DOCSLIB.ORG

Operator Theoretic Aspects of Ergodic Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Operator Theoretic Aspects of Ergodic Theory T. Eisner, B. Farkas, M. Haase, R. Nagel Operator Theoretic Aspects of Ergodic Theory July 21, 2012 Springer Contents Preface ............................................................ ix 1 What is Ergodic Theory? ....................................... 1 2 Topological Dynamical Systems .................................. 7 2.1 Some Basic Examples . .7 2.2 Basic Constructions . 11 2.3 Topological Transitivity . 17 2.4 Transitivity of Subshifts . 21 Exercises . 23 3 Minimality and Recurrence ..................................... 27 3.1 Minimality . 28 3.2 Topological Recurrence . 30 3.3 Recurrence in Extensions . 32 Exercises . 35 4 The C∗-algebra C(K) and the Koopman Operator . 37 4.1 Continuous Functions on Compact Spaces . 38 4.2 The Space C(K) as a Commutative C∗-Algebra . 43 4.3 The Koopman Operator . 45 4.4 The Theorem of Gelfand–Naimark . 49 Supplement: Proof of the Gelfand–Naimark Theorem . 50 Exercises . 56 5 Measure-Preserving Systems .................................... 59 5.1 Examples . 61 5.2 Measures on Compact Spaces . 66 5.3 Haar Measure and Rotations . 70 Exercises . 71 v vi Contents 6 Recurrence and Ergodicity ...................................... 75 6.1 The Measure Algebra and Invertible MDS . 75 6.2 Recurrence . 77 6.3 Ergodicity . 82 Supplement: Induced Transformation and Kakutani–Rokhlin Lemma . 85 Exercises . 88 7 The Banach Lattice Lp and the Koopman Operator . 91 7.1 Banach Lattices and Positive Operators . 92 7.2 The Space Lp(X) as a Banach Lattice . 97 7.3 The Koopman Operator and Ergodicity . 99 Supplement: Interplay between Lattice and Algebra Structure . 102 Exercises . 104 8 The Mean Ergodic Theorem .....................................107 8.1 Von Neumann’s Mean Ergodic Theorem . 108 8.2 The Fixed Space and Ergodicity . 111 8.3 Perron’s Theorem and the Ergodicity of Markov Shifts . 114 8.4 Mean Ergodic Operators . 116 Supplement: Operations with Mean Ergodic Operators . 120 Exercises . 123 9 Mixing Dynamical Systems ......................................125 9.1 Strong Mixing . 126 9.2 Weakly Mixing Systems . 131 9.3 Weak Mixing of All Orders . 138 Exercises . 143 10 Mean Ergodic Operators on C(K) ................................147 10.1 Invariant Measures . 147 10.2 Uniquely and Strictly Ergodic Systems . 150 10.3 Mean Ergodicity of Group Rotations . 152 10.4 Furstenberg’s Theorem on Group Extensions . 154 10.5 Application: Equidistribution . 157 Exercises . 160 11 The Pointwise Ergodic Theorem .................................163 11.1 Pointwise Ergodic Operators . 164 11.2 Banach’s Principle and Maximal Inequalities . 166 11.3 Applications . 170 Exercises . 173 Contents vii 12 Isomorphisms and Topological Models . 175 12.1 Point Isomorphisms and Factor Maps . 175 12.2 Algebra Isomorphisms of Measure Preserving Systems . 179 12.3 Topological Models . 183 12.4 The Stone Representation . 186 12.5 Mean Ergodic Subalgebras . 188 Exercises . 191 13 Markov Operators .............................................193 13.1 Examples and Basic Properties . 194 13.2 Embeddings, Factor Maps and Isomorphisms . 196 13.3 Markov Projections . 199 13.4 Factors and their Models . 203 Supplement: Operator Theory on Hilbert Space . 206 Exercises . 210 14 Compact Semigroups and Groups ................................211 14.1 Compact Semigroups . 211 14.2 Compact Groups . 217 14.3 The Character Group. 219 14.4 The Pontryagin Duality Theorem . 223 14.5 Application: Ergodic Rotations and Kronecker’s Theorem . 225 Exercises . 227 15 Topological Dynamics Revisited ..................................229 15.1 The Cech–Stoneˇ Compactification . 229 15.2 The Space of Ultrafilters . 229 15.3 Multiple Recurrence . 229 15.4 Applications to Ramsey Theory. 229 Exercises . 229 16 The Jacobs–de Leeuw–Glicksberg Decomposition . 231 16.1 Weakly Compact Operator Semigroups . 232 16.2 The Jacobs–de Leeuw–Glicksberg Decomposition . 237 16.3 The Almost Weakly Stable Part . 240 16.4 Compact Group Actions on Banach Spaces . 244 16.5 The Reversible Part Revisited . 250 Supplement: Unitary Representations of Compact Groups . 252 Exercises . 257 17 Dynamical Systems with Discrete Spectrum . 261 17.1 Operators with Discrete Spectrum . 261 17.2 Monothetic Groups . 263 17.3 Dynamical Systems with Discrete Spectrum . 265 17.4 Examples . 269 Exercises . 272 viii Contents 18 A Glimpse at Arithmetic Progressions ............................275 18.1 The Furstenberg Correspondence Principle . ..
Load more

Recommended publications
  • Measure Theory (Graduate Texts in Mathematics)
    PaulR. Halmos Measure Theory Springer-VerlagNewYork-Heidelberg-Berlin Managing Editors P. R. Halmos C. C. Moore Indiana University University of California Department of Mathematics at Berkeley Swain Hall East Department of Mathematics Bloomington, Indiana 47401 Berkeley, California 94720 AMS Subject Classifications (1970) Primary: 28 - 02, 28A10, 28A15, 28A20, 28A25, 28A30, 28A35, 28A40, 28A60, 28A65, 28A70 Secondary: 60A05, 60Bxx Library of Congress Cataloging in Publication Data Halmos, Paul Richard, 1914- Measure theory. (Graduate texts in mathematics, 18) Reprint of the ed. published by Van Nostrand, New York, in series: The University series in higher mathematics. Bibliography: p. 1. Measure theory. I. Title. II. Series. [QA312.H26 1974] 515'.42 74-10690 ISBN 0-387-90088-8 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. © 1950 by Litton Educational Publishing, Inc. and 1974 by Springer-Verlag New York Inc. Printed in the United States of America. ISBN 0-387-90088-8 Springer-Verlag New York Heidelberg Berlin ISBN 3-540-90088-8 Springer-Verlag Berlin Heidelberg New York PREFACE My main purpose in this book is to present a unified treatment of that part of measure theory which in recent years has shown itself to be most useful for its applications in modern analysis. If I have accomplished my purpose, then the book should be found usable both as a text for students and as a source of refer­ ence for the more advanced mathematician. I have tried to keep to a minimum the amount of new and unusual terminology and notation.
    [Show full text]
  • Weak Compactness in the Space of Operator Valued Measures and Optimal Control Nasiruddin Ahmed
    Weak Compactness in the Space of Operator Valued Measures and Optimal Control Nasiruddin Ahmed To cite this version: Nasiruddin Ahmed. Weak Compactness in the Space of Operator Valued Measures and Optimal Control. 25th System Modeling and Optimization (CSMO), Sep 2011, Berlin, Germany. pp.49-58, 10.1007/978-3-642-36062-6_5. hal-01347522 HAL Id: hal-01347522 https://hal.inria.fr/hal-01347522 Submitted on 21 Jul 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License WEAK COMPACTNESS IN THE SPACE OF OPERATOR VALUED MEASURES AND OPTIMAL CONTROL N.U.Ahmed EECS, University of Ottawa, Ottawa, Canada Abstract. In this paper we present a brief review of some important results on weak compactness in the space of vector valued measures. We also review some recent results of the author on weak compactness of any set of operator valued measures. These results are then applied to optimal structural feedback control for deterministic systems on infinite dimensional spaces. Keywords: Space of Operator valued measures, Countably additive op- erator valued measures, Weak compactness, Semigroups of bounded lin- ear operators, Optimal Structural control.
    [Show full text]
  • Math 259A Lecture 7 Notes
    Math 259A Lecture 7 Notes Daniel Raban October 11, 2019 1 WO and SO Continuity of Linear Functionals and The Pre-Dual of B 1.1 Weak operator and strong operator continuity of linear functionals Lemma 1.1. Let X be a vector space with seminorms p1; : : : ; pn. Let ' : X ! C be a Pn linear functional such that j'(x)j ≤ i=1 pi(x) for all x 2 X. Then there exist linear P functionals '1;:::;'n : X ! C such that ' = i 'i with j'i(x)j ≤ pi(x) for all x 2 X and for all i. Proof. Let D = fx~ = (x; : : : ; x): x 2 Xg ⊆ Xn, which is a vector subspace. On Xn, n P take p((xi)i=1) = i pi(xi). We also have a linear map' ~ : D ! C given by' ~(~x) = '(x). This map satisfies j~(~x)j ≤ p(~x). By the Hahn-Banach theorem, there exists an n ∗ extension 2 (X ) of' ~ such that j (x1; : : : ; xn)j ≤ p(x1; : : : ; xn). Now define 'k(x) := (0; : : : ; x; 0;::: ), where the x is in the k-th position. Theorem 1.1. Let ' : B! C be linear. ' is weak operator continuous if and only if it is it is strong operator continuous. Proof. We only need to show that if ' is strong operator continuous, then it is weak Pn operator continuous. So assume there exist ξ1; : : : ; ξn 2 X such that j'(x)j ≤ i=1 kxξik P for all x 2 B. By the lemma, we can split ' = 'k, such that j'k(x)j ≤ kxξkk for all x and k.
    [Show full text]
  • Derivations on Metric Measure Spaces
    Derivations on Metric Measure Spaces by Jasun Gong A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2008 Doctoral Committee: Professor Mario Bonk, Chair Professor Alexander I. Barvinok Professor Juha Heinonen (Deceased) Associate Professor James P. Tappenden Assistant Professor Pekka J. Pankka “Or se’ tu quel Virgilio e quella fonte che spandi di parlar si largo fiume?” rispuos’io lui con vergognosa fronte. “O de li altri poeti onore e lume, vagliami ’l lungo studio e ’l grande amore che m’ha fatto cercar lo tuo volume. Tu se’ lo mio maestro e ’l mio autore, tu se’ solo colui da cu’ io tolsi lo bello stilo che m’ha fatto onore.” [“And are you then that Virgil, you the fountain that freely pours so rich a stream of speech?” I answered him with shame upon my brow. “O light and honor of all other poets, may my long study and the intense love that made me search your volume serve me now. You are my master and my author, you– the only one from whom my writing drew the noble style for which I have been honored.”] from the Divine Comedy by Dante Alighieri, as translated by Allen Mandelbaum [Man82]. In memory of Juha Heinonen, my advisor, teacher, and friend. ii ACKNOWLEDGEMENTS This work was inspired and influenced by many people. I first thank my parents, Ping Po Gong and Chau Sim Gong for all their love and support. They are my first teachers, and from them I learned the value of education and hard work.
    [Show full text]
  • Course Structure for M.Sc. in Mathematics (Academic Year 2019 − 2020)
    Course Structure for M.Sc. in Mathematics (Academic Year 2019 − 2020) School of Physical Sciences, Jawaharlal Nehru University 1 Contents 1 Preamble 3 1.1 Minimum eligibility criteria for admission . .3 1.2 Selection procedure . .3 2 Programme structure 4 2.1 Overview . .4 2.2 Semester wise course distribution . .4 3 Courses: core and elective 5 4 Details of the core courses 6 4.1 Algebra I .........................................6 4.2 Real Analysis .......................................8 4.3 Complex Analysis ....................................9 4.4 Basic Topology ...................................... 10 4.5 Algebra II ......................................... 11 4.6 Measure Theory .................................... 12 4.7 Functional Analysis ................................... 13 4.8 Discrete Mathematics ................................. 14 4.9 Probability and Statistics ............................... 15 4.10 Computational Mathematics ............................. 16 4.11 Ordinary Differential Equations ........................... 18 4.12 Partial Differential Equations ............................. 19 4.13 Project ........................................... 20 5 Details of the elective courses 21 5.1 Number Theory ..................................... 21 5.2 Differential Topology .................................. 23 5.3 Harmonic Analysis ................................... 24 5.4 Analytic Number Theory ............................... 25 5.5 Proofs ........................................... 26 5.6 Advanced Algebra ...................................
    [Show full text]
  • Borel Extensions of Baire Measures
    FUNDAMENTA MATHEMATICAE 154 (1997) Borel extensions of Baire measures by J. M. A l d a z (Madrid) Abstract. We show that in a countably metacompact space, if a Baire measure admits a Borel extension, then it admits a regular Borel extension. We also prove that under the special axiom ♣ there is a Dowker space which is quasi-Maˇr´ıkbut not Maˇr´ık,answering a question of H. Ohta and K. Tamano, and under P (c), that there is a Maˇr´ıkDowker space, answering a question of W. Adamski. We answer further questions of H. Ohta and K. Tamano by showing that the union of a Maˇr´ıkspace and a compact space is Maˇr´ık,that under “c is real-valued measurable”, a Baire subset of a Maˇr´ıkspace need not be Maˇr´ık, and finally, that the preimage of a Maˇr´ıkspace under an open perfect map is Maˇr´ık. 1. Introduction. The Borel sets are the σ-algebra generated by the open sets of a topological space, and the Baire sets are the smallest σ- algebra making all real-valued continuous functions measurable. The Borel extension problem asks: Given a Baire measure, when can it be extended to a Borel measure? Whenever one deals with Baire measures on a topological space, it is assumed that the space is completely regular and Hausdorff, so there are enough continuous functions to separate points and closed sets. In 1957 (see [Ma]), J. Maˇr´ıkproved that all normal, countably paracompact spaces have the following property: Every Baire measure extends to a regular Borel measure.
    [Show full text]
  • On Measure-Compactness and Borel Measure-Compactness
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Osaka City University Repository Okada, S. and Okazaki, Y. Osaka J. Math. 15 (1978), 183-191 ON MEASURE-COMPACTNESS AND BOREL MEASURE-COMPACTNESS SUSUMU OKADA AND YOSHIAKI OKAZAKI (Received December 6, 1976) 1. Introduction Moran has introduced "measure-compactness' and "strongly measure- compactness' in [7], [8], For measure-compactness, Kirk has independently presented in [4]. For Borel measures, "Borel measure-compactness' and "property B" are introduced by Gardner [1], We study, defining "a weak Radon space", the relation among these proper- ties at the last of §2. Furthermore we investigate the stability of countable unions in §3 and the hereditariness in §4. The authors wish to thank Professor Y. Yamasaki for his suggestions. 2. Preliminaries and fundamental results All spaces considered in this paper are completely regular and Hausdorff. Let Cb(X) be the space of all bounded real-valued continuous functions on a topological space X. By Z(X] we mean the family of zero sets {/" \0) / e Cb(X)} c and U(X) the family of cozero sets {Z \ Z<=Z(X)}. We denote by &a(X) the Baίre field, that is, the smallest σ-algebra containing Z(X) and by 1$(X) the σ-algebra generated by all open subsets of X, which is called the Borel field. A Baire measure [resp. Borel measure} is a totally finite, non-negative coun- tably additive set function defined on Άa(X) [resp. 3)(X)]. For a Baire measure μ, the support of μ is supp μ = {x^Xm, μ(U)>0 for every U^ U(X) containing x} .
    [Show full text]
  • Spectral Radii of Bounded Operators on Topological Vector Spaces
    SPECTRAL RADII OF BOUNDED OPERATORS ON TOPOLOGICAL VECTOR SPACES VLADIMIR G. TROITSKY Abstract. In this paper we develop a version of spectral theory for bounded linear operators on topological vector spaces. We show that the Gelfand formula for spectral radius and Neumann series can still be naturally interpreted for operators on topological vector spaces. Of course, the resulting theory has many similarities to the conventional spectral theory of bounded operators on Banach spaces, though there are several im- portant differences. The main difference is that an operator on a topological vector space has several spectra and several spectral radii, which fit a well-organized pattern. 0. Introduction The spectral radius of a bounded linear operator T on a Banach space is defined by the Gelfand formula r(T ) = lim n T n . It is well known that r(T ) equals the n k k actual radius of the spectrum σ(T ) p= sup λ : λ σ(T ) . Further, it is known −1 {| | ∈ } ∞ T i that the resolvent R =(λI T ) is given by the Neumann series +1 whenever λ − i=0 λi λ > r(T ). It is natural to ask if similar results are valid in a moreP general setting, | | e.g., for a bounded linear operator on an arbitrary topological vector space. The author arrived to these questions when generalizing some results on Invariant Subspace Problem from Banach lattices to ordered topological vector spaces. One major difficulty is that it is not clear which class of operators should be considered, because there are several non-equivalent ways of defining bounded operators on topological vector spaces.
    [Show full text]
  • Weak Operator Topology, Operator Ranges and Operator Equations Via Kolmogorov Widths
    Weak operator topology, operator ranges and operator equations via Kolmogorov widths M. I. Ostrovskii and V. S. Shulman Abstract. Let K be an absolutely convex infinite-dimensional compact in a Banach space X . The set of all bounded linear operators T on X satisfying TK ⊃ K is denoted by G(K). Our starting point is the study of the closure WG(K) of G(K) in the weak operator topology. We prove that WG(K) contains the algebra of all operators leaving lin(K) invariant. More precise results are obtained in terms of the Kolmogorov n-widths of the compact K. The obtained results are used in the study of operator ranges and operator equations. Mathematics Subject Classification (2000). Primary 47A05; Secondary 41A46, 47A30, 47A62. Keywords. Banach space, bounded linear operator, Hilbert space, Kolmogorov width, operator equation, operator range, strong operator topology, weak op- erator topology. 1. Introduction Let K be a subset in a Banach space X . We say (with some abuse of the language) that an operator D 2 L(X ) covers K, if DK ⊃ K. The set of all operators covering K will be denoted by G(K). It is a semigroup with a unit since the identity operator is in G(K). It is easy to check that if K is compact then G(K) is closed in the norm topology and, moreover, sequentially closed in the weak operator topology (WOT). It is somewhat surprising that for each absolutely convex infinite- dimensional compact K the WOT-closure of G(K) is much larger than G(K) itself, and in many cases it coincides with the algebra L(X ) of all operators on X .
    [Show full text]
  • An Introduction to Some Aspects of Functional Analysis, 7: Convergence of Operators
    An introduction to some aspects of functional analysis, 7: Convergence of operators Stephen Semmes Rice University Abstract Here we look at strong and weak operator topologies on spaces of bounded linear mappings, and convergence of sequences of operators with respect to these topologies in particular. Contents I The strong operator topology 2 1 Seminorms 2 2 Bounded linear mappings 4 3 The strong operator topology 5 4 Shift operators 6 5 Multiplication operators 7 6 Dense sets 9 7 Shift operators, 2 10 8 Other operators 11 9 Unitary operators 12 10 Measure-preserving transformations 13 II The weak operator topology 15 11 Definitions 15 1 12 Multiplication operators, 2 16 13 Dual linear mappings 17 14 Shift operators, 3 19 15 Uniform boundedness 21 16 Continuous linear functionals 22 17 Bilinear functionals 23 18 Compactness 24 19 Other operators, 2 26 20 Composition operators 27 21 Continuity properties 30 References 32 Part I The strong operator topology 1 Seminorms Let V be a vector space over the real or complex numbers. A nonnegative real-valued function N(v) on V is said to be a seminorm on V if (1.1) N(tv) = |t| N(v) for every v ∈ V and t ∈ R or C, as appropriate, and (1.2) N(v + w) ≤ N(v) + N(w) for every v, w ∈ V . Here |t| denotes the absolute value of a real number t, or the modulus of a complex number t. If N(v) > 0 when v 6= 0, then N(v) is a norm on V , and (1.3) d(v, w) = N(v − w) defines a metric on V .
    [Show full text]
  • On the Cyclicity and Synthesis of Diagonal Operators on the Space of Functions Analytic on a Disk
    ON THE CYCLICITY AND SYNTHESIS OF DIAGONAL OPERATORS ON THE SPACE OF FUNCTIONS ANALYTIC ON A DISK Ian Deters A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of A DISSERTATION May 2009 Committee: Steven Seubert, Advisor Sachi Sakthivel, Graduate Faculty Representative Neal Carothers Juan Bes ii ABSTRACT Steven Seubert, Advisor A diagonal operator on the space of functions holomorphic on a disk of finite radius is a continuous linear operator having the monomials as eigenvectors. In this dissertation, necessary and sufficient conditions are given for a diagonal operator to be cyclic. Necessary and sufficient conditions are also given for a cyclic diagonal operator to admit spectral synthesis, that is, to have as closed invariant subspaces only the closed linear span of sets of eigenvectors. In particular, it is shown that a cyclic diagonal operator admits synthesis if and only if one vector, not depending on the operator, is cyclic. It is also shown that this is equivalent to existence of sequences of polynomials which seperate and have minimum growth on the eigenvalues of the operator. iii This dissertation is dedicated to my Lord and Savior Jesus Christ. Worthy are You, our Lord and our God, to receive glory and honor and power; for You created all things, and because of Your will they existed, and were created. -Revelation 4:11 We give You thanks, O Lord God, the Alighty, Who are and Who were, because You have taken Your great power and have begun to reign. -Revelation 11:17 Great and marvelous are Your works, O Lord God, the Almighty; Righteous and true are Your ways, King of the nations! Who will not fear, O Lord, and glorify Your name? For You alone are holy; For all the nations will come and worship before You, for Your righteous acts have been revealed.
    [Show full text]
  • Locality in GNS Representations of Deformation Quantization
    Locality in GNS Representations of Deformation Quantization Stefan Waldmann∗ Fakult¨at f¨ur Physik Universit¨at Freiburg Hermann-Herder-Str. 3 79104 Freiburg i. Br., F. R. G FR-THEP-99/2 March 1999 Abstract In the framework of deformation quantization we apply the formal GNS construction to find representations of the deformed algebras in pre-Hilbert spaces over C[[λ]] and establish the notion of local operators in these pre-Hilbert spaces. The commutant within the local operators is used to distinguish ‘thermal’ from ‘pure’ representations. The computation of the local commutant is exemplified in various situations leading to the physically reasonable distinction between thermal representations and pure ones. Moreover, an analogue of von Neumann’s double commutant theorem is proved in the particular situation of a GNS representation with respect to a KMS functional and for the Schr¨odinger representation on cotangent bundles. Finally we prove a formal version of the Tomita-Takesaki theorem. arXiv:math/9903053v1 [math.QA] 9 Mar 1999 ∗[email protected] 1 Contents 1 Introduction 3 2 Basic definitions 4 3 Locality in GNS representations and commutants 5 4 Faithful positive linear functionals 10 5 Basic examples 11 6 Strong topologies and von Neumann algebras 17 7 Tomita-Takesaki theory 21 A Pre-Hilbert spaces over ordered rings and the GNS construction 23 B Formal series and λ-adic topology 24 2 1 Introduction The concept of deformation quantization and star products has been introduced by Bayen, Flato, Frønsdal, Lichnerowicz, and Sternheimer in [4] and is now a well-established and successful quan- tization procedure.
    [Show full text]