Substitution Dynamical Systems - Spectral Analysis (Lecture Notes in Mathematics, 1294) 3642112110, 9783642112119

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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1294

Martine Queffélec

Substitution Dynamical Systems – Spectral Analysis Second Edition

123

Martine Queffélec Université Lille 1 Laboratoire Paul Painlevé 59655 Villeneuve d’Ascq CX France [email protected]

ISBN: 978-3-642-11211-9 e-ISBN: 978-3-642-11212-6 DOI: 10.1007/978-3-642-11212-6 Springer Heidelberg Dordrecht London NewYork Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2010920970 Mathematics Subject Classification (2000): 37, 42, 11, 28, 47, 43, 46, 82 c Springer-Verlag Berlin Heidelberg 2010  This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publisher Services Printed on acid-free paper springer.com

Preface for the Second Edition

This revised edition initially intended to correct the misprints of the first one. But why does it happen now, while the subject extensively expanded in the past twenty years, and after the publication of two major books (among other ones) devoted to dynamical systems [88] and automatic sequences [14]? Let us try to explain why we got convinced to do this new version. On the one hand, the initial account of the LNM 1294 offered a basis on which much has been built and, for this reason, it is often referred to as a first step. On the other hand, the two previously quoted books consist in impressive and complete compilations on the subject [14, 88]; this was not the spirit of our LNM, almost self-contained and “converging” to the proof of a specific result in spectral theory of dynamical systems. From this point of view, those three books might appear as complementary ones. This having been said, reproducing the corrected LNM identically would have been unsatisfactory : a lot of contributions have concurred to clarify certain aspects of the subject and to fix notations and definitions; also a great part of the raised questions have now been solved. Mentioning these improvements seemed to us quite necessary. Therefore, we chose to add some material to the first introductory chapters, which of course does not (cannot) reflect the whole progress in the field but some interesting directions. Moreover, two applications of substitutions more generally of combinatorics of words - to discrete Schr¨odinger operators and to continued fraction expansions clearly deserved to take place in this new version : two additional appendices summarize the main results in those fields. The initial bibliography has been inflated to provide a much more up-to-date list of references. This renewed bibliography is still far from being exhaustive and we should refer the interested reader to the two previously cited accounts. In recent contributions, the terminology has changed, emphasizing on the morphism property. However, we chose to keep to the initial terminology, bearing in mind the fact that this is definitely a second edition. Lille December 2009

Martine Queff´elec

v

Preface for the First Edition

Our purpose is a complete and unified description of the spectrum of dynamical systems arising from substitution of constant length (under mild hypotheses). The very attractive feature of this analysis is the link between several domains : combinatorics, ergodic theory and harmonic analysis of measures. The rather long story of these systems begins perhaps in 1906, with the construction by A. Thue [234] of a sequence with certain non-repetition properties (rediscovered in 1921 by M. Morse [190]): 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 ··· This sequence (called from now on the Thue-Morse sequence) can be obtained by an obvious iteration of the substitution 0 → 01, 1 → 10, or else, as an infinite block product : 01 × 01 × 01 × · · ·, where B × 01, for any 0 − 1 block B, means : repeat B ˜ the block deduced from B by exchanging 0 and 1. Also, if S2 (n) denotes and then B, the sum of digits of n in the 2-adic expansion, u = (un ) with un = eiπ S2 (n) is the ±1 Thue-Morse sequence. The Thue-Morse sequence admits a strictly ergodic (= minimal and uniquely ergodic) orbit closure and a simple singular spectrum, as observed by M. Keane [143]. The various definitions of the Thue-Morse sequence lead to various constructions of sequences, and thus, of dynamical systems: – substitution sequences [55, 63, 68, 104, 188] then [71, 119, 132, 135, 173, 174, 189, 208], . . . – a class of 0 − 1 sequences introduced by M. Keane, called generalized Morse sequences [143], admitting in turn extensions [175, 176] then [102, 154–156, 162], ... – q-multiplicative sequences, with q = (qn ), qn integer ≥ 2 [59] then [166, 202], ... In this account, we restrict our attention to the first category of sequences, but, in case of bijective substitutions (chapter 9), we deal with particular G-Morse sequences and q-multiplicative sequences. vii

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Preface for the First Edition

Ergodic and topological properties of substitution dynamical systems have been extensively studied; criteria for strict ergodicity [68,188], zero entropy [68,209], rational pure point spectrum [68,173,174], conditions for presence of mixed spectrum [68] and various mixing properties [71] are main investigations and results in these last years. But, except in some examples ([135, 143], . . .), no descriptive spectral analysis of the continuous part of the spectrum has been carried out. Indeed, not so many dynamical systems lead themselves to a comprehensive computation of spectral invariants. I mean, mainly, maximal spectral type and spectral (global) multiplicity (see [214] for a rather complete historical survey). Of course, transformations with purely discrete spectrum are quite well-known [240], and in this case, the spectrum is simple. In the opposite direction, a countable Lebesgue spectrum occurs in ergodic automorphisms of compact abelian groups as in K-automorphisms (see [61]). A very important class of dynamical systems, with respect to the spectral analysis, consists of gaussian dynamical systems. Guirsanov proved a conjecture of Kolmogorov [110]: the maximal spectral type of a gaussian dynamical system is equivalent to eσ , where σ denotes the spectral measure of the process; and its spectral multiplicity has been shown by Vershik to be either one - with singular spectrum - or infinite ([237, 238], see also [89]). Then arose the question of whether finite multiplicity ≥ 2 (or ≥ 1 for Lebesgue spectrum) was possible, and the last results in multiplicity theory have been mostly constructions of suitable examples. I just quote the last three important ones : Robinson E.A. Jr in [214] exhibits, for every m ≥ 1, a measure-preserving transformation with singular spectrum and spectral multiplicity m. On the other hand, Mathew and Nadkarni in [177, 178] construct, for every N ≥ 2, a measure-preserving transformation with a Lebesgue spectrum of multiplicity N φ (N) (φ Euler totient function). In these examples, the transformations are group extensions. Recently, M. Lemanczyk obtained every even Lebesgue multiplicity [160]. Turning back to substitution dynamical systems, we prove the following : for a substitution of length q over the alphabet A (or q-automaton [55]), the spectrum is generated by k ≤ Card A probability measures which are strongly mixing with respect to the q-adic transformation on T; in most examples, these measures are specific generalizations of Riesz products, which is not so surprising because of the self-similarity property inherent in this study. (Note that such Riesz products play a prominent part in distinguishing normal numbers to different bases [136]; see also [50, 198], . . .). Earlier Ledrappier and Y. Meyer already realized classical Riesz products as the maximal spectral type of some dynamical system. The generating measures of the spectrum of some q-automaton are computable from a matrix of correlation measures, indeed a matrix Riesz product, whose rank gives rise to the spectral multiplicity. For example, the continuous part of the RudinShapiro dynamical system is Lebesgue with multiplicity 2, while, by using the mutual singularity of generalized Riesz products (analyzed in chapter 1), we get various singular spectra with multiplicity 1 or 2, as obtained by Kwiatkowski and Sikorski ([156], see also [101, 102]). For substitutions of nonconstant length, no

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spectral description seems accessible at present but we state a recent characterization of eigenvalues established by B. Host [119] and list some problems. We have aimed to a self-contained text, accessible to non-specialists who are not familiar with the topic and its notations. For this reason, we have developed with all details the properties of the main tools such that Riesz products, correlation measures, matrices of measures, nonnegative matrices and even basic notions of spectral theory of unitary operators and dynamical systems, with examples and applications. More precisely, the text gets gradually more specialized, beginning in chapter 1 with generalities on the algebra M(T) and its Gelfand spectrum Δ . We introduce generalized Riesz products and give a criterion for mutual singularity. Chapter 2 is devoted to spectral analysis of unitary operators, where all fundamental definitions, notations and properties of spectral objects can be found. We prove the representation theorem and two versions of the spectral decomposition theorem. We restrict ourselves, in chapter 3, to the unitary operator associated with some measure-preserving transformation and we deduce, from the foregoing chapter, spectral characterizations of ergodicity and of various mixing properties (strong, mild, weak). As an application of D-ergodicity (ergodicity with respect to a group of translations [47]), we discuss spectral properties of some skew products over the irrational rotation [100, 103, 140, 212]. In chapter 4, we investigate shift invariant subsets of the shift space (subshifts), such like the orbit closure of some sequence. Strict ergodicity can be read from the given sequence, if taking values in a finite alphabet. The correlation measure of some sequence - when unique - belongs to the spectral family; hence, from earlier results, we derive spectral properties of the sequence. We give a classical application to uniform distribution modulo 2π (Van der Corput’s lemma) and we discuss results around sets of recurrence [25, 35, 93, 219]. From now on we are concerned with substitution sequences. All previously quoted results regarding substitution dynamical systems are proved in chapters 5–6, sometimes with a different point of view and unified notations (strict ergodicity, zero entropy, eigenvalues and mixing properties). We are needing the Perron-Frobenius theorem and, for sake of completeness, we give too a proof of it. Till the end of the account, the substitution is supposed to have a constant length. We define, in chapter 7, the matrix of correlation measures Σ and we show how to deduce the maximal spectral type from it. Then we prove elementary results about matrices of measures which will be used later. In chapter 8, we realize Σ as a matrix Riesz product and this fact provides a quite simple way to compute it explicitly. Applying the techniques immediately, we treat the first examples : Morse sequence, Rudin-Shapiro sequence, and a class of sequences arising from commutative substitutions (particular G-Morse sequences), admitting generalized Riesz product as generating measures. An important class of substitutions is studied in chapter 9 without complete success. It would be interesting in this case to get a more precise estimate of the spectral multiplicity, which is proved to be at least 2 for substitutions over a nonabelian group.

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Preface for the First Edition

Finally, the main results on spectral invariants in the general case are obtained in chapters 10–11 by using all the foregoing. We have to consider a bigger matrix of correlation measures, involving occurrences of pairs of given letters instead of simple ones, which enjoys the fundamental strong mixing property and provides the maximal spectral type of the initial substitution. The spectral multiplicity can be read from the matrix Σ , as investigated with the Rudin-Shapiro sequence and some bijective substitution. We obtain in both cases a Lebesgue multiplicity equal to 2, while N-generalized Rudin-Shapiro sequences admit a Lebesgue multiplicity N φ (N) [203, 211]. In an appendix, we suggest an extension to automatic sequences over a compact nondiscrete alphabet. We give conditions ensuring strict ergodicity of the orbit closure. As explained before, we preferred to develop topics involving spectral properties of measures and for this reason, the reader will not find in this study a complete survey of substitutions. A lot of relevant contributions have been ignored or perhaps forgotten : we apologize the mathematicians concerned. Paris July 1987

Martine Queff´elec

Contents

1

The Banach Algebra M(T) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 1 1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 1 1.2 The Gelfand Spectrum Δ of M(T) . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 7 1.2.1 Generalized Characters .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 7 1.2.2 Basic Operations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 8 1.2.3 Topologies on Δ (μ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 8 1.2.4 Constants in Δ (μ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 9 1.3 Generalized Riesz Products .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 10 1.4 Idempotents in Δ and Decompositions of M(T) . . . . . . . . . . .. . . . . . . . . . . 14 1.5 Dirichlet Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 17

2

Spectral Theory of Unitary Operators . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.1 Representation Theorem of Unitary Operators . . . . . . . . . . . .. . . . . . . . . . . 2.1.1 Construction of Spectral Measures . . . . . . . . . . . . . . .. . . . . . . . . . . 2.1.2 Properties of the Spectral Family . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.1.3 Spectral Representation Theorem .. . . . . . . . . . . . . . . .. . . . . . . . . . . 2.1.4 Invariant Subspaces and Spectral Projectors . . . . .. . . . . . . . . . . 2.2 Operators with Simple Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.2.1 Basic Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.2.2 Simple Lebesgue Spectrum.. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.3 Spectral Decomposition Theorem and Maximal Spectral Type .. . . . . 2.4 Spectral Decomposition Theorem and Spectral Multiplicity.. . . . . . . . 2.4.1 Multiplicity Function .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.4.2 Global Multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.5 Eigenvalues and Discrete Spectrum . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.5.1 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.5.2 Discrete Spectral Measures . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.5.3 Basic Examples .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2.6 Application to Ergodic Sequences .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .

3

Spectral Theory of Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 49 3.1 Notations and Definitions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 49

21 21 21 22 25 26 29 29 30 31 34 35 37 40 40 41 43 44

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3.2

3.3

3.4

3.5

3.6

Ergodic Dynamical Systems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.2.1 First Examples .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.2.2 Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.2.3 Quasi-Invariant Systems. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . Mixing Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.3.1 Weak and Strong Mixing Properties .. . . . . . . . . . . . .. . . . . . . . . . . 3.3.2 Mild Mixing Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.3.3 Multiple Mixing Properties . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . Discrete Ergodic Systems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.4.1 Von Neumann’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.4.2 The Kronecker Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . Purity Law and D-Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.5.1 Extremal Properties of Ergodic Probabilities . . . .. . . . . . . . . . . 3.5.2 D-Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.5.3 Applications of Purity Laws . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . Group Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.6.1 Two-Points Extensions of an Irrational Rotation.. . . . . . . . . . . 3.6.2 Two-Points Extensions of an Odometer . . . . . . . . . .. . . . . . . . . . .

51 51 53 56 59 59 62 66 69 69 71 73 74 75 77 80 81 85

4

Dynamical Systems Associated with Sequences . . . . . . . . . . . . . . . .. . . . . . . . . . . 87 4.1 Topological Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 87 4.1.1 Minimality and Topological Transitivity . . . . . . . . .. . . . . . . . . . . 87 4.1.2 Invariant Measures and Unique Ergodicity .. . . . . .. . . . . . . . . . . 90 4.1.3 Examples and Application to Asymptotic Distribution .. . . . 93 4.2 Dynamical Systems Associated with Finitely Valued Sequences .. . . 97 4.2.1 Subshifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 97 4.2.2 Minimality and Unique Ergodicity . . . . . . . . . . . . . . .. . . . . . . . . . . 98 4.2.3 Examples .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .101 4.3 Spectral Properties of Bounded Sequences . . . . . . . . . . . . . . . .. . . . . . . . . . .102 4.3.1 Correlation Measures .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .103 4.3.2 Applications of Correlation Measures .. . . . . . . . . . .. . . . . . . . . . .108 4.3.3 Examples of Correlation Measures . . . . . . . . . . . . . . .. . . . . . . . . . .120 4.3.4 Back to Finite-Valued Sequences . . . . . . . . . . . . . . . . .. . . . . . . . . . .123

5

Dynamical Systems Arising from Substitutions . . . . . . . . . . . . . . . .. . . . . . . . . . .125 5.1 Definitions and Notations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .125 5.2 Minimality of Primitive Substitutions .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .128 5.3 Nonnegative Matrices and ζ -Matrix . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .131 5.3.1 Perron Frobenius’ Theorem for Nonnegative Matrices .. . . .132 5.3.2 Frequency of Letters .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .135 5.3.3 Perron Numbers, Pisot Numbers .. . . . . . . . . . . . . . . . .. . . . . . . . . . .136 5.4 Unique Ergodicity .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .137 5.4.1 Frequency of Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .137 5.4.2 The Unique Invariant Measure .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . .140 5.4.3 Matrices M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .143

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5.5

Combinatorial Aspects .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .146 5.5.1 Complexity Function and Topological Entropy ... . . . . . . . . . .146 5.5.2 Recognizability Properties . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .150 Structure of Substitution Dynamical Systems . . . . . . . . . . . . .. . . . . . . . . . .152 5.6.1 Consequences of the Recognizability.. . . . . . . . . . . .. . . . . . . . . . .152 5.6.2 ζ (X)-Induced Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .154 5.6.3 Metric Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .156 5.6.4 Bilateral Substitution Sequences .. . . . . . . . . . . . . . . . .. . . . . . . . . . .159

5.6

6

Eigenvalues of Substitution Dynamical Systems . . . . . . . . . . . . . . .. . . . . . . . . . .161 6.1 Eigenvalues of a Constant-Length Substitution .. . . . . . . . . . .. . . . . . . . . . .161 6.1.1 Continuous Eigenvalues .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .161 6.1.2 L2 -Eigenvalues .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .164 6.2 Eigenvalues of a Nonconstant-Length Substitution.. . . . . . .. . . . . . . . . . .169 6.2.1 Host’s Theorem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .169 6.2.2 Application to Mixing Property .. . . . . . . . . . . . . . . . . .. . . . . . . . . . .174 6.3 Pure Point Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .176 6.3.1 Discrete Constant-Length Substitutions .. . . . . . . . .. . . . . . . . . . .176 6.3.2 Discrete Nonconstant-Length Substitutions .. . . . .. . . . . . . . . . .180 6.3.3 Pisot Substitutions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .187

7

Matrices of Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .193 7.1 Correlation Matrix.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .193 7.1.1 Spectral Multiplicity .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .193 7.1.2 Maximal Spectral Type .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .195 7.1.3 The Correlation Matrix Σ . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .197 7.2 Positive Definite Matrices of Measures . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .199 7.2.1 Definitions and Properties .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .199 7.2.2 Decompositions of Matrices of Measures . . . . . . . .. . . . . . . . . . .202 7.3 Characters on a Matrix of Measures.. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .204

8

Matrix Riesz Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .209 8.1 Σ as a Matrix Riesz Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .209 8.2 Examples of Maximal Spectral Types .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .211 8.2.1 Thue-Morse Sequence .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .211 8.2.2 Rudin-Shapiro Sequence . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .212 8.2.3 Q-mirror Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .215 8.3 Commutative Automata .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .216 8.4 Automatic Sums .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .220

9

Bijective Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .225 9.1 Structure of Bijective Substitution Systems. . . . . . . . . . . . . . . .. . . . . . . . . . .226 9.1.1 Extension of the q-Odometer . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .226 9.1.2 Group Automaton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .230

xiv

Contents

9.2

Spectral Study of Bijective Substitutions . . . . . . . . . . . . . . . . . .. . . . . . . . . . .232 9.2.1 Abelian Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .233 9.2.2 Non-Abelian Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .235

10 Maximal Spectral Type of General Automata .. . . . . . . . . . . . . . . . .. . . . . . . . . . .243 10.1 The Coincidence Matrix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .243 10.1.1 Properties of C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .244 10.1.2 Examples .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .247 10.2 The Projection Operator P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .248 10.2.1 A Property of P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .250 10.2.2 Examples .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .252 10.3 The Bi-Correlation Matrix Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .254 10.4 Main Theorem on Maximal Spectral Type .. . . . . . . . . . . . . . . .. . . . . . . . . . .259 11 Spectral Multiplicity of General Automata .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .265 11.1 More About the Spectrum of Constant-Length Substitutions .. . . . . . .265 11.1.1 The Convex Set K .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .265 11.1.2 Examples .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .268 11.2 Spectral Multiplicity of Constant-Length Substitutions . . .. . . . . . . . . . .271 11.2.1 Main Theorem on Spectral Multiplicity . . . . . . . . .. . . . . . . . . . .272 11.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .275 11.2.3 More About Lebesgue Multiplicity . . . . . . . . . . . . . .. . . . . . . . . . .279 12 Compact Automata .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .281 12.1 Strictly Ergodic Automatic Flows . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .281 12.2 Application to Bounded Remainder Sets. . . . . . . . . . . . . . . . . . .. . . . . . . . . . .286 A

Schr¨odinger Operators with Substitutive Potential.. . . . . . . . . . .. . . . . . . . . . .293 A.1 Classical Facts on 1D Discrete Schr¨odinger Operators.. . .. . . . . . . . . . .293 A.1.1 Preliminaries .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .293 A.1.2 Schr¨odinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .295 A.1.3 Periodic Potential .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .299 A.2 Ergodic Family of Schr¨odinger Operators . . . . . . . . . . . . . . . . .. . . . . . . . . . .302 A.2.1 General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .303 A.2.2 Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .305 A.2.3 Results from Pastur, Kotani, Last and Simon . . . .. . . . . . . . . . .307 A.3 Substitutive Schr¨odinger Operators . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .308 A.3.1 The Trace Map Method . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .310 A.3.2 The Palindromic Density Method.. . . . . . . . . . . . . . . .. . . . . . . . . . .315

B

Substitutive Continued Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .319 B.1 Overview on Continued Fraction Expansions . . . . . . . . . . . . .. . . . . . . . . . .319 B.1.1 The Gauss Dynamical System . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .321 B.1.2 Diophantine Approximation and BAD . . . . . . . . . . .. . . . . . . . . . .322

Contents

B.2

B.3

xv

Morphic Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .324 B.2.1 Schmidt’s Theorem on Non-Quadratic Numbers . . . . . . . . . . .324 B.2.2 The Thue-Morse Continued Fraction .. . . . . . . . . . . .. . . . . . . . . . .328 Schmidt Subspace Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .331 B.3.1 Transcendence and Repetitions . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .332 B.3.2 Transcendence and Palindromes .. . . . . . . . . . . . . . . . .. . . . . . . . . . .333

References .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .335 Glossary . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .345 Index . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .347

Chapter 1

The Banach Algebra M(T)

This first chapter is devoted to the study of the Banach algebra M(T). This study will be brief because we need only little about M(T), and there exist excellent books on the subject, in which all the proofs will be found [123, 141, 218, 232]. We introduce the technics of generalized characters to precise the spectral properties of measures such as generalized Riesz products, which will nicely appear later as maximal spectral type of certain dynamical systems.

1.1 Basic Definitions We consider U the multiplicative compact group of complex numbers of modulus one, and T = R\2π Z that we identify with U by the map λ → eiλ ; T is equipped with the Haar measure m, identified this way with the normalized Lebesgue measure 1 2π dx on [−π , π ]. ˆ isomorphic to Z, will be considered 1. The elements of the character group Γ = T, sometimes as integers, with addition, sometimes as multiplicative functions on T, and, in this case, we denote by γn instead of n the element t → eint . 2. M(T) is the algebra of the regular Borel complex measures on T, equipped with the convolution product of measures, defined by

μ ∗ ν (E) =

 T

μ (E − t) d ν (t)

for μ , ν ∈ M(T) and E any measurable subset of T. M(T) is a Banach algebra for the norm ||μ || =



d|μ |,

|μ | being the total variation of μ .

M. Queff´elec, Substitution Dynamical Systems – Spectral Analysis: Second Edition, Lecture Notes in Mathematics 1294, DOI 10.1007/978-3-642-11212-6 1, c Springer-Verlag Berlin Heidelberg 2010 

1

2

1 The Banach Algebra M(T)

The Fourier coefficients of μ ∈ M(T) are, by definition,

μˆ (n) =

 T

eint d μ (t) =



γn d μ , n ∈ Z

and satisfy : ||μˆ ||∞ := supn∈Z |μˆ (n)| ≤ ||μ ||. The Fourier spectrum of μ is the set of integers n ∈ Z for which μˆ (n) = 0. 3. The measure μ is positive if μ (E) ≥ 0 for every measurable set E, and, in this case, the sequence (μˆ (n)) is positive definite, namely

∑

zi z j μˆ (i − j) ≥ 0

1≤i, j≤n

for any finite complex sequence (zi )1≤i≤n . Conversely, the Bochner theorem asserts that a positive definite sequence (an )n∈Z is the Fourier transform of a positive measure on T. Positive measures of total mass one are probability measures. 4. We recall that μ is a discrete measure if μ = ∑ a j δt j , (δt being the unit mass at t ∈ T) and that μ is a continuous measure if μ {t} = 0 for all t ∈ T. Md (T) is the sub-algebra of discrete measures in M(T) and Mc (T), the convolution-ideal of all continuous measures on T. Every μ ∈ M(T) can be uniquely decomposed into a sum μ = μd + μc where μd ∈ Md (T) and μc ∈ Mc (T) respectively are the discrete part and the continuous part of μ . There is a necessary and sufficient condition for a measure μ to be continuous, which involves the Fourier transform of μ : Lemma 1.1 (Wiener). Let μ ∈ M(T). Then : N 1 |μˆ (n)|2 = 0 ∑ N→∞ 2N + 1 n=−N

μ ∈ Mc (T) ⇐⇒ lim and in this case, we have limN→∞

1 2N+1

ˆ (n)|2 = 0 uniformly in K. ∑N+K n=−N+K | μ

5. Let μ , ν ∈ M(T); we say that μ is absolutely continuous with respect to ν and we write μ  ν if |μ |(E) = 0 as soon as |ν |(E) = 0, for any measurable set E. Then, by the Radon-Nikodym property, μ = f · ν where f ∈ L1 (ν ) is referred to as the density of μ with respect to ν , usually denoted by d μ /d ν . Let us define L(ν ) = {μ ∈ M(T); μ  ν }

(1.1)

So we are allowed to identify L(ν ) with L1 (ν ). The measures μ , ν are said to be equivalent, and we write μ ∼ ν , if μ  ν and ν  μ . In the opposite direction, we say that μ and ν are mutually singular, and we write μ ⊥ ν , if there exists a measurable set E such that

1.1 Basic Definitions

3

|μ (E)| = ||μ || and |ν |(E) = 0. The measure μ is said to be singular if μ ⊥ m, m the Haar measure on T. Every μ ∈ M(T) can be uniquely decomposed into a sum

μ = μa + μs where μa  m and μs is singular (respectively the absolutely continuous part and the singular part of μ ). Affinity between two measures. Let μ and ν be two positive measures in M(T) and λ ∈ M(T) be such that both μ  λ and ν  λ . The affinity ρ (μ , ν ) of the measures μ and ν is the quantity

ρ (μ , ν ) =

  d μ 1/2  d ν 1/2 T

dλ

dλ

dλ ,

(1.2)

obviously independent of the choice of λ . Note that

ρ (μ , ν ) = 0 if and only if μ ⊥ ν . The measure μ is said to have independent powers if μ n ⊥ μ m whenever n = m, n, m ∈ N, where μ n := μ ∗ · · · ∗ μ n times. Such a measure is singular : if not, the absolutely continuous part ν of μ satisfies 0 = ν n  μ n for all n ∈ N; thus, if m ≥ n, ν m  μ m , ν m  ν n  μ n and μ m ⊥ μ n . It is less obvious to give conditions on the Fourier transform of μ , ensuring the absolute continuity of μ (with respect to m). Of course, it is necessary for μ to satisfy : lim|n|→∞ μˆ (n) = 0. This condition is not sufficient and we shall use the notation M0 (T) for the ideal of all measures μ whose Fourier transform vanishes at infinity. Sometimes, those measures are called Rajchman measures, and a nice survey on them appears in [172]. 6. M(T) is identified with the dual space C(T)∗ of the continuous functions on T. Let (μn ) and μ in M(T). From Fejer’s theorem, μn converges to μ in the weakstar topology of M(T), σ (M(T),C(T)), if and only if

μˆ n (γ ) → μˆ (γ ) for every γ ∈ Γ . We shall write : w∗ − limn→∞ μn = μ . Recall that the unit ball of M(T) is a weakstar compact set. The following proposition will be used in chapter 4 (see [59]). Proposition 1.1. Let (μn ) and (νn ) be two sequences of positive measures on T such that μn → μ and νn → ν in the weak-star topology of M(T). Then lim sup ρ (μn , νn ) ≤ ρ (μ , ν ), n

where ρ denotes the affinity defined in (1.2)

4

1 The Banach Algebra M(T)

Proof. The proof involves the Cauchy-Schwarz inequality applied with some suitable partition of unity. We assume, without loss of generality, that μ and ν are probability measures and we fix a probability measure λ dominating both μ and ν . We put M0 = { dd λμ = 0}, N0 = { ddλν = 0}\M0 and we consider for j ∈ Z and some fixed ε > 0 U j = {x ∈ T\(M0 ∪ N0 ), (1 + ε ) j

dμ dμ dν (x) < (x) ≤ (1 + ε ) j+1 (x)}. (1.3) dλ dλ dλ

Clearly, the sequence (U j ), supplemented by M0 and N0 , provides an infinite partition of T. In particular, ∑ j μ (U j ) < ∞ and we fix J such that

∑

μ (U j ) ≤ ε 2 .

(1.4)

| j|≥J

From now on, we denote by V0 ,V1 ,V2 , . . . ,V2J ,V2J+1 the finite Borel partition M0 , N0 ,U−J+1 , . . . ,UJ−1 , ∪| j|≥J U j of T. Note that, for every 2 ≤ j ≤ 2J, (1 + ε ) j−1−J μ (V j ) ≤ ν (V j ) ≤ (1 + ε ) j−J μ (V j ),

(1.5)

by integrating the inequalities (1.3) on U j with respect to λ . For each j, 0 ≤ j ≤ 2J + 1, let us choose by regularity an open set ω j ⊃ V j such that

μ (ω j ) ≤ (1 + ε )1/2 μ (V j ), ν (ω j ) ≤ (1 + ε )1/2ν (V j ); let ( f j )0≤ j≤2J+1 be a continuous partition of unity subordinate to the open covering (ω j )0≤ j≤2J+1 . Clearly we have  T



as well as

T

f j d μ ≤ μ (ω j ) ≤ (1 + ε )1/2 μ (V j )

(1.6)

f j d ν ≤ ν (ω j ) ≤ (1 + ε )1/2ν (V j ).

(1.7)

We deduce that ρ (μn , νn ) :=  T

(

2J+1 d μn 1/2 d νn 1/2 ) ( ) dλ = ∑ dλ dλ j=0

≤

2J+1

 T



∑(

j=0

(fj

d μn 1/2 d νn 1/2 ) (fj ) dλ dλ dλ 

T

f j d μn )1/2 (

T

f j d νn )1/2

1.1 Basic Definitions

5

by the Cauchy-Schwarz inequality, and the weak* convergence hypothesis, combined with (1.6) and (1.7), leads to lim sup ρ (μn , νn ) ≤ lim

2J+1

n

n

=

2J+1

∑

j=0



∑(

j=0



(

T



T

f j d μn )1/2 ( 

f j d μ )1/2 (

≤ (1 + ε )1/2

2J+1 

∑

T

T

f j d νn )1/2

f j d ν )1/2

μ (V j )ν (V j ).

j=0

We are left to compare this latter term with the affinity ρ (μ , ν ). Actually, since V0 = M0 and V1 = N0 , and by the choice of J in (1.4), we have 2J+1 

∑

μ (V j )ν (V j ) ≤

j=0



2J

∑

μ (V j )ν (V j ) + ε .

j=2

Now, from (1.5) and (1.3), 2J

∑



μ (V j )ν (V j ) ≤

j=2

2J

∑ (1 + ε )( j−J)/2 μ (V j )

j=2

2J

≤ (1 + ε )1/2 ∑



j=2 V j

(

d μ 1/2 d ν 1/2 ) ( ) dλ dλ dλ

≤ (1 + ε )1/2ρ (μ , ν ). Finally,

lim sup ρ (μn , νn ) ≤ (1 + ε )(ρ (μ , ν ) + ε ) n

and the result follows by letting ε tend to zero.

 

A very important example of measure is given as a weak-star limit point of some sequence of absolutely continuous measures. Fundamental Example (Classical Riesz products). Consider a sequence (a j ) of real numbers of modulus ≤ 1; the polynomial 1 + a j cos(3 j t) is non-negative for every t ∈ R, hence PN (t) =

N−1

∏ (1 + a j cos(3 j t)) ≥ 0,

for every N ≥ 1

j=0

j  and ||PN ||L1 = P N (0). The sequence (3 ) is dissociate i.e. every integer n has at most one representation of the form

n = ∑ ε j3 j,

with ε j = −1, 0, 1.

6

1 The Banach Algebra M(T)

This is due to the following fact : if η j ∈ {0, ±1, ±2}, | ± 3n + ηn−1 3n−1 + · · · + η1 3 + η0| ≥ 3n − 2(3n−1 + · · · + 3 + 1) = 1. It follows that

 ||PN ||L1 = P N (0) = 1,    P / ∑ ε j 3 j , ε j = −1, 0, 1 N (n) = 0 if n ∈ j ε },

ε > 0 and μ in the ideal.

Chapter 2

Spectral Theory of Unitary Operators

We wish to classify isometric operators on Hilbert spaces, up to unitary equivalence (or spectral equivalence). We introduce for this purpose different notions of the spectral theory of unitary operators, such as : spectral measure, maximal spectral type, spectral multiplicity, multiplicity function, etc.; we establish two versions of the spectral decomposition theorem for these operators, with our familiar notations. The definitions and results will be used in the next chapter where we focus on dynamical systems, and later, when we study the spectral properties of substitutive sequences.

2.1 Representation Theorem of Unitary Operators Let U be a unitary operator on the separable Hilbert space H, endowed with the inner product ·, ·. In chapter 3, H will be L2 (X, B, μ ) and U = UT , the unitary operator associated to an automorphism of the probability space (X, B, μ ), defined by UT f (x) = f (T x) if f ∈ L2 (X, B, μ ). Recall that the spectrum of an operator A on H is the set sp(A) of complex numbers z such that A − zI is not invertible. If A = U is a unitary operator, sp(U) is a compact subset of the circle. But we need more to distinguish unitary operators.

2.1.1 Construction of Spectral Measures For each f ∈ H, the sequence (tn )n∈Z where tn = U n f , f  is positive definite since ∑i, j zi z j ti− j = ∑i, j zi z j U i− j f , f  = ∑i, j ziU i f , z j U j f  = || ∑i ziU i f ||2H ≥ 0.

M. Queff´elec, Substitution Dynamical Systems – Spectral Analysis: Second Edition, Lecture Notes in Mathematics 1294, DOI 10.1007/978-3-642-11212-6 2, c Springer-Verlag Berlin Heidelberg 2010 

21

22

2 Spectral Theory of Unitary Operators

By the Bochner theorem (section 1.1 §3), we can associate to the element f ∈ H a positive measure on T, denoted by σ f , that we call the spectral measure of f ; σ f is characterized by its Fourier coefficients

σ f (n) = U n f , f , and its total mass is :

n ∈ Z,

||σ f || = || f ||2H .

(2.1)

Now, if f and g are two elements of H, we consider an = U n f , g,

n ∈ Z,

so that U n g, f  = a−n . The elementary identity 4U n f , g = U n ( f + g), f + g − U n( f − g), f − g + iU n( f + ig), f + ig − iU n( f − ig), f − ig

(2.2)

proves that (an ) is the Fourier transform of a complex measure on T : an = U n f , g =: σ f ,g (n),

n ∈ Z,

where, by (2.2), 1 σ f ,g = (σ f +g − σ f −g + iσ f +ig − iσ f −ig ). 4 We have σ f , f = σ f ,

σ f ,g = σg, f and σ f +g = σ f + σg + σ f ,g + σg, f

(2.3)

Definition 2.1. The family of measures (σ f ,g ) f ,g∈H is referred to as the spectral family of the operator U.

2.1.2 Properties of the Spectral Family With the help of the spectral family, we shall give an integral representation of the operator U. Definition 2.2. If f ∈ H, we write [U, f ] for the cyclic subspace generated by f , which is the closure in H of the linear span of {U n f , n ∈ Z}. More generally, [U, f1 , . . . , fk ] will denote the cyclic subspace generated by f1 , . . . , fk ∈ H.

2.1 Representation Theorem of Unitary Operators

23

We have the simple but fundamental following propositions. Proposition 2.1. Let R be a trigonometric polynomial on T, then ||R(U) f ||H = ||R||L2 (σ f ) .

(2.4)

ikt , then ˆ Proof. If R(t) = ∑k R(k)e k ˆ ˆ j)U j f  f , ∑ R( ||R(U) f ||2H = ∑ R(k)U j

k

ˆ R( ˆ j)U k− j f , f  = ∑ R(k) j,k

=

ˆ R( ˆ j)σˆ f (k − j) ∑ R(k) j,k

= =

  T



T

ˆ R( ˆ j)eikt e−i jt ∑ R(k)



d σ f (t)

j,k

|R(t)|2 d σ f (t).

 

Proposition 2.2. Let R be a trigonometric polynomial on T, then

σR(U) f ,g = R · σ f ,g

and σR(U) f = |R|2 σ f .

(2.5)

Proof. The second assertion is a direct consequence of the first one, and for this first identity, it suffices to write

σR(U) f ,g (k) = U k R(U) f , g = = (R · σ f ,g )(k).

 T

eikt R(t) d σ f ,g (t)  

Proposition 2.3. The map ( f , g) → σ f ,g from H × H into M(T) is bilinear and continuous. Proof. We shall prove the continuity by showing the inequality ||σ f ,g || ≤ || f ||H ||g||H , the bilinearity being obvious by construction. For a trigonometric polynomial R on T, we have,  T

R(t) d σ f ,g (t) = |R(U) f , g| ≤ ||R(U) f ||H ||g||H = ||R||L2 (σ f ) ||g||H

by (2.5) by (2.4)

≤ ||R||∞ || f ||H ||g||H since ||σ f || = || f ||2H . From this inequality, we derive the claimed one by taking the sup on the trigonometric polynomials R with norm ||R||∞ ≤ 1.  

24

2 Spectral Theory of Unitary Operators

Proposition 2.4. For all f , g ∈ H, the measure σ f ,g is absolutely continuous with respect to σ f and σg ; more precisely,   σ f (B) σg (B)

|σ f ,g |(B) ≤ for every Borel set B in T.

Proof. Fix a Borel set B ⊂ T; applying the Schwarz inequality to the positive bilinear form ( f , g) → σ f ,g (B), we obtain |σ f ,g (B)| ≤

  σ f (B) σg (B).

Now, by definition of the total variation measure |σ f ,g |, for any fixed ε > 0 there exists a partition (Bn ) of B such that

∑ |σ f ,g (Bn )| ≥ |σ f ,g |(B) − ε . n

One more application of the Schwarz inequality leads to ∑n |σ f ,g (Bn )| ≤ ∑n

  σ f (Bn ) σg (Bn )

≤ (∑n σ f (Bn ))1/2 (∑n σg (Bn ))1/2 , so that, finally, |σ f ,g |(B) ≤

  σ f (B) σg (B) + ε , and the result follows.

 

Remark 2.1. Actually, this property is equivalent to the previous one. But it can be slightly improved in the following way : If both of σ f and σg are absolutely continuous with respect to a same positive measure ω , so is σ f ,g , and we have  d σ f ,g  ≤  dω



dσ f dω



d σg . dω

As a consequence of (2.3), note that we have Corollary 2.1. For f , g ∈ H,

σ f +g  σ f + σg . Also Corollary 2.2. If ( fn ) converges to f in H, σ fn converges to σ f in M(T). More precisely, we have the inequalities ||σ fn − σ f || ≤ ||σ fn − f || + 2||σ f , fn− f || ≤ || f − fn ||2H + 2|| f ||H || fn − f ||H .

2.1 Representation Theorem of Unitary Operators

25

2.1.3 Spectral Representation Theorem Identity (2.4) defines, by extension, an isometry W from [U, f ] onto L2 (σ f ) with the following properties : W f = 1 and, if h ∈ [U, f ] then (i) ||h||H = ||W h||L2 (σ f ) (ii)

eit W h(t) = WUh(t)

(2.6)

In other words Theorem 2.1. There exists a unitary equivalence W between U restricted to the cyclic subspace [U, f ] and V , the operator of multiplication by eit on L2 (σ f ), so that the following diagram commutes : [U, f ]

W

U

 [U, f ]

/ L2 (σ f ) V

W

 / L2 (σ f )

This theorem has many consequences. Note first the following : Corollary 2.3. If f , g ∈ H, then U

if σ f ∼ σg .

[U, f ]

is unitarily equivalent to U

[U,g]

if and only

Also, we can extend identity (2.5) to the cyclic subspace [U, f ]. Corollary 2.4. Let f ∈ H; then h ∈ [U, f ] if and only if σh = |φ |2 σ f for some φ ∈ L2 (σ f ). Proof. By using the above theorem and (2.6), we get for h ∈ [U, f ] and g ∈ H

σh,g (n) = U n h, gH = WU n h,W gL2 (σ f ) = eint W h,W gL2 (σ f ) =

= W h W g σ f (n)



T

eint W h W g d σ f

so that σh,g = W h W g σ f ; in particular σh = |W h|2 σ f . Conversely, if σ  σ f , we may write σ = |φ |2 σ f with φ ∈ L2 (σ f ); since W is onto, there exists h ∈ [U, f ] such that φ = W h, and σ = σh by reverse calculations.   Note that when φ is a trigonometric polynomial R, h = R(U) f by (2.5). In case φ ∈ L2 (σ f ), we denote this element h by φ (U) f and the corollary becomes :

26

2 Spectral Theory of Unitary Operators

Corollary 2.5. For every f ∈ H and φ ∈ L2 (σ f ), we can define an element φ (U) f ∈ [U, f ], satisfying : σφ (U) f ,g = φ σ f ,g , ∀g ∈ H. (2.7) In particular, σφ (U) f = |φ |2 σ f . We are now in a position to extend the functional calculus to L∞ (T). Proposition 2.5. If φ is a bounded Borel function on T, the map f ∈ H → φ (U) f defines a linear operator on H, bounded by ||φ ||∞ . Proof. When φ ∈ L∞ (T), according to the previous corollary, φ (U) f is quite well defined for each f ∈ H; as a consequence of (2.7) and thanks to the bilinearity property of σ f ,g , we check that

σφ (U)( f +g)−φ (U) f −φ (U)g = |φ |2 (σ f +g + σ f + σg − σ f +g, f − σ f , f +g − σ f +g,g − σg, f +g + σ f ,g + σg, f ) = 0 so that ||φ (U)( f + g) − φ (U) f − φ (U)g||2H = ||σφ (U)( f +g)−φ (U) f −φ (U)g || = 0 In the same way, ||φ (U)(λ f ) − λ φ (U) f ||H = ||σφ (U)(λ f )−λ φ (U) f || = 0, and φ (U) is linear. Finally, ||φ (U) f ||2H = ||σφ (U) f || = |||φ |2 σ f || ≤ ||φ ||2∞ || f ||2H  

gives the claimed bound.

We have proved, in passing, the so-called spectral representation theorem for unitary operators, namely : Theorem 2.2 (Spectral representation theorem). Let U be a unitary operator on the separable Hilbert space H. Then there exists a family of measures on T, (σ f ,g ) f ,g∈H , such that, for every bounded Borel function ψ on T, we have ψ (U) f , g =



ψ d σ f ,g .

(2.8)

2.1.4 Invariant Subspaces and Spectral Projectors If the subspace H0 of H is invariant under U, so is H0⊥ , and, if P0 is the orthogonal projection onto H0 , then P0U = UP0 . We would like to identify the operators

2.1 Representation Theorem of Unitary Operators

27

commuting with U, and, in view of theorem 2.1, it is natural to begin with the description of operators commuting with a multiplication operator. Proposition 2.6. Let σ be a probability measure on T, and V the operator of multiplication by eit on L2 (σ ). Then 1) The bounded operator Q commutes with V on L2 (σ ) if and only if Q is the operator of multiplication by some real function ϕ ∈ L∞ (σ ). 2) The subspace M of L2 (σ ) is invariant under V if and only if M = 1B L2 (σ ), for some Borel set B in T and 1B its indicator function. Proof. The assertion 2) can be derived from 1) : if PM is the orthogonal projection 2 2 onto M, PM f = ϕ f with ϕ ∈ L∞ R (σ ) by 1). But PM = PM and ϕ = ϕ whence ϕ = 1B for some Borel set B. Let us prove the first assertion. By assumption, QV n = V n Q so that, putting ϕ = Q1 ∈ L2 (σ ), we have eint ϕ (t) = Q(eint ) for every n ≥ 0. This identity extends readily to the trigonometric polynomials P : Q(P(t)) = P(t)ϕ (t)

(2.9)

We must show that ϕ ∈ L∞ (σ ). If ϕ is not zero, consider a > 0 such that the set B = {|ϕ | ≥ a} has positive measure. Integrating (2.9) on B, we get  B

|Pϕ |2 d σ =

 B

|Q(P)|2 d σ ≤ C2 ||P||2L2 (σ )

if C := ||Q|| > 0. Hence a||1BP||L2 (σ ) ≤ C||P||L2 (σ ) for every trigonometric polynomial P; in particular for Pn where Pn tends to 1B in L2 (σ ). Taking the limit on n in the inequality a||1B Pn ||L2 (σ ) ≤ C||Pn ||L2 (σ ) , we deduce that a ≤ C and ϕ ∈ L∞ (σ ).   This proposition gives an answer to the commutator problem when the space H is cyclic under U. In the general case, we shall need a decomposition of H into cyclic subspaces, which is the purpose of the next section. A first reduction arises from the spectral projectors : Proposition 2.7. Let B be a Borel set in T; we denote by 1B (U) the associated operator (proposition 2.5). Then, 1B (U) is the orthogonal projection onto the following subspace HB of H : HB = { f ∈ H, σ f (Bc ) = 0}. These projectors are called spectral projectors and, from now on, denoted by EB , B ∈ BT .

28

2 Spectral Theory of Unitary Operators

Proof. Thanks to the absolute continuity relation σ f +g  σ f + σg , HB is clearly a subspace of H. It is closed in H since ||σ fn − σ f || tends to zero as soon as fn tends to f in H (corollary 2.2). If we put P = EB , we have ||P|| ≤ 1 (proposition 2.5) and it remains to prove that P2 = P. By (2.7), for f , g ∈ H,

σP f ,g = 1B σ f ,g and σP f ,g is supported on B. This implies P(P f ), g = (1B σP f ,g )(0) = (1B σ f ,g )(0) = σ f ,g (B) = P f , g and P2 = P. Thus P is the orthogonal projection onto HB .

(2.10) (2.11) (2.12)  

The following corollary will be useful later: Corollary 2.6. Let h ∈ H and σh = μ + ν be a decomposition of σ f into mutually singular positive measures. Then there exist f and g in [U, h] such that σ f = μ and σg = ν . Proof. By mutual singularity of μ and ν , there exists a Borel set B ⊂ T such that 1B μ = μ and 1Bc ν = ν . If we set f = EB h and g = EBc h, then f , g ∈ [U, h] with

σ f = 1B σh = 1B (μ + ν ) = μ , and

σg = 1Bc σh = 1Bc (μ + ν ) = ν .  

Turning back to our problem, we can show : Proposition 2.8. Q commutes with U if and only if Q commutes with all the spectral projectors. Proof. If f , g ∈ H and if B is a Borel set in T, QEB f , g = EB f , Q∗ g = σ f ,Q∗ g (B) while

EB Q f , g = σQ f ,g (B).

But

σQ f ,g (n) = U n Q f , g = QU n f , g = σ f ,Q∗ g (n) for all n, f , g if and only if Q commutes with U. This gives the result.

 

2.2 Operators with Simple Spectrum

29

2.2 Operators with Simple Spectrum Before establishing the spectral decomposition theorem in the general case, we dwell upon the cyclic case, from the operator point of view. Definition 2.3. The unitary operator U is said to have simple spectrum if there exists h ∈ H such that [U, h] = H. In this case, H is isometrically isomorphic to L2 (σh ) and U is unitarily equivalent to the multiplication operator V defined on L2 (σh ) by (V φ )(t) = eit φ (t) (theorem 2.1). We begin with the following general remark, obvious from σ f ,g (n) = U n f , g : Lemma 2.1. For f , g ∈ H,

σ f ,g = 0 ⇐⇒ [U, f ] ⊥ [U, g].

(2.13)

and we say, in this case, that f and g are U-orthogonal. So we always have : σ f ⊥ σg =⇒ σ f ,g = 0 ⇐⇒ [U, f ] ⊥ [U, g]; but the reverse implication may be false and this is related to multiplicity (see section 2.4).

2.2.1

Basic Examples

1. Consider H = L2 (T, m) =: L2 (T) and U the unitary operator associated to the rotation Rθ : x → x + θ mod 2π , θ /π irrational. We claim that U has simple spectrum. In other words, we can exhibit a function f ∈ H, for which the linear span of Rnθ f , n ∈ Z, is dense in H. It is well-known that the candidates are exactly those functions f whose Fourier transform never vanishes. Let us sketch the proof rapidly : suppose that φ ∈ H is orthogonal to the Rnθ f , n ∈ Z; this means that (φ ∗ fˇ)(nθ ) = 0

∀n ∈ Z,

where fˇ(x) = f (−x). By continuity of the L2 ∗L2 - functions, necessarily φ ∗ fˇ = 0 and φˆ · fˆ = 0. This implies φ = 0 unless fˆ vanishes somewhere, whence the claim. (We shall construct explicitly the isometry W between H and L2 (σ f ) in subsection 2.5.2) 2. Consider now H = L2 ([0, 1]) and V the pointwise multiplication by φ , where φ is any continuous, one-to-one and unimodular function on [0, 1]. Once more, V has simple spectrum : indeed, the sub-algebra of C([0, 1]) generated by φ is dense theorem), so that H = [V, f ] where because φ is one-to-one (Stone-Weierstrass  f = 1. We deduce that σˆ f (n) = φ n dm, or equivalently, σ f is the pull-back of m under φ . 3. Let us go back to the previous example, but with φ (x) = e2ix ; φ is π -periodic. A new application of the Stone-Weierstrass theorem shows that [V, 1[0,π ] ] =

30

2 Spectral Theory of Unitary Operators

L2 ([0, π ], m) and [V, 1[π ,2π ] ] = L2 ([π , 2π ], m); hence H admits the decomposition H = [V, f ]⊕ [V, g] where f and g are V -orthogonal functions with σ f = σg = m/2. V has no longer simple spectrum and we say that the multiplicity of V is equal to 2 (see next subsection 2.4.2). The following characterization is in fact a consequence of a result which will be proved later. Proposition 2.9. The following properties are equivalent : (a) U has simple spectrum. (b) For every f , g ∈ H, σ f ,g = 0 =⇒ σ f ⊥ σg . Proof. Suppose that (a) holds and H = [U, h]. If W denotes the natural isometry of H onto L2 (σh ), we have

σ f = |W f |2 σh , σg = |W g|2 σh , σ f ,g = W f ·W g σh . If σ f ,g = 0, then W f ·W g = 0 σh − a.e. and the supports of W f and W g are disjoint in L2 (σh ); hence σ f ⊥ σg . Under assumption (b) we have to prove that U has simple spectrum. If this is not the case, for any h ∈ H, there would exist g ∈ H such that [U, g] ⊥ [U, h], so that, by (b), σh ⊥ σg . But this would contradict the following result ensuring the existence of maximal elements : There exists h ∈ H such that

σg  σh

for every g ∈ H.

This is a simplified version of the forthcoming lemma 2.4

2.2.2

 

Simple Lebesgue Spectrum

Definition 2.4. U has simple Lebesgue spectrum on H if H = [U, h] for some h ∈ H with σh ∼ m. Proposition 2.10. U has simple Lebesgue spectrum on H if and only if there exists h ∈ H such that (U n h)n∈Z forms an orthonormal basis of H. We shall prove an apparently stronger result: Given f ∈ H, the following assertions are equivalent: a) There exists h ∈ [U, f ] such that (U n h)n∈Z forms an orthonormal basis of [U, f ]. b) σ f ∼ m. Proof. If (U n h)n∈Z is complete in [U, f ], then [U, h] = [U, f ] and σh ∼ σ f (corollary 2.4). In addition, (U n h) is an orthonormal system if and only if ||h|| = 1 and U n h, h = 0 for all n = 0. This is exactly the property σh = m. Whence the first implication a)=⇒ b).

2.3 Spectral Decomposition Theorem and Maximal Spectral Type

31

Conversely, suppose that σ f ∼ m. Then, m = ϕσ f with ϕ ∈ L1 (σ f ), ϕ ≥ 0; putting ϕ = |φ |2 , φ ∈ L2 (σ f ) and we get σh := σφ (U) f = m. It follows that (U n h) is an orthonormal system with, clearly, h ∈ [U, f ]. But σ f ∼ σh = m and [U, f ] = [U, h].   This proves that (U n h)n∈Z forms an orthonormal basis of [U, f ]. Via theorem 2.1, this property means that H is isometrically isomorphic to L2 (T, m) and that U is unitarily equivalent to the operator V of multiplication by eit , on L2 (T, m). The isometry W of H onto L2 (T, m) is defined by setting W (U n h) = eint if (U n h) is an orthonormal basis of H.

2.3 Spectral Decomposition Theorem and Maximal Spectral Type Let H and H  be separable Hilbert spaces. Recall that unitary operators U on H and U  on H  are said to be spectrally equivalent or just conjugate if there exists an isometric isomorphism W : H → H  such that WU = U W . We already achieved the description of a unitary operator (up to equivalence) in the cyclic case (spectral representation theorem 2.1), and now, we consider the general case. Definition 2.5. The class of measures equivalent to a fixed measure μ ∈ M(T) is called the type of μ and will be denoted by [μ ]. Theorem 2.3 (First formulation of the spectral decomposition theorem). Let U be a unitary operator on a separable Hilbert space H. There exists a (possibly finite) sequence (hn )n≥1 of elements of H, such that (a) (b)

H = ⊕∞ n=1 [U, hn ] and [U, hi ] ⊥ [U, h j ] for i = j. σh1  σh2  · · ·

and for any other sequence (hn )n≥1 of elements of H satisfying (a) and (b), we have σhi ∼ σhi for each i ≥ 1. Remark 2.2. This formulation means that U is unitarily equivalent to the operator V , 2 defined on the space ⊕∞ n=1 L (T, σhn ) by V ( f1 , f2 , . . .)(t1 ,t2 , . . .) = (eit1 f1 (t1 ), eit2 f2 (t2 ), . . .),

ti ∈ T,

the sequence ([σn ]) being uniquely determined by U. Proof. We shall construct inductively a sequence satisfying (a); then, we shall modify it to realize (b) in addition. 1. Lemma 2.2. There exists a sequence (en ) of elements of H such that the cyclic subspaces H j = [U, e j ] satisfy (a). Proof. We start with an orthonormal basis (ε j ) of the separable space H, and we put e1 = ε1 , H1 = [U, e1 ]. Suppose e1 , . . . , e j have been so constructed that

32

2 Spectral Theory of Unitary Operators

the spaces H1 , . . . , H j are orthogonal and ε1 , . . . , ε j belong to ⊕i≤ j Hi . We now proceed this way : suppose that n j is the first index n for which εn is not in this sum, if such exists. Hence, n j ≥ j + 1, and we put e j+1 = εn j − P⊕i≤ j Hi (εn j ), (where, in general, PM is the orthogonal projection onto the closed subspace M). Clearly, H j+1 = [U, e j+1 ] is orthogonal to ⊕i≤ j Hi . If ε j+1 is already in H1 ⊕ · · · ⊕ H j , we are done; otherwise, n j = j + 1 and therefore ε j+1 = e j+1 + P⊕i≤ j Hi (ε j+1 ) belongs to H1 ⊕ · · · ⊕ H j+1 . The lemma follows by induction on j.   We write H = ⊕⊥ n≥1 [U, en ] for the orthogonal sum. Note that the so-constructed sequence (en ) satisfy σei ,e j = 0 for i = j, and Hn is isomorphic to L2 (σen ). 2. Lemma 2.3. If H = ⊕⊥ n≥1 [U, en ] and if the positive measure σ is such that σen  σ for every n, then σx  σ for every x ∈ H. (σ is maximal for H, in relation to the absolute continuity property of measures.) Proof. Every x ∈ H can be decomposed into a sum ∑n≥1 xn where xn ∈ [U, en ], and the sequence of measures (σ∑n≤N xn )N converges in norm to σx (corollary 2.2). In addition, σ∑n≤N xn = ∑ σxn since σxi ,x j = 0 for i = j, and σxn  σen  σ n≤N

for every n; thus ∑n≤N σxn  σ for every N, and σx  σ for every x ∈ H.

 

3. Combining these two lemmas, we shall construct maximal spectral measures. Lemma 2.4. For every e ∈ H, there exists h ∈ H such that e ∈ [U, h] and σx  σh

∀x ∈ H.

Proof. Given e ∈ H, we construct, according to the first lemma, an orthonormal basis (en ) such that e1 = e. H = ⊕⊥ n≥1 [U, en ], We put f1 = e1 and g1 = 0. For n ≥ 2, we can choose fn , gn ∈ [U, en ] such that : f n + g n = en ,

σ fn ⊥ σe

and σgn  σe ;

indeed, remembering that the σen are not necessarily mutually singular, we decompose σen = μ + ν , with μ ⊥ σe and ν  σe , and corollary 2.6 provides both functions fn and gn . We claim that h=

fn

fn

∑ || fn ||2n−1 = e + ∑ || fn ||2n−1 = e + f

n≥1

n≥2

will do the job. First of all, σe  σh and e ∈ [U, h] (corollary 2.4); then, σe ⊥ σ f so that σh = σe + σ f and for n ≥ 2, σen = σ fn + σgn  σ f + σe = σh . We conclude with the help of the second lemma.  

2.3 Spectral Decomposition Theorem and Maximal Spectral Type

33

4. We are now in a position to finish the proof of the theorem. Assume, as before, H = ⊕⊥ n≥1 [U, en ] where (en ) is an orthonormal basis of H. Applying lemma 2.4 to e1 , we get h1 ∈ H such that e1 ∈ [U, h1 ] and σh1 is maximal for H. We consider H1 = [U, h1 ]; H1⊥ is invariant under U. Let k be the first index n ≥ 2 for which en ∈ / H1 ; then ek = ek − PH1 (ek ) is in H1⊥ , and [U, e1 , . . . , ek−1 , ek ] = [U, e1 , . . . , ek−1 , ek ]. So we choose h2 ∈ H1⊥ such that ek ∈ [U, h2 ] and σh2 is maximal for H1⊥ . Clearly, σh2  σh1 , and we put H2 = [U, h2 ]. We repeat this process indefinitely to obtain σhn+1  σhn for all ⊥ n ≥ 1 and H = ⊕⊥ n≥1 Hn = ⊕n≥1 [U, hn ].  5. Unicity : Suppose that H = ⊕⊥ n≥1 [U, hn ] is another decomposition of H with property (b). By maximality of both, σh1 ∼ σh ; hence, the spaces L2 (σh1 ) and 1 L2 (σh ) are isomorphic, and there exists an isometry conjugating the multiplica1 tion operators V on L2 (σh1 ) and V  on L2 (σh1 ) (take the operator of multiplication and U  by ϕ 1/2 if σh1 = ϕσh1 ). Now, by theorem 2.1, the restrictions U [U,h1 ] [U,h1 ] are conjugate too (we write in short U  U  ). The unicity will follow [U,h1 ]

[U,h1 ]

easily by induction if we prove the following ultimate lemma : Lemma 2.5. Fix f , g ∈ H. If U

[U, f ]

 U

[U,g]

, then U

[U, f ]⊥

 U

[U,g]⊥

.

 Proof. First of all, we may assume that H = [U, f ] + [U, g] since U [U, f ]+[U,g] U obviously. [U, f ]+[U,g] By corollary 2.6, we can find g0 and g1 in [U, g] such that

σg = σg0 + σg1 ,

σg0  σ f

and σg1 ⊥ σ f .

Thanks to the canonical isometry W , we see that [U, g] is exactly [U, g0 ] ⊕⊥ [U, g1 ]. Next, we decompose σ f . If σ f = μ + ν with μ  σg0 and ν ⊥ σg0 , necessarily μ ∼ σg0 since σg0  σ f must be  μ . This implies that μ = |φ |2 σg0 = σφ (U)g0 for some φ ∈ L2 (σg0 ). Thus g0 = φ (U)g0 is such that : σg ≤ σ f and [U, g0 ] = 0 [U, g0 ]. We may suppose already σg0 ≤ σ f . Whence the decomposition

σ f = σg0 + σ f0 ,

σ f0 ⊥ σ f0

and

f0 ∈ [U, f ].

Moreover [U, f ] = [U, g0 ] ⊕⊥ [U, f0 ] and finally we obtain a new decomposition for H : H = [U, f0 ] ⊕⊥ [U, g0 ] ⊕⊥ [U, g1 ]. Turning back to the hypothesis, we proceed now by equivalence :  U ⇐⇒ σ f ∼ σg U [U, f ] [U,g]

34

2 Spectral Theory of Unitary Operators

(corollary 2.3), and, thanks to our decomposition,

σ f ∼ σg ⇐⇒ σ f0 ∼ σg1 ; but this means U

[U, f 0 ]

 U

[U,g1 ]

, which was to be proved.

 

The proof of the theorem is complete.   We can now precise some notions. Keeping in mind the notations of the theorem, Definition 2.6. [σh1 ] is called the maximal spectral type of the operator U acting on H, and, from now on, denoted by [σmax ]. U possesses a discrete–continuous–singular–absolutely continuous–or Lebesgue spectrum if the measure σmax is discrete–continuous–singular–absolutely continuous (with respect to the Lebesgue measure)– or equivalent to the Lebesgue measure (respectively). U has a simple spectrum if 0 = σh2 = σh2 = · · · , and U has a countable spectrum if all the σh j are = 0. Obviously, from the construction of the σh j , we have the following characterization : Corollary 2.7. σmax is characterized, up to equivalence, by the following properties (i) σ f  σmax for every f ∈ H, (ii) If σ ∈ M(T) satisfies 0 ≤ σ  σmax , then σ = σ f for some f ∈ H (in particular σmax = σ f0 for some f0 ∈ H). The next corollary has already been noticed (subsection 2.2.1) Corollary 2.8. U possesses a simple spectrum if and only if, for every pair f , g ∈ H, σ f ,g = 0 implies σ f ⊥ σg . We can now exhibit, using the decomposition, a candidate h with [U, h] = H : if H = ⊕⊥ n≥1 [U, en ], any h = ∑n≥1 an en , where an = 0 for all n, will work.

2.4 Spectral Decomposition Theorem and Spectral Multiplicity A mathematical object, related to Hilbert space operator theory, is called a spectral invariant if it is preserved under unitary equivalence. A system of spectral invariants is said to be complete if it characterizes unitary operators up to equivalence. The unicity assertion in the last theorem means that the spectral types [σhn ] constitute a complete system of spectral invariants; actually, in view to describing U up to equivalence, we need only know the maximal spectral type [σmax ] and the multiplicity function that we define now.

2.4 Spectral Decomposition Theorem and Spectral Multiplicity

35

2.4.1 Multiplicity Function We will write more briefly σ for σmax . According to the proof of lemma 2.5, each hn may be replaced by an fn such that [U, fn ] = [U, hn ] and σ fn = 1Bn σ , where (Bn ) is a non-increasing sequence of Borel sets. This leads to a re-formulation of theorem 2.3 : There exist a sequence ( fn ) of elements of H and a non-increasing sequence of Borel sets (Bn ) in T such that

σ f n = 1Bn σ

and H = ⊕⊥ n≥1 [U, f n ].

Definition 2.7. The function M = ∑n≥1 1Bn is called the multiplicity function. Note that M takes its values in N ∪ {∞} and M(t) = sup{n ≥ 1, t ∈ Bn } σ − a.e. Remark 2.3. The following notation may be convenient. For a separable Hilbert space with an orthonormal basis (en )n≥1 , one denotes by Hm the subspace of H generated by the m first vectors e1 , . . . , em , H∞ = H. If τ is some positive measure on T, L2 (τ , H) is the space of square-integrable vector-valued functions : {ϕ , T ||ϕ (t)||2 d τ (t) < ∞}. Let now t → m(t) with m(t) ∈ N ∪ {∞} : then,  ⊕

Hm(t) d τ (t) = {ϕ ∈ L2 (τ , H),

ϕ (t) ∈ Hm(t) }.

With those notations, the spectral decomposition theorem says: There exists an  isomorphism of Hilbert spaces: W : H → ⊕ HM(t) d σ (t) conjugating U with the operator V of multiplication by eit . This allows us to put the finishing touches on the commutator problem. Proposition 2.11. Let σ be the maximal spectral type of the unitary operator U on H. Then the operator Q commutes with U on H if and only if Q is, up to equivalence, a multiplication operator on L2 (σ , H), i.e. (QF)(t) = Q(t)F(t) where, for σ -a.e. t, Q(t) is an operator on HM(t) with t → Q(t) weakly measurable and t → ||Q(t)|| in L∞ (σ ). For the purpose of determinating the multiplicity function, a new formulation of the spectral decomposition theorem will be needed. Theorem 2.4 (Second formulation of the spectral decomposition theorem). Let U be a unitary operator on a separable Hilbert space H. For n ∈ N ∪ {∞}, one can (k) find a Borel set An in T, and a finite sequence (hn )1≤k≤n of elements of H such that

36

2 Spectral Theory of Unitary Operators

(a) (An ) is a partition of T. (k) (k) (k ) (b) H = ⊕n≥1 ⊕1≤k≤n [U, hn ], and [U, hn ] ⊥ [U, hn ] for (n, k) = (n , k ). (c) σ (k) = σ (n) for 1 ≤ k ≤ n, and σ (n) (Acn ) = 0. hn

Moreover, for any other sequence (hn (k) )n≥1 of elements of H satisfying (a), (b) and (c), we have σ (k) ∼ σh (k) , for each (n, k). hn

n

Proof. Suppose that (a), (b) and (c) hold; if we set

∑ hn

(k)

hn =

,

k≥n

the elements hn fulfill conditions (a) and (b) of theorem 2.3. Conversely, let us assume that there exist a sequence ( fn ) of elements of H and a non-increasing sequence of Borel sets (Bn ) in T such that

σ f n = 1Bn σ

and H = ⊕⊥ n≥1 [U, f n ].

We set An = Bn \Bn+1 ,

A∞ = ∩n≥1 Bn ;

(An ) is a Borel partition of T whence (a). Now, for 1 ≤ k ≤ n, 1An σ fk is the spectral measure of some element in [U, fk ] that (k)

we call hn . Since Bk ⊃ Bn ,

σ

(k)

hn

= 1Bn \Bn+1 1Bk σ = 1Bn \Bn+1 σ

and σ (k) does not depend on k, 1 ≤ k ≤ n; so we put σ (n) = 1An σ and (c) holds. hn Clearly, for each k ≥ 1, (k)

[U, fk ] = ⊕⊥ n≥k [U, hn ]. It follows that (k)

⊥ ⊥ H = ⊕⊥ n≥1 [U, f n ] = ⊕n≥1 ⊕1≤k≤n [U, hn ],

 

and the theorem is proved. The multiplicity function is now more readable : with the same notations, Corollary 2.9. The multiplicity function M is defined on T by the relation : M(t) = n if t ∈ An .

2.4 Spectral Decomposition Theorem and Spectral Multiplicity

37

2.4.2 Global Multiplicity We start with the following notions and notations : Definition 2.8. [σ (n) ] is referred to as the spectral type of multiplicity n. One says that U has finite multiplicity on H if σ (n) = 0 for n > n0 ; if σ (n) = 0 for n = n0 , i.e. M is constant, one says that U has a homogeneous spectrum of multiplicity n0 . Definition 2.9. The essential supremum of the function M is called the spectral multiplicity of U on H or global multiplicity, and denoted by m(U). Hence, U has a simple spectrum if m(U) = 1. Corollary 2.10. The global multiplicity of U is also the cardinality of the maximal set of U-orthogonal elements in H with identical spectral measures. Proof. If g1 , . . . , g p are p elements of H, U-orthogonal and such that σg1 = · · · = σg p = τ , we shall prove that m(U) ≥ p. We decompose for that, H = H(τ ) ⊕ H(τ )⊥ where H(τ ) = { f ∈ H, σ f  τ }. U has [τ ] as its maximal spectral type on H(τ ). H(τ ), in turn, admits the decomposition H(τ ) = [U, g1 ] ⊕⊥ · · · ⊕⊥ [U, g p ] ⊕⊥ H  H  being U-invariant too. Applying now theorem 2.3 to H  and U we get: H(τ ) = [U, g1 ] ⊕⊥ · · · ⊕⊥ [U, g p] ⊕⊥ n≥1 [U, hn ] has multiplicity ≥ p, and the with τ  σh1  σh2  · · · . We deduce that U H(τ ) corollary follows.   This notion of global multiplicity turns out to be very important : in the homogeneous case, the operator is determined, up to equivalence, by the pair of invariants σmax and m(U). The following results deal with estimations of m(U) and will be used in a forthcoming section. Proposition 2.12. Assume (Hn ) to be a non-decreasing sequence of U-invariant subspaces of H, such that H = ∪n≥1 Hn . Let r be fixed. If the spectral multiplicity of U on Hn is ≤ r for all n, so is the global multiplicity. Proof. If the global multiplicity m(U) is at least r + 1, by corollary 2.10 we can find r + 1 U-orthogonal vectors f1 , . . . , fr+1 with identical spectral measure σ . We have to estimate the multiplicity of U on Hn .

38

2 Spectral Theory of Unitary Operators

Denote by Pn the orthogonal projection onto Hn . Since H = ∪n≥1 Hn and by the continuity property of the map f → σ f (corollary 2.2), for ε > 0 we can choose n(ε ) such that ||σPn f j − σ || ≤ ε ,

for every 1 ≤ j ≤ r + 1, and n ≥ n(ε ).

But σPn f j ≤ σ , and making ε small enough, one can find τ = 0 such that

τ  σPn f j , 1 ≤ j ≤ r + 1, n = n(ε ). Now, for each j = 1, . . . , r + 1, we choose g j ∈ [U, Pn f j ] ⊂ [U, f j ], with σg j = τ . It follows that g1 , . . . , gr+1 are U-orthogonal elements in Hn with identical spectral measures. Thus, the multiplicity of U on Hn is ≥ r + 1, which was to be proved.   This proposition will be convenient in aim of establishing the simplicity of the spectrum. The next one (referred to as Chacon’s theorem) provides a lower bound for the multiplicity [54, 106]. Proposition 2.13. The global multiplicity m(U) is ≥ m if and only if there exists an orthonormal family f1 , . . . , fm in H such that, for any cyclic subspace H0 , we have : m

∑ d( fi , H0 )2 ≥ m − 1

(2.14)

i=1

(d(., H0 ) denoting the distance to H0 ) Proof.  If the global multiplicity m(U) ≥ m we can find, as above, m U-orthogonal vectors f1 , . . . , fm with σ f j = σ , for j = 1, . . . , m. Without loss of generality, we assume || f j || = 1 for all j so that σ is a probability measure. We claim that the vectors ( f j ) do the job. We are led to prove the following : if h ∈ H is fixed, m

S := ∑ || fi − gi ||2 ≥ m − 1 i=1

for all g1 , . . . , gm in the cyclic space [U, h], and more simply, for all g1 , . . . , gm in the linear span of the U n h, n ∈ Z. We thus consider, for i = 1, . . . , m, gi = Pi (U)h where Pi is a polynomial in C[X, X −1 ] and we decompose h into the form h=

m

∑ h, f j  f j + h

j=1

with h ⊥ [U, f j ] for j = 1, . . . , m. We deduce S=

m

∑

i=1



m || fi − ∑ h, f j Pi (U) f j ||2 + ||Pi (U)h ||2 j=1

2.4 Spectral Decomposition Theorem and Spectral Multiplicity

≥ =

m

m

i=1 m

j=1

39

∑ || fi − ∑ h, f j Pi (U) f j ||2 ∑ || fi − αi Pi(U) fi ||2 + ∑

i=1

|α j |2 ||Pi (U) f j ||2

1≤i, j≤m j=i

by U-orthogonality, where α j = h, f j . Using now the isometric isomorphism between each cyclic space [U, fi ] and L2 (σ ), we obtain  m

S≥

=

∑ |1 − αiPi(t)|2 + ∑

T



i=1

m

∑



T i=1

|α j |2 |Pi (t)|2 d σ (t)

1≤i, j≤m j=i

∑

|1 − αi Pi (t)|2 + |Pi(t)|2 (

|α j |2 ) d σ (t)

1≤ j≤m j=i

where P(t) as usual denotes the associated trigonometric polynomial. We end the m proof by establishing that Q(t) := ∑i=1 |1 − αi Pi (t)|2 + |Pi(t)|2 (∑ 1≤ j≤m |α j |2 ) − j=i

m + 1 is a positive polynomial.

1/2 2 Indeed, putting a = ∑m | α | and expanding the sum in Q, we readily get i=1 i via the Cauchy-Schwarz inequality m

m

m

i=1

i=1

Q(t) = 1 + ∑ |αi |2 |Pi (t)|2 − 2 ∑ ℜ(αi Pi (t)) + ∑ |Pi (t)|2 (a2 − |αi |2 ) i=1

m

m

i=1 m

i=1

= 1 + a2 ∑ |Pi (t)|2 − 2 ∑ ℜ(αi Pi (t)) ≥ 1 + a2 ∑ |Pi (t)|2 − 2a i=1



m

∑ |Pi(t)|2

1/2

≥ 0.

i=1

The necessity of the condition is proved.  Let f1 , . . . , fm be an orthonormal family in H such that, for any cyclic sub2 space H0 , we have : ∑m i=1 d( f i , H0 ) ≥ m− 1. If P0 is the orthogonal projection on H0 , d( fi , H0 )2 = || fi ||2 − ||P0 fi ||2 = 1 − ||P0 fi ||2 and

m

∑ ||P0 fi ||2 ≤ 1.

i=1

Taking H0 = [U, f j ], we deduce that fi ⊥ [U, f j ] for i = j. We shall prove that the spectral measures σ j := σ f j are equivalent. Otherwise, we may assume that σ1 ∼ σ2 ; let A be a Borel set satisfying

σ1 (A) > 0 and σ2 (A) = 0.

40

2 Spectral Theory of Unitary Operators

We shall get a contradiction by considering H0 = [U, g] with g = EA f 1 + f 2 . Indeed, EA g = EA f1 so that EA f1 and f2 are in H0 , and m

∑ ||P0 fi ||2 ≥ || f2 ||2 + ||P0 f1 ||2 ≥ 1 + ||EA f1 ||2 ≥ 1 + σ1(A) > 1,

i=1

which is impossible. This proves the sufficiency of condition (2.11).

 

2.5 Eigenvalues and Discrete Spectrum In this section, U is an unitary operator on a separable Hilbert space.

2.5.1 Eigenvalues Recall that, U being unitary, the spectrum of U is a compact subset of the circle. Definition 2.10. The complex number eiλ , λ ∈ T, is an eigenvalue of U if U f = eiλ f for some f ∈ H, f = 0, which is an eigenvector corresponding to eiλ . We establish now the connection between the eigenvalues of U and the maximal spectral type [σmax ]. Lemma 2.6. If λ ∈ T, the operator 1{λ } (U) =: Eλ is the orthogonal projection onto the eigen-subspace associated to the eigenvalue eiλ . Proof. By the definition-proposition 2.7, Hλ := Eλ H is the subspace of h ∈ H whose spectral measure σh is supported by {λ }. We shall now identify Hλ and the eigensubspace associated with eiλ . Suppose that Uh = eiλ h; then σˆ h (n) = einλ ||h||2 , n ∈ Z, and σh = ||h||2 δλ . Conversely, suppose there is h ∈ H with σh = αδλ ; if R is a trigonometric polynomial, ||R(U)h||H = ||R||L2 (σh ) = α R(λ ). Choose R(t) = eit − eiλ ; then ||R(U)h||H = ||Uh − eiλ h||H = 0 and h is an eigenvector corresponding to eiλ .   As a consequence, with the same notations,

2.5 Eigenvalues and Discrete Spectrum

41

Proposition 2.14. eiλ is an eigenvalue of U if and only if σmax {λ } = 0. Proof. This means : eiλ is an eigenvalue as soon as one can find f ∈ H with σ f {λ } = 0. But, by (2.8)

σ f {λ } = Eλ f , f  = ||Eλ f ||2 , It follows that

∀ f ∈ H.

(2.15)

f ∈ Hλ⊥ ⇐⇒ Eλ f = 0 ⇐⇒ σ f {λ } = 0,  

whence the claim.

Remark 2.4. We are able to precise the link between the spectral set sp(U) and the maximal spectral type. Let h ∈ H be such that σmax = σh . If eiλ ∈ / sp(U), there exists a positive constant C such that ||U f − eiλ f || ≥ C|| f || for every f ∈ H. Fix ε > 0 and consider the function 1]λ −ε ,λ +ε [ ∈ L2 (σh ). If gε is the associated element in [U, h]  L2 (σh ), then we have

σgε = 1]λ −ε ,λ +ε [σh , ||gε ||2 = σh (]λ − ε , λ + ε [), and

||Ugε − eiλ gε ||2 =



]λ −ε ,λ +ε [ |e

it

− eiλ |2 d σh (t)

≤ ε 2 σh (]λ − ε , λ + ε [); Since ||Ugε − eiλ gε ||2 ≥ C2 ||gε ||2 , this imposes C ≤ ε for every positive ε and a contradiction, unless gε = 0 and σmax (]λ − ε , λ + ε [) = 0. For any f ∈ H, the equation Ug − eiλ g = f has a solution g ∈ H if and only if one can solve the equation eit ϕ − eiλ ϕ = 1 in L2 (σ f ), and this is possible if 1/(eit − eiλ ) belongs to L2 (σ f ). Suppose now that eiλ is avoiding the topological support of σh ; / sp(U). in this case, 1/(eit − eiλ ) is bounded σ f -a.e. and consequently, eiλ ∈ The link between the spectrum of the unitary operator U and its maximal spectral type is now clear : sp(U) is exactly the topological support of the measure σmax .

2.5.2 Discrete Spectral Measures To any decomposition of σmax into a sum of mutually singular measures, there corresponds a decomposition of H into a sum of U-invariant orthogonal closed subspaces: If H(τ ) = { f ∈ H, σ f  τ },

42

2 Spectral Theory of Unitary Operators

H = H(σ1 ) ⊕⊥ H(σ2 ) for any decomposition σmax = σ1 + σ2 with σ1 ⊥ σ2 . In particular, we introduce the following definition : Definition 2.11. Hd is the closed subspace of elements in H with discrete spectral measure, and Hc , the closed subspace of elements in H with continuous spectral measure; also H = Hd ⊕⊥ Hc . We have an almost obvious description of Hd : Proposition 2.15. Hd is the closure of the linear span of eigenvectors of U. Proof. Let f ∈ H; σ f is discrete if and only if ||σ f || = ∑λ σ f {λ } = || f ||2H ; but, using (2.15), ∑ σ f {λ } = ∑ ||Eλ f ||2 , λ

λ

so that f ∈ Hd if and only if || f ||2H = ∑ ||Eλ f ||2 . λ

By the converse of the Parseval identity, this condition means exactly that f =   ∑λ Eλ f which was to be proved. Definition 2.12. U is said to have discrete (or pure-point) spectrum when σmax is a (purely) discrete measure. If Hd = {0} (resp. Hc = {0}), U is said to possess a discrete component (resp. a continuous component). If U possesses a discrete as well as a continuous component, U has a mixed spectrum. Whence this simple remark. Corollary 2.11. U has a discrete spectrum if and only if the linear span of eigenvectors of U is dense in H. We conclude this subsection by a useful observation made by Solomyak [230]. Proposition 2.16. Let U be a unitary operator on a Hilbert space H. Suppose that there exists a total set F ⊂ H, and a linearly recurrent sequence of integers (qn ) such that limn→∞ qn = ∞ and

∑ U qn f − f 2 < ∞,

n≥1

Then U has purely discrete spectrum.

for every f ∈ F .

2.5 Eigenvalues and Discrete Spectrum

43

Proof. Since U qn f − f 2 =

   eitqn − 12 d σ f (t) T

for any f ∈ H, the assumption in the proposition implies that 



∑ σ f ({t, eitqn − 1 ≥ 2−}) < ∞, for every

f ∈ F and every  > 0.

n≥1

By the Borel-Cantelli lemma, σ f (T\{t, eitqn → 1}) = 0 for those f and, F being total, one deduces that σmax (T\{t, eitqn → 1}) = 0. But, the sequence (qn ) satisfies a linear integral recurrence relation so that the set {t, eitqn → 1} is countable and σmax is a discrete measure. This is the claim.  

2.5.3 Basic Examples 1. Irrational rotation : we turn back to the operator U defined on H = L2 (T) by U f (t) = f (t + θ ),

θ /π ∈ / Q,

with simple spectrum as observed in 2.2.1. We claim that : U has discrete spectrum and σmax ∼ ∑k∈Z 2−|k| δkθ . Indeed, for each k ∈ Z, eikt is obviously an eigenvector of U with eigenvalue ik θ e , and the family (eikt )k∈Z is complete in H; whence the discrete spectrum. Let now f ∈ H be such that fˆ(k) = 0 for every k ∈ Z; as we already observed in subsection 2.2.1, σ f ∼ σmax and H = [U, f ]  L2 (σ f ); moreover, σ f {kθ } = ||Ekθ f ||2 so that σ f = ∑ | fˆ(k)|2 δkθ . k∈Z

We wish to determine an isometry W conjugating U on L2 (T, m) and V on L2 (σ f ), such that W f = 1. If g ∈ L2 (T, m), there exists (an ) ∈ 2 (Z) such that g=

∑ anU n f

n∈Z

and, if k ∈ Z,

g(k) ˆ =

∑ an(U n f )ˆ(k) = ∑ an einkθ fˆ(k)

n∈Z

n∈Z

ˆ fˆ(k) for all k ∈ Z, gives We shall see that the formula W g = h with h(kθ ) = g(k)/ the result; indeed, such an h satisfies 

|h|2 d σ =

2 ˆ < ∞, ∑ |h(kθ )|2 | fˆ(k)|2 = ∑ |g(k)|

k∈Z

k∈Z

44

2 Spectral Theory of Unitary Operators

and W is an isometry from L2 (T, m) into L2 (σ f ). Also, W is clearly invertible, and W (U n f ) = hn with hn (t) = eint so that, finally, the conjugation identity WU = eit W, valid on {U n f , n ∈ Z}, extends to L2 (T, m) = [U, f ]. 2. Let H = L2 (T) once more, and U be the unitary operator defined on f ∈ H by U f (t) = φ (t) f (t + θ ),

θ ∈ [0, 2π ],

φ being unimodular. We claim that : σmax ∼ ∑k∈Z 2−|k| σ ∗ δkθ , where σ is the spectral measure of the function 1. If f ∈ H, U n f (t) = φ (n) (t) f (t + nθ ), where the cocycle (φ (n) ) is defined by ⎧ ⎪ ⎨ φ (t)φ (t + θ ) · · · φ (t + (n − 1)θ ) if n ≥ 1, (n) φ (t) = φ (−n) (t + nθ ) if n ≤ −1, ⎪ ⎩ 1 if n = 0; since

σ f (n) = we see that

 T

φ (n) (x) f (t + nθ ) f (t) dm(t), σ f eit = σ f ∗ δθ .

(2.16)

It follows that the measure ν := ∑k∈Z 2−|k| σ ∗ δkθ is  σmax . Conversely, if f (t) = eikt , σ f = σ ∗ δkθ by (2.16) and σ f  ν for every trigonometric polynomial f . Approximating any f ∈ H by a trigonometric polynomial and invoking the continuity of the map f → σ f (corollary 2.2), we get σ f  ν for all f ∈ H; whence σmax ∼ ν . 3. Let U be the operator defined by U f (t) = f (2t) on H = L2 (T); then U has infinite multiplicity. Actually, for every finite set f1 , f2 , . . . , f p in H, [U, ( f j )1≤ j≤p ]⊥ is infinite dimensional.

2.6 Application to Ergodic Sequences 1. By using the isometry machinery, we can now easily prove the following classical theorem due to von Neumann : Theorem 2.5 (von Neumann). If U is an isometry of the separable Hilbert space H, then, for all f ∈ H, 1 N−1 n ∑ U f → Pf, N n=0 where P is the orthogonal projection onto the subspace of U-invariant vectors.

2.6 Application to Ergodic Sequences

45

Proof. Suppose first that U is an unitary operator. n For any fixed f ∈ H, the sequence (MN f )N≥1 , with MN = N1 ∑N−1 n=0 U , stays inside the cyclic subspace [U, f ]; so, we are reduced to study the limit of the int in L2 (σ ). But it is easily sequence of trigonometric polynomials N1 ∑N−1 f n=0 e 1 N−1 int eint = 1{0} seen that limN→∞ N ∑n=0 e = 1{0} , and even, limN→∞ N1 ∑N+K−1 n=K 1 N+K−1 int uniformly with respect to K. We conclude that limN→∞ N ∑n=K e = 1{0} in L2 (σ f ) by the dominated convergence theorem, and N1 ∑N+K−1 U n f → E0 f in H n=K uniformly in K. In particular, MN f → P f , for all f ∈ H. When the isometry U is no longer an unitary operator, the spectral family, associated to U, is defined in this way : if f ∈ H, we put 

γn =

U n f , f  if n ≥ 0  f ,U −n f  if n < 0

Since γn = γ−n for n < 0, (γn )n∈Z is still a positive definite sequence and the spectral measure σ f is well defined by

σ f (n) = γn ,

n ∈ Z.

We easily check that the fundamental identity (2.4) : ||R(U) f ||H = ||R||L2 (σ f ) , remains valid for trigonometric polynomials on T with nonnegative frequences : R(t) = ∑k≥0 ak eikt . This is sufficient to conclude in this case.   2. Coboundaries. Consider an isometry U of the separable Hilbert space H and g ∈ H. Then g is a U-coboundary if there exists f ∈ H such that g = U f − f . Note that in this case, (U n g) is a bounded sequence in H; in fact the converse is true and holds in a more general context [167]. Theorem 2.6. Let U be an isometry of the separable Hilbert space H. Then g ∈ H is a U-coboundary if and only if (U n g) is a bounded sequence in H. Proof. The weak closure of convex hull of the U n g, n ≥ 0, is a weak compact set K, which is stable under the continuous map ϕ : f → U f − g. By the MarkovKakutani theorem, ϕ has a fixed point in K : there exists f ∈ K such that ϕ ( f ) = f or U f − f = g.   3. The following result may be viewed as a consequence of von Neumann’s theorem. Proposition 2.17. Let U be an isometry of the separable Hilbert space H. Then, for every f ∈ H and every ε > 0, the set of non-negative integers {n ∈ N,

|σ f (n)| ≥ σ f {0} − ε }

is relatively dense. We recall that a set of integers is said to be relatively dense or syndetic if it has bounded gaps.

46

2 Spectral Theory of Unitary Operators

Proof. Observe first that σ f {0} = ||σP f || = ||P f ||2 = P f , f  with P = E0 . Now 1 N+K−1 n ∑ U f , f  N→∞ N n=K

P f , f  = lim

uniformly in K. Let us fix ε > 0; there exists N > 0 such that |

1 N+K−1 n ∑ U f , f  − P f , f | ≤ ε , N n=K

hence 1 N+K−1 ∑ |σ f (n)| ≥ σ f {0} − ε N n=K for all K ≥ 0. This means that {K, . . . , K + N − 1} contains at least one integer n for which |σ f (n)| ≥ σ f {0} − ε , whatever K ≥ 0, and the property is established.   Remark 2.5. Actually, this property is shared by any positive measure σ ∈ M(T). 4. More generally, we consider sequences (U kn ) for some increasing sequence of non-negative integers (kn ). Proposition 2.18. The two assertions (a) and (b) are equivalent : (a) The sequence ( N1 ∑n