bounded linear operator T1 ⊗ T2 on F1 ⊗ F2 such that (T1 ⊗ T2)(x1 ⊗ x2 ) = (T1 x1 ) ⊗ (T2 x2 ) (x1 ∈ F1 , x2 ∈ F2 ). The norm of this operator is kT1 ⊗ T2 k = kT1 k · kT2 k. Tensor products can be iterated to yield a space F1 ⊗ F2 ⊗ · · ·⊗Fn associated with a finite number of Hilbert spaces F j ( j = 1, 2, . . . , n). In particular, for any integer n ≥ 1 we can form the tensor power F⊗n = F ⊗ F ⊗ · · · ⊗ F {z } | n times

of a Hilbert space F, and the tensor power

T ⊗n = T ⊗ T ⊗ · · · ⊗ T {z } | n times

of an operator acting on F. The map T → T ⊗n is multiplicative, contractive, (T ∗ )⊗n = (T ⊗n )∗ , and T ⊗n is isometric if T is isometric. Moreover, sequential strong convergence Tk → T implies Tk⊗n → T ⊗n . We are interested in a subspace F∧n ⊂ F⊗n , called the nth exterior power of F. This space is the range of the orthogonal projection Pn on F⊗n which satisfies Pn (x1 ⊗ x2 ⊗ · · · ⊗ xn ) =

1 ∑ ε (σ )xσ (1) ⊗ xσ (2) ⊗ · · · ⊗ xσ (n), n! σ ∈Sn

where Sn denotes the group of permutations of {1, 2, . . . , n}, and ε (σ ) is the sign of the permutation σ , that is, ε (σ ) = 1 if σ is even and ε (σ ) = −1 if σ is odd. It is obvious that Pn commutes with every tensor power T ⊗n , and therefore F∧n is a reducing space for such operators. We write T ∧n = T ⊗n |F∧n . The map T → T ∧n is multiplicative. We use the notation √ x1 ∧ x2 ∧ · · · ∧ xn = n!Pn (x1 ⊗ x2 ⊗ · · · ⊗ xn). For the remainder of this chapter, the symbol ∧ is only used in relation to exterior powers, and no longer designates greatest common inner divisors, even when applied to functions. Observe that xσ (1) ∧ xσ (2) ∧ · · · ∧ xσ (n) = ε (σ )x1 ∧ x2 ∧ · · · ∧ xn

(σ ∈ Sn ),

and (x1 ∧ x2 ∧ · · · ∧ xn , y1 ∧ y2 ∧ · · · ∧ yn ) = det[(xi , y j )]1≤i, j≤n .

If F is separable and {ei : 0 ≤ i < dim(F)} is an orthonormal basis for F, the system {ei1 ∧ ei2 ∧ · · · ∧ ein : 0 ≤ i1 < i2 < · · · < in < dim(F)}

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is an orthonormal basis for F∧n . In particular, F∧n = {0} if dim(F) < n. There is a continuous bilinear map ∧ : F∧n × F∧m → F∧n+m

such that

(x1 ∧ x2 ∧ · · · ∧ xn ) ∧ (xn+1 ∧ xn+2 ∧ · · · ∧ xn+m ) = x1 ∧ x2 ∧ · · · ∧ xn+m

(x j ∈ F).

Clearly we have (T ∧n u) ∧ (T ∧m v) = T ∧n+m (u ∧ v) (u ∈ F∧n , v ∈ F∧m ), for every linear operator T on F. Given a vector e ∈ F, the operator u → e ∧ u (u ∈ F∧n ), is a multiple of a partial isometry from F∧n to F∧n+1 . Its adjoint is denoted v → e∗ ∨ v,

v ∈ F∧n+1 .

This is sometimes called a contraction operation, and n+1

e∗ ∨ ( f1 ∧ f2 ∧ · · · ∧ fn+1 ) = ∑ (−1) j−1 ( f j , e) f1 ∧ · · · ∧ b f j ∧ · · · ∧ fn+1 j=1

for f1 , . . . , fn+1 ∈ F, where the symbol b indicates that the corresponding factor is omitted. This formula is easily verified using an orthonormal basis (in F) including the vector e. If T is a bounded operator on F, we have e∗ ∨ T ∧n+1 ( f1 ∧ f2 ∧ · · · ∧ fn+1 )

(7.1)

n+1

= T ∧n ∑ (−1) j−1 (T f j , e) f1 ∧ · · · ∧ b f j ∧ · · · ∧ fn+1 . j=1

2. The continuity of the multilinear operations described above implies that, given an inner function {F, F, Θ (λ )} in D, then {F∧n , F∧n , Θ (λ )∧n } is also an inner function. Analogously, if f j : D → F, j = 1, 2, . . . , n, are analytic functions, then the function λ → f1 (λ ) ∧ f2 (λ ) ∧ · · · ∧ fn (λ ) is analytic as well, and it is denoted f1 ∧ f2 ∧ · · · ∧ fn . Such products do not necessarily belong to H 2 (F∧n ) if f j ∈ H 2 (F). Let us denote by H ∞ (F) the collection of all bounded functions in H 2 (F). The following lemma is easily verified. Lemma 7.1. Given functions f2 , f3 , . . . , fn ∈ H ∞ (F), the map f → f ∧ f2 ∧ · · · ∧ fn is a bounded linear operator from H 2 (F) to H 2 (F∧n ).

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3. Now let F be a separable Hilbert space, and let {F, F, Θ (λ )} be an inner function. We denote by TΘ the functional model associated with this function. Thus TΘ is a c.n.u. contraction whose characteristic function coincides with Θ . According to Theorem VI.5.1, TΘ is of class C0 if and only if Θ has a scalar multiple, that is,

Θ (λ )Ω (λ ) = Ω (λ )Θ (λ ) = u(λ )IF

(λ ∈ D),

where {F, F, Ω (λ )} is another inner function, and u ∈ H ∞ is inner as well.

Lemma 7.2. If u ∈ H ∞ is a scalar multiple of {F, F, Θ (λ )}, and n ≤ dim(F) is a positive integer, then un is a scalar multiple of {F∧n , F∧n , Θ (λ )∧n }. Proof. If Ω satisfies the relation above, we have

Θ (λ )∧n Ω (λ )∧n = Ω (λ )∧nΘ (λ )∧n = u(λ )n IF∧n

(λ ∈ D).

The lemma follows. The least inner scalar multiple of Θ is precisely the minimal function of TΘ (cf. Theorem VI.5.1). We denote by dn,Θ ∈ H ∞ the least inner scalar multiple of Θ ∧n . Note that Θ ∧n acts on the space {0} if n > dim(F), so we do not use the notation dn,Θ for such values of n. Proposition 7.3. Assume that the inner function {F, F, Θ (λ )} has a scalar multiple. Then dn,Θ divides dn+1,Θ for all n < dim(F). Proof. The function u = dn+1,Θ is a scalar multiple of Θ ∧n+1 . Equivalently, u f ∈ Θ ∧n+1 H 2 (F∧n+1 ) for every f ∈ H 2 (F∧n+1 ). We need to prove that this requirement is also satisfied with n in place of n + 1. In fact, because Θ ∧n H 2 (F∧n ) is invariant for the unilateral shift on H 2 (F∧n ), it suffices to show that ue1 ∧ e2 ∧ · · · ∧ en ∈ Θ ∧n H 2 (F∧n ) for any orthonormal system {e j : 1 ≤ j ≤ n} in F. Consider such an orthonormal system, and choose a unit vector e0 ∈ F orthogonal to e j for 1 ≤ j ≤ n. The hypothesis implies the existence of a function f ∈ H 2 (F∧n+1 ) such that u(λ )e0 ∧ e1 ∧ · · · ∧ en = Θ (λ )∧n+1 f (λ )

(λ ∈ D).

We have then u(λ )e1 ∧ · · · ∧ en = e∗0 ∨ (u(λ )e0 ∧ e1 ∧ · · · ∧ en ) = e∗0 ∨ Θ (λ )∧n+1 f (λ ),

and (7.1) shows that this function is of the form Θ (λ )∧n g(λ ) for some analytic function g. The function g must in fact be bounded because Θ ∧n is inner, and therefore g ∈ H 2 (F∧n ), as desired.

We now show that all the functions dn,Θ can be calculated in terms of the Jordan model of TΘ .

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Theorem 7.4. Let {F, F, Θ (λ )} and {F′ , F′ , Θ ′ (λ )} be inner functions, both having scalar multiples. If TΘ ≺ TΘ ′ then dn,Θ ′ divides dn,Θ provided that n ≤ dim(F) and n ≤ dim(F′ ). Proof. Let X : H → H′ be a quasi-affinity such that XTΘ = TΘ ′ X ; here H = H 2 (F) ⊖ Θ ′ H 2 (F),

H′ = H 2 (F′ ) ⊖ Θ H 2(F′ ).

By Theorem VI.3.6, there exists a bounded analytic function {F, F′ , Ψ (λ )} such that ΨΘ H 2 (F) ⊂ Θ ′ H 2 (F′ ) and

X f = PH′Ψ f

( f ∈ H(Θ )).

The first requirement on Ψ implies the existence of a bounded analytic function {F, F′ , Ψ ′ (λ )} such that ΨΘ = Θ ′Ψ ′ . The range of X is equal to (Ψ H 2 (F) + Θ ′ H 2 (F′ )) ⊖ Θ ′ H 2 (F′ ), and therefore the space Ψ H 2 (F) + Θ ′ H 2 (F′ ) is dense in H 2 (F′ ). Fix now n ≤ min{dim(F), dim(F′ )}, and set u = dn,Θ , so that uH 2 (F∧n ) ⊂ Θ ∧n H 2 (F∧n ). It suffices to show that ux1 ∧ x2 ∧ · · · ∧ xn ∈ Θ ′∧n H 2 (F′∧n ) for any choice of vectors x1 , x2 , . . . , xn ∈ F′ . For each j = 1, 2, . . . , n, there exist functions f jk ∈ H 2 (F) and g jk ∈ H 2 (F′ ) such that lim (Ψ f jk + Θ ′ g jk ) = x j

k→∞

( j = 1, 2, . . . , n),

in the H 2 norm. Moreover, the functions f jk and g jk can be assumed to be bounded. It follows that for each k, the function hk1 k2 ...kn = (Ψ f1k1 + Θ ′ g1k1 ) ∧ (Ψ f2k2 + Θ ′ g2k2 ) ∧ · · · ∧ (Ψ fnkn + Θ ′ gnkn ) belongs to H 2 (F′∧n ). Moreover, Lemma 7.1 allow us to let the indices k j tend to infinity one at a time, allowing us to conclude that the vector x1 ∧ x2 ∧ · · · ∧ xn belongs to the norm closure of the set S = {hk1 k2 ...kn : k1 , k2 , . . . , kn = 1, 2, . . . }. Thus it suffices to show that uh ∈ Θ ′∧n H 2 (F′∧n ) when h ∈ S. Any such function h is a finite sum of functions of the form F0 = Θ ′ g1 ∧ Θ ′ g2 ∧ · · · ∧ Θ ′ gn , or

Fn = Ψ f1 ∧ Ψ f2 ∧ · · · ∧ Ψ fn

Fj = Ψ f1 ∧ Ψ f2 ∧ · · · ∧ Ψ f j ∧ Θ ′ g1 ∧ Θ ′ g2 ∧ · · · ∧ Θ ′ gn− j ,

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where 1 ≤ j ≤ n − 1, f1 , f2 , . . . , fn ∈ H ∞ (F), and g1 , g2 , . . . , gn ∈ H ∞ (F′ ). It therefore suffices to show that uF ∈ Θ ′∧n H 2 (F′∧n ) for such functions F. When j = 0 there is nothing to prove. Assume therefore that j ≥ 1. By Proposition 7.3, the function d j,Θ divides u, and therefore u f1 ∧ f2 ∧ · · · ∧ f j = Θ ∧ j f for some f ∈ H ∞ (F∧ j ). For j < n, we conclude that

uFj = Ψ ∧ j (u f1 ∧ f2 ∧ · · · ∧ f j ) ∧ Θ ′∧n− j (g1 ∧ g2 ∧ · · · ∧ gn− j ) = Ψ ∧ jΘ ∧ j f ∧ Θ ′∧n− j (g1 ∧ g2 ∧ · · · ∧ gn− j ) = Θ ′∧ jΨ ′∧ j f ∧ Θ ′∧n− j (g1 ∧ g2 ∧ · · · ∧ gn− j ) = Θ ′∧n (Ψ ′∧ j f ∧ (g1 ∧ g2 ∧ · · · ∧ gn− j )).

For j = n the same equation holds, except that there are no factors of the form gi . The theorem is proved. We can now calculate the Jordan model of TΘ in terms of the functions dn,Θ . Theorem 7.5. Let F be a separable, infinite-dimensional Hilbert space, and let {F, F, Θ (λ )} be an inner function such that the operator T = TΘ is of class C0 . Define inner functions ϕ0 = d1,Θ and ϕ j = d j+1,Θ /d j,Θ for j ≥ 1. (1) We have ϕ j+1 |ϕ j for all j. L (2) The Jordan model of T is ∞j=0 S(ϕ j ). L

Proof. Let ∞j=0 S(ψ j ) be the Jordan model of T , and denote by Θ ′ the inner function diag(ψ j )∞j=0 . The function Θ ′∧n is diagonal as well, with diagonal entries {ψ j1 ψ j2 · · · ψ jn : 0 ≤ j1 < j2 < · · · < jn }. Clearly, dn,Θ ′ is the least common multiple of these entries; that is, dn,Θ ′ = ψ0 ψ1 · · · ψn−1 . Theorem 7.4 implies the equality dn,Θ = dn,Θ ′ for all n ≥ 1. The theorem follows immediately from this observation. Corollary 7.6. Let T be a contraction of class C0 on a Hilbert space, and let   ′ T ∗ T= O T ′′ be the triangulation associated with an invariant subspace of T . Denote by S(Φ ), S(Φ ′ ), and S(Φ ′′ ) the Jordan models of T, T ′ , and T ′′ , respectively, where Φ = {ϕ j : j ≥ 0}, Φ ′ = {ϕ ′j : j ≥ 0}, and Φ ′′ = {ϕ ′′j : j ≥ 0}. Then we have ′ ′′ ϕ0 ϕ1 · · · ϕn−1 |ϕ0′ ϕ1′ · · · ϕn−1 · ϕ0′′ ϕ1′′ · · · ϕn−1

for all n ≥ 1. Proof. According to Theorem VII.1.1, there exist inner functions {F, F, Θ (λ )}, {F, F, Θ ′ (λ )}, and {F, F, Θ ′′ (λ )} whose pure parts coincide with the characteristic functions of T, T ′ , and T ′′ , respectively, and such that Θ (λ ) = Θ ′′ (λ )Θ ′ (λ ). By the preceding theorem, the corollary is equivalent to the statement that dn,Θ |dn,Θ ′ dn,Θ ′′ .

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Thus we must prove that dn,Θ ′ dn,Θ ′′ is a scalar multiple of Θ ∧n . Choose inner functions Ω ′ , Ω ′′ such that Θ ′∧n Ω ′ = Ω ′Θ ′∧n = dn,Θ ′ IF∧n and Θ ′′∧n Ω ′′ = Ω ′′Θ ′′∧n = dn,Θ ′′ IF∧n , and set Ω = Ω ′ Ω ′′ . It is easy to see that Θ ∧n Ω = dn,Θ ′ dn,Θ ′′ IF∧n , as desired.

8 Weak contractions of class C0 1. We have seen in Proposition V.6.1 how algebraic adjoints can be used to show that det(Θ (λ )) is a scalar multiple of the function {F, F, Θ (λ )} when dim(F) < ∞. There is an extension of the notion of algebraic adjoint that allows us to show that det(Θ (λ )) is a scalar multiple of Θ ∧n as well. Proposition 8.1. Assume that F is a Hilbert space of finite dimension N. For 1 ≤ n < N, there exists a continuous map A → AAd n which associates with each operator A on F an operator AAd n on F∧n such that: (1) AAd n A∧n = A∧n AAd n = det(A)IF∧n . (2) (AB)Ad n = BAd n AAd n . (3) kAAd n k = kA∧N−n k. Proof. Fix n, and an orthonormal basis {e j : 1 ≤ j ≤ N} in F. The bilinear form B(h, g) = (h ∧ g, e1 ∧ e2 ∧ · · · ∧ eN )

(h ∈ F∧n , g ∈ F∧N−n ),

is nondegenerate; that is, the equality B(h, g) = 0 for every g (resp., every h) implies that h = 0 (resp., g = 0). In fact B(eσ (1) ∧ · · · ∧ eσ (n) , eσ (n+1) ∧ · · · ∧ eσ (N) ) = ε (σ ) if σ ∈ SN , and this implies that there is an isometric transformation C from F∧N−n onto the dual space (F∧n )∗ such that (Cg)(h) = B(h, g) (h ∈ F∧n , g ∈ F∧N−n ). Given an operator A on F, we can then define the operator X on (F∧n )∗ by X = CA∧N−nC−1 . The operator AAd n is then defined so that its dual (in the Banach space sense) is equal to X. In other words, B(AAd n h, g) = B(h, A∧N−n g) (h ∈ F∧n , g ∈ F∧N−n ). Clearly, the map A → AAd n is continuous, and equality (3) follows immediately. Equality (2) is due to the fact that taking duals reverses the order of the factors. To prove (1) note that for every h ∈ F∧n , g ∈ F∧N−n we have B(AAd n A∧n h, g) = B(A∧n h, A∧N−n g) = (A∧N (h ∧ g), e1 ∧ e2 ∧ · · · ∧ eN ) = det(A)B(h, g),

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where we used the fact that A∧N = det(A)IF∧N . Because B is a nondegenerate form, we obtain AAd n A∧n = det(A)IF∧n . Equality (1) follows immediately for invertible A, and it extends to arbitrary A by continuity. Observe that the construction given above depends on the choice of basis, but (1) shows that the result is independent of this choice. It is implicit in the proof above that AAd n is unitarily equivalent to A∧N−n , so its matrix entries of AAd n are minors of order N − n of the matrix A.

Corollary 8.2. Let A be a linear operator on the Hilbert space F of dimension N, and let {e1 , e2 , . . . , en } be an orthonormal system in F. If P denotes the orthogonal projection onto the space generated by {e1 , e2 , . . . , en }, we have (AAd n (e1 ∧ e2 ∧ · · · ∧ en ), e1 ∧ e2 ∧ · · · ∧ en ) = det(P + (I − P)A(I − P)). Proof. Complete the given vectors to an orthonormal basis {e j : 1 ≤ j ≤ N} for F, and set f = e1 ∧ e2 ∧ · · · ∧ en , g = en+1 ∧ · · · ∧ eN . Using the notation in the preceding proof, we have (AAd n f , f ) = ((AAd n f ) ∧ g, f ∧ g) = B(AAd n f , g) = B( f , A∧N−n g)

= ( f ∧ A∧N−n g, f ∧ g). Because (P + A(I − P))e j = e j for j = 1, 2, . . . , n and (P + A(I − P))e j = Ae j for j = n + 1, n + 2, . . ., N, we have (P + A(I − P))∧N ( f ∧ g) = f ∧ A∧N−n g, and thus (AAd n f , f ) = ((P + A(I − P))∧N ( f ∧ g), f ∧ g) = det(P + A(I − P)).

To conclude the proof, observe that det(P + A(I − P)) = det(P + (I − P)A(I − P)) because P + A(I − P) has a block upper triangular form.

Remark. The above formula can be extended as follows. Let {e1 , e2 , . . . , en } and { f1 , f2 , . . . , fn } be orthonormal systems in F, with 1 ≤ n < dim(F). Choose a unitary operator U on F such that det(U) = 1 and U f j = e j for j = 1, 2, . . . , n. Denote by P the orthogonal projection onto the space generated by {e1 , e2 , . . . , en }, and set e = e1 ∧ e2 ∧ · · · ∧ en , f = f1 ∧ f2 ∧ · · · ∧ fn . We have U ∧n f = e, so that U Ad n e = f , and thus (AAd n f , e) = ((UA)Ad n e, e) = det(P + (I − P)UA(I − P)).

(8.1)

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2. In order to extend algebraic adjoints to infinite dimensions, we need a few additional facts about the trace class whose definition we now recall. An operator X on a Hilbert space F is of trace class if the positive operator |X | = (X ∗ X )1/2 has finite trace as defined in Section VIII.1. In other words, |X| is compact with summable eigenvalues. We use the notation S1 (F) for the collection of trace class operators, and we set kX k1 = tr(|X|) (X ∈ S1 (F)). Then S1 (F) is a Banach space with this norm, and the operators with finite rank are dense in S1 (F). With each X ∈ S1 (F) one can associate a complex number det(I + X) that behaves like the usual determinant. More precisely, (a) (b) (c) (d) (e)

det(AB) = det(A) det(B) if A, B ∈ I + S1(F). A is invertible if and only if det(A) 6= 0. | det(A)| = 1 if A is unitary. | det(A)| ≤ 1 if kAk ≤ 1. If A(λ ) is analytic (resp., continuous), then det(A(λ )) is analytic (resp., continuous). (f) If F has an orthonormal basis {e j : j = 1, 2, . . . }, then det(A) = lim det[(Aei , e j )]ni, j=1 . n→∞

(g) If B ∈ S1 (F) then eB ∈ I + S1 (F) and det(eB ) = etrB . For these facts the reader can consult G OHBERG AND K RE˘I N [4]. The following result is essentially proved in Sec. VIII.4 for the finite-dimensional case. Corollary 8.3. Let A(λ ) ∈ I + S1 (F) be an analytic function in a neighborhood of 0 such that A(0) = I. Then d det(A(λ )) = trA′ (0). dλ λ =0

Proof. Property (g) above shows that det(A(λ )) = exp(tr log A(λ )) for λ close to zero. Here log A(λ ) is calculated using the Riesz and Dunford calculus for the principal branch of the logarithm. The lemma follows immediately from this observation via the chain rule for differentiation.

Lemma 8.4. Assume that dim(F) < ∞, and let A be a linear operator on F. Then we have kAAd n k ≤ exp((1 + kA − Ik1)2 − 1). for 1 ≤ n < dim(F).

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Proof. Assume first that A ≥ O, and λ1 ≥ λ2 ≥ · · · ≥ λN ≥ 0 are its eigenvalues, repeated according to their multiplicities. Then AAd n is positive, and its eigenvalues are {λi1 λi2 · · · λiN−n : 1 ≤ i1 < i2 < · · · < iN−n ≤ N}. Therefore kAAd n k = λ1 λ2 · · · λN−n ≤ (1 + |λ1 − 1|)(1 + |λ2 − 1|) · · · (1 + |λN − 1|) ≤ exp(|λ1 − 1|) exp(|λ2 − 1|) · · ·exp(|λN − 1|) = exp(tr(|A − I|)) = exp(kA − Ik1).

In the general case we use the polar decomposition A = U|A|: kAAd n k = kU Ad n |A|Ad n k ≤ k|A|Ad n k ≤ exp(k|A| − Ik1), and the easy estimate k|A| − Ik1 ≤ kA∗ A − Ik1 ≤ kA∗ (A − I)k1 + kA − Ik1 ≤ (kAk + 1)kA − Ik1 ≤ (kA − Ik1 + 2)kA − Ik1

= (1 + kA − Ik1)2 − 1. The lemma follows.

Theorem 8.5. Assume that F is a separable, infinite dimensional Hilbert space, and n ≥ 1. For every operator A ∈ I + S1 (F) there exists an operator AAd n on F∧n with the following properties. (1) AAd n A∧n = A∧n AAd n = det(A)IF∧n . (2) (AB)Ad n = BAd n AAd n . (3) If P denotes the orthogonal projection onto the space generated by an orthonormal system {e1 , e2 , . . . , en }, we have (AAd n (e1 ∧ e2 ∧ · · · ∧ en ), e1 ∧ e2 ∧ · · · ∧ en ) = det(P + (I − P)A(I − P)). (4) kAAd n k ≤ exp((1 + kA − Ik1)2 − 1). (5) kAAd n k ≤ 1 if kAk ≤ 1. (6) If A(λ ) is an analytic function, then A(λ )Ad n is also analytic. Proof. Fix n, choose an orthonormal basis { f j : j ≥ 1} for F, and denote by Pk the orthogonal projection onto the space Fk generated by { f j : 1 ≤ j ≤ k}. Given A ∈ I + S1 (F), set Ak = Pk APk , and define an operator Xk on F∧n by setting Ad n Xk |F∧n k = (Ak |Fk )

⊥ and Xk |(F∧n k ) = O.

Because kAk − IFk k1 ≤ kA − Ik1, Lemma 8.4 implies that the operators Xk are uniformly bounded. We show that they converge weakly. Given integers 1 ≤ i1 < i2 < · · · < in and 1 ≤ j1 < j2 < · · · < jn , there exists a unitary operator U on F such that

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U fk = fk for k > max{in , jn } and U f jℓ = fiℓ for ℓ = 1, 2, . . . , n. We deduce from (8.1) that (Xk ( f j1 ∧ · · · ∧ f jn ), fi1 ∧ · · · ∧ fin ) = det(P + (I − P)UAk (I − P)) for k > max{in , jn }, where P denotes the orthogonal projection onto the space generated by { fiℓ : 1 ≤ ℓ ≤ n}. Now, limk→∞ kA − Ak k1 = 0, and property (e) of the determinants implies that the determinants in the right-hand side of the above equation tend to det(P + (I − P)UA(I − P)) as k → ∞. We denote by AAd n the weak limit of the operators Xk . Properties (4) and (5) are trivially verified, and property (6) follows from Dunford’s theorem because the above argument, along with property (e) of determinants, shows that the matrix entries of A(λ )Ad n are analytic functions. To verify (1), observe that n Ad n ∧n Xk AAd k h = Ak Xk h = det(Ak + I − Pk )h (h ∈ Fk ),

and let k → ∞. Let {Am }∞ m=1 ⊂ I + S(F) be a sequence such that limm→∞ kA − Am k1 = 0. The above considerations show that Am → A in the weak operator topology. Property (2) follows from (1) if A and B are invertible. For the general case, choose invertible operators Am , Bm ∈ I + S(F) such that limm→∞ kA − Am k1 = n Ad n yields the desired relimm→∞ kB − Bmk1 = 0. The equality (Am Bℓ )Ad n = BAd ℓ Am sult as m → ∞ and ℓ → ∞. Finally, observe that the operators AAd n do not depend on the basis { f j }∞j=1 . This follows from (1) when A is invertible, and from approximation by invertible operators in general. Note also that (3) has already been verified if e j = f j for 1 ≤ j ≤ n. The general case follows if we construct AAd n using a basis for F that contains the elements {e j }nj=1 . For applications to weak contractions, the following result is very useful. Theorem 8.6. Let {F, F, Θ (λ )} be an inner function such that I − Θ (λ ) ∈ S1 (F) for all λ ∈ D. If det(Θ (λ )) is not identically zero, then it is an inner scalar multiple of Θ ∧n for all integers n, 1 ≤ n ≤ dim F. Proof. Assume that the determinant is not identically zero. The relation

Θ (λ )∧nΘ (λ )Ad n = Θ (λ )Ad nΘ (λ )∧n = det(Θ (λ ))IF∧n shows that det(Θ (λ )) is a scalar multiple of Θ (λ )∧n . Write det(Θ (λ )) = di do , with di inner and do outer. Then di is also a scalar multiple of Θ (λ )∧n . Thus Θ (λ )∧n Ωn (λ ) = di IF∧n for some contractive analytic function Ωn (λ ). The relation

Θ (λ )∧n (Θ (λ )Ad n − do (λ )Ωn (λ )) = O (λ ∈ D),

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and the fact that Θ (λ )∧n is inner, imply the equality Θ (λ )Ad n = do (λ )Ωn (λ ). In particular, kΘ (0)Ad n k ≤ |do (0)|, and therefore |(Θ (0)Ad n e1 ∧ e2 ∧ · · · ∧ en , e1 ∧ e2 ∧ · · · ∧ en )| ≤ |do (0)|, where {e j : j = 1, 2, . . . } is any orthonormal basis in F. Denoting by Pn the orthogonal projection onto the space generated by {e j : 1 ≤ j ≤ n}, Theorem 8.5(3) implies that | det(Pn + (I − Pn)Θ (0)(I − Pn )| ≤ |do (0)| (n ≥ 1).

The operators Pn + (I − Pn )Θ (0)(I − Pn ) tend to I in the trace norm, and therefore their determinants tend to 1. We deduce that |do (0)| ≥ 1, and the maximum modulus principle implies that do is constant. Thus det(Θ (λ )) = do (0)di (λ ) is inner, as claimed. We need one more result about the trace class. Proposition 8.7. Let A be an invertible operator on F such that O ≤ A ≤ I. If the sequence {k(A∧n )−1 k : n ≥ 1} is bounded, then I − A ∈ S1 (F). R

Proof. Write the spectral integral I − A = 01 t dE(t). We show first that I − A is compact. Indeed, in the contrary case there exists ε > 0 such that the space E((ε , 1])F is infinite dimensional. Choose an orthonormal system { f j : j ≥ 1} in this space, and note that kA∧n ( f1 ∧ f2 ∧ · · · ∧ fn )k ≤ (1 − ε )n . Thus k(A∧n )−1 k ≥ (1 − ε )−n , contrary to the hypothesis. Hence I − A is compact. If I − A has finite rank, we are done. If not, let λ1 ≥ λ2 ≥ · · · be the eigenvalues of I − A, repeated according to multiplicity, and let {e j : j ≥ 1} be an orthonormal system such that Ae j = (1 �