DOCSLIB.ORG

Linear Functional Analysis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Linear Functional Analysis Linear Functional Analysis Joan Cerdà Graduate Studies in Mathematics Volume 116 American Mathematical Society Real Sociedad Matemática Española http://dx.doi.org/10.1090/gsm/116 Linear Functional Analysis Linear Functional Analysis Joan Cerdà Graduate Studies in Mathematics Volume 116 American Mathematical Society Providence, Rhode Island Real Sociedad Matemática Española Madrid, Spain Editorial Board of Graduate Studies in Mathematics David Cox (Chair) Rafe Mazzeo Martin Scharlemann Gigliola Staffilani Editorial Committee of the Real Sociedad Matem´atica Espa˜nola Guillermo P. Curbera, Director Luis Al´ıas Alberto Elduque Emilio Carrizosa Rosa Mar´ıa Mir´o Bernardo Cascales Pablo Pedregal Javier Duoandikoetxea Juan Soler 2010 Mathematics Subject Classification. Primary 46–01; Secondary 46Axx, 46Bxx, 46Exx, 46Fxx, 46Jxx, 47B15. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-116 Library of Congress Cataloging-in-Publication Data Cerd`a, Joan, 1942– Linear functional analysis / Joan Cerd`a. p. cm. — (Graduate studies in mathematics ; v. 116) Includes bibliographical references and index. ISBN 978-0-8218-5115-9 (alk. paper) 1. Functional analysis. I. Title. QA321.C47 2010 515.7—dc22 2010006449 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2010 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 151413121110 To Carla and Marc Contents Preface xi Chapter 1. Introduction 1 §1.1. Topological spaces 1 §1.2. Measure and integration 8 §1.3. Exercises 21 Chapter 2. Normed spaces and operators 25 §2.1. Banach spaces 26 §2.2. Linear operators 39 §2.3. Hilbert spaces 45 §2.4. Convolutions and summability kernels 52 §2.5. The Riesz-Thorin interpolation theorem 59 §2.6. Applications to linear differential equations 63 §2.7. Exercises 69 Chapter 3. Fr´echet spaces and Banach theorems 75 §3.1. Fr´echet spaces 76 §3.2. Banach theorems 82 §3.3. Exercises 88 Chapter 4. Duality 93 §4.1. The dual of a Hilbert space 93 §4.2. Applications of the Riesz representation theorem 98 §4.3. The Hahn-Banach theorem 106 vii viii Contents §4.4. Spectral theory of compact operators 114 §4.5. Exercises 122 Chapter 5. Weak topologies 127 §5.1. Weak convergence 127 §5.2. Weak and weak* topologies 128 §5.3. An application to the Dirichlet problem in the disc 132 §5.4. Exercises 138 Chapter 6. Distributions 143 §6.1. Test functions 144 §6.2. The distributions 146 §6.3. Differentiation of distributions 150 §6.4. Convolution of distributions 154 §6.5. Distributional differential equations 161 §6.6. Exercises 175 Chapter 7. Fourier transform and Sobolev spaces 181 §7.1. The Fourier integral 182 §7.2. The Schwartz class S 186 §7.3. Tempered distributions 189 §7.4. Fourier transform and signal theory 195 §7.5. The Dirichlet problem in the half-space 200 §7.6. Sobolev spaces 206 §7.7. Applications 213 §7.8. Exercises 222 Chapter 8. Banach algebras 227 §8.1. Definition and examples 228 §8.2. Spectrum 229 §8.3. Commutative Banach algebras 234 §8.4. C∗-algebras 238 §8.5. Spectral theory of bounded normal operators 241 §8.6. Exercises 250 Chapter 9. Unbounded operators in a Hilbert space 257 §9.1. Definitions and basic properties 258 §9.2. Unbounded self-adjoint operators 262 Contents ix §9.3. Spectral representation of unbounded self-adjoint operators 273 §9.4. Unbounded operators in quantum mechanics 277 §9.5. Appendix: Proof of the spectral theorem 287 §9.6. Exercises 295 Hints to exercises 299 Bibliography 321 Index 325 Preface The aim of this book is to present the basic facts of linear functional anal- ysis related to applications to some fundamental aspects of mathematical analysis. If mathematics is supposed to show common general facts and struc- tures of particular results, functional analysis does this while dealing with classical problems, many of them related to ordinary and partial differential equations, integral equations, harmonic analysis, function theory, and the calculus of variations. In functional analysis, individual functions satisfying specific equations are replaced by classes of functions and transforms which are determined by each particular problem. The objects of functional analysis are spaces and operators acting between them which, after systematic studies intertwining linear and topological or metric structures, appear to be behind classical problems in a kind of cleaning process. In order to make the scope of functional analysis clearer, I have chosen to sacrifice generality for the sake of an easier understanding of its methods, and to show how they clarify what is essential in analytical problems. I have tried to avoid the introduction of cold abstractions and unnecessary terminology in further developments and, when choosing the different topics, I have included some applications that connect functional analysis with other areas. The text is based on a graduate course taught at the Universitat de Barcelona, with some additions, mainly to make it more self-contained. The material in the first chapters could be adapted as an introductory course on functional analysis, aiming to present the role of duality in analysis, and xi xii Preface also the spectral theory of compact linear operators in the context of Hilbert and Banach spaces. In this first part of the book, the mutual influence between functional analysis and other areas of analysis is shown when studying duality, with von Neumann’s proof of the Radon-Nikodym theorem based on the Riesz representation theorem for the dual of a Hilbert space, followed by the rep- resentations of the duals of the Lp spaces and of C(K), in this case by means of complex Borel measures. The reader will also see how to deal with initial and boundary value problems in ordinary linear differential equations via the use of integral operators. Moreover examples are included that illustrate how functional analytic methods are useful in the study of Fourier series. In the second part, distributions provide a natural framework extend- ing some fundamental operations in analysis. Convolution and the Fourier transform are included as useful tools for dealing with partial differential operators, with basic notions such as fundamental solutions and Green’s functions. Distributions are also appropriate for the introduction of Sobolev spaces, which are very useful for the study of the solutions of partial differential equations. A clear example is provided by the resolution of the Dirichlet problem and the description of the eigenvalues of the Laplacian, in combi- nation with Hilbert space techniques. The last two chapters are essentially devoted to the spectral theory of bounded and unbounded self-adjoint operators, which is presented by us- ing the Gelfand transform for Banach algebras. This spectral theory is illustrated with an introduction to the basic axioms of quantum mechanics, which motivated many studies in the Hilbert space theory. Some very short historical comments have been included, mainly by means of footnotes. For a good overview of the evolution of functional analysis, J. Dieudonn´e’s and A. F. Monna’s books, [10] and [31], are two good references. The limitation of space has forced us to leave out many other important topics that could, and probably should, have been included. Among them are the geometry of Banach spaces, a general theory of locally convex spaces and structure theory of Fr´echet spaces, functional calculus of nonnormal operators, groups and semigroups of operators, invariant subspaces, index theory, von Neumann algebras, and scattering theory. Fortunately, many excellent texts dealing with these subjects are available and a few references have been selected for further study. Preface xiii A small number of references have been gathered at the end of each chapter to focus the reader’s attention on some appropriate items from a general bibliographical list of 44 items. Almost 240 exercises are gathered at the end of the chapters and form an important part of the book. They are intended to help the reader to develop techniques and working knowledge of functional analysis. These exercises are highly nonuniform in difficulty. Some are very simple, to aid in better understanding of the concepts employed, whereas others are fairly challenging for the beginners. Hints and solutions are provided at the end of the book. The prerequisites are very standard. Although it is assumed that the reader has some a priori knowledge of general topology, integral calculus with Lebesgue measure, and elementary aspects of normed or Hilbert spaces, a review of the basic aspects of these topics has been included in the first chapters. I turn finally to the pleasant task of thanking those who helped me during the writing. Particular thanks are due to Javier Soria, who revised most of the manuscript and proposed important corrections and suggestions. I have also received valuable advice and criticism from Mar´ıa J. Carro and Joaquim Ortega-Cerd`a. I have been very fortunate to have received their assistance.
Load more

Recommended publications
  • Complex Measures 1 11
    Tutorial 11: Complex Measures 1 11. Complex Measures In the following, (Ω, F) denotes an arbitrary measurable space. Definition 90 Let (an)n≥1 be a sequence of complex numbers. We a say that ( n)n≥1 has the permutation property if and only if, for ∗ ∗ +∞ 1 all bijections σ : N → N ,theseries k=1 aσ(k) converges in C Exercise 1. Let (an)n≥1 be a sequence of complex numbers. 1. Show that if (an)n≥1 has the permutation property, then the same is true of (Re(an))n≥1 and (Im(an))n≥1. +∞ 2. Suppose an ∈ R for all n ≥ 1. Show that if k=1 ak converges: +∞ +∞ +∞ + − |ak| =+∞⇒ ak = ak =+∞ k=1 k=1 k=1 1which excludes ±∞ as limit. www.probability.net Tutorial 11: Complex Measures 2 Exercise 2. Let (an)n≥1 be a sequence in R, such that the series +∞ +∞ k=1 ak converges, and k=1 |ak| =+∞.LetA>0. We define: + − N = {k ≥ 1:ak ≥ 0} ,N = {k ≥ 1:ak < 0} 1. Show that N + and N − are infinite. 2. Let φ+ : N∗ → N + and φ− : N∗ → N − be two bijections. Show the existence of k1 ≥ 1 such that: k1 aφ+(k) ≥ A k=1 3. Show the existence of an increasing sequence (kp)p≥1 such that: kp aφ+(k) ≥ A k=kp−1+1 www.probability.net Tutorial 11: Complex Measures 3 for all p ≥ 1, where k0 =0. 4. Consider the permutation σ : N∗ → N∗ defined informally by: φ− ,φ+ ,...,φ+ k ,φ− ,φ+ k ,...,φ+ k ,..
    [Show full text]
  • Appendix A. Measure and Integration
    Appendix A. Measure and integration We suppose the reader is familiar with the basic facts concerning set theory and integration as they are presented in the introductory course of analysis. In this appendix, we review them briefly, and add some more which we shall need in the text. Basic references for proofs and a detailed exposition are, e.g., [[ H a l 1 ]] , [[ J a r 1 , 2 ]] , [[ K F 1 , 2 ]] , [[ L i L ]] , [[ R u 1 ]] , or any other textbook on analysis you might prefer. A.1 Sets, mappings, relations A set is a collection of objects called elements. The symbol card X denotes the cardi- nality of the set X. The subset M consisting of the elements of X which satisfy the conditions P1(x),...,Pn(x) is usually written as M = { x ∈ X : P1(x),...,Pn(x) }.A set whose elements are certain sets is called a system or family of these sets; the family of all subsystems of a given X is denoted as 2X . The operations of union, intersection, and set difference are introduced in the standard way; the first two of these are commutative, associative, and mutually distributive. In a { } system Mα of any cardinality, the de Morgan relations , X \ Mα = (X \ Mα)and X \ Mα = (X \ Mα), α α α α are valid. Another elementary property is the following: for any family {Mn} ,whichis { } at most countable, there is a disjoint family Nn of the same cardinality such that ⊂ \ ∪ \ Nn Mn and n Nn = n Mn.Theset(M N) (N M) is called the symmetric difference of the sets M,N and denoted as M #N.
    [Show full text]
  • Fundamentals of Functional Analysis Kluwer Texts in the Mathematical Sciences
    Fundamentals of Functional Analysis Kluwer Texts in the Mathematical Sciences VOLUME 12 A Graduate-Level Book Series The titles published in this series are listed at the end of this volume. Fundamentals of Functional Analysis by S. S. Kutateladze Sobolev Institute ofMathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia Springer-Science+Business Media, B.V. A C.I.P. Catalogue record for this book is available from the Library of Congress ISBN 978-90-481-4661-1 ISBN 978-94-015-8755-6 (eBook) DOI 10.1007/978-94-015-8755-6 Translated from OCHOBbI Ij)YHK~HOHaJIl>HODO aHaJIHsa. J/IS;l\~ 2, ;l\OIIOJIHeHHoe., Sobo1ev Institute of Mathematics, Novosibirsk, © 1995 S. S. Kutate1adze Printed on acid-free paper All Rights Reserved © 1996 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1996. Softcover reprint of the hardcover 1st edition 1996 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Contents Preface to the English Translation ix Preface to the First Russian Edition x Preface to the Second Russian Edition xii Chapter 1. An Excursion into Set Theory 1.1. Correspondences . 1 1.2. Ordered Sets . 3 1.3. Filters . 6 Exercises . 8 Chapter 2. Vector Spaces 2.1. Spaces and Subspaces ... ......................... 10 2.2. Linear Operators . 12 2.3. Equations in Operators ........................ .. 15 Exercises . 18 Chapter 3. Convex Analysis 3.1.
    [Show full text]
  • The Uniform Boundedness Theorem in Asymmetric Normed Spaces
    Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2012, Article ID 809626, 8 pages doi:10.1155/2012/809626 Research Article The Uniform Boundedness Theorem in Asymmetric Normed Spaces C. Alegre,1 S. Romaguera,1 and P. Veeramani2 1 Instituto Universitario de Matematica´ Pura y Aplicada, Universitat Politecnica` de Valencia,` 46022 Valencia, Spain 2 Department of Mathematics, Indian Institute of Technology Madras, Chennai 6000 36, India Correspondence should be addressed to C. Alegre, [email protected] Received 5 July 2012; Accepted 27 August 2012 Academic Editor: Yong Zhou Copyright q 2012 C. Alegre et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We obtain a uniform boundedness type theorem in the frame of asymmetric normed spaces. The classical result for normed spaces follows as a particular case. 1. Introduction Throughout this paper the letters R and R will denote the set of real numbers and the set of nonnegative real numbers, respectively. The book of Aliprantis and Border 1 provides a good basic reference for functional analysis in our context. Let X be a real vector space. A function p : X → R is said to be an asymmetric norm on X 2, 3 if for all x, y ∈ X,andr ∈ R , i pxp−x0 if and only if x 0; ii prxrpx; iii px y ≤ pxpy. The pair X, p is called an asymmetric normed space. Asymmetric norms are also called quasinorms in 4–6, and nonsymmetric norms in 7.
    [Show full text]
  • FUNCTIONAL ANALYSIS 1. Banach and Hilbert Spaces in What
    FUNCTIONAL ANALYSIS PIOTR HAJLASZ 1. Banach and Hilbert spaces In what follows K will denote R of C. Definition. A normed space is a pair (X, k · k), where X is a linear space over K and k · k : X → [0, ∞) is a function, called a norm, such that (1) kx + yk ≤ kxk + kyk for all x, y ∈ X; (2) kαxk = |α|kxk for all x ∈ X and α ∈ K; (3) kxk = 0 if and only if x = 0. Since kx − yk ≤ kx − zk + kz − yk for all x, y, z ∈ X, d(x, y) = kx − yk defines a metric in a normed space. In what follows normed paces will always be regarded as metric spaces with respect to the metric d. A normed space is called a Banach space if it is complete with respect to the metric d. Definition. Let X be a linear space over K (=R or C). The inner product (scalar product) is a function h·, ·i : X × X → K such that (1) hx, xi ≥ 0; (2) hx, xi = 0 if and only if x = 0; (3) hαx, yi = αhx, yi; (4) hx1 + x2, yi = hx1, yi + hx2, yi; (5) hx, yi = hy, xi, for all x, x1, x2, y ∈ X and all α ∈ K. As an obvious corollary we obtain hx, y1 + y2i = hx, y1i + hx, y2i, hx, αyi = αhx, yi , Date: February 12, 2009. 1 2 PIOTR HAJLASZ for all x, y1, y2 ∈ X and α ∈ K. For a space with an inner product we define kxk = phx, xi . Lemma 1.1 (Schwarz inequality).
    [Show full text]
  • The Index of Normal Fredholm Elements of C* -Algebras
    proceedings of the american mathematical society Volume 113, Number 1, September 1991 THE INDEX OF NORMAL FREDHOLM ELEMENTS OF C*-ALGEBRAS J. A. MINGO AND J. S. SPIELBERG (Communicated by Palle E. T. Jorgensen) Abstract. Examples are given of normal elements of C*-algebras that are invertible modulo an ideal and have nonzero index, in contrast to the case of Fredholm operators on Hubert space. It is shown that this phenomenon occurs only along the lines of these examples. Let T be a bounded operator on a Hubert space. If the range of T is closed and both T and T* have a finite dimensional kernel then T is Fredholm, and the index of T is dim(kerT) - dim(kerT*). If T is normal then kerT = ker T*, so a normal Fredholm operator has index 0. Let us consider a generalization of the notion of Fredholm operator intro- duced by Atiyah. Let X be a compact Hausdorff space and consider continuous functions T: X —>B(H), where B(H) is the set of bounded linear operators on a separable infinite dimensional Hubert space with the norm topology. The set of such functions forms a C*- algebra C(X) <g>B(H). A function T is Fredholm if T(x) is Fredholm for each x . Atiyah [1, Appendix] showed how such an element has an index which is an element of K°(X). Suppose that T is Fredholm and T(x) is normal for each x. Is the index of T necessarily 0? There is a generalization of this question that we would like to consider.
    [Show full text]
  • Functional Analysis Lecture Notes Chapter 2. Operators on Hilbert Spaces
    FUNCTIONAL ANALYSIS LECTURE NOTES CHAPTER 2. OPERATORS ON HILBERT SPACES CHRISTOPHER HEIL 1. Elementary Properties and Examples First recall the basic definitions regarding operators. Definition 1.1 (Continuous and Bounded Operators). Let X, Y be normed linear spaces, and let L: X Y be a linear operator. ! (a) L is continuous at a point f X if f f in X implies Lf Lf in Y . 2 n ! n ! (b) L is continuous if it is continuous at every point, i.e., if fn f in X implies Lfn Lf in Y for every f. ! ! (c) L is bounded if there exists a finite K 0 such that ≥ f X; Lf K f : 8 2 k k ≤ k k Note that Lf is the norm of Lf in Y , while f is the norm of f in X. k k k k (d) The operator norm of L is L = sup Lf : k k kfk=1 k k (e) We let (X; Y ) denote the set of all bounded linear operators mapping X into Y , i.e., B (X; Y ) = L: X Y : L is bounded and linear : B f ! g If X = Y = X then we write (X) = (X; X). B B (f) If Y = F then we say that L is a functional. The set of all bounded linear functionals on X is the dual space of X, and is denoted X0 = (X; F) = L: X F : L is bounded and linear : B f ! g We saw in Chapter 1 that, for a linear operator, boundedness and continuity are equivalent.
    [Show full text]
  • Notes on Linear Functional Analysis by M.A Sofi
    Linear Functional Analysis Prof. M. A. Sofi Department of Mathematics University of Kashmir Srinagar-190006 1 Bounded linear operators In this section, we shall characterize continuity of linear operators acting between normed spaces. It turns out that a linear operator is continuous on a normed linear space as soon as it is continuous at the origin or for that matter, at any point of the domain of its definition. Theorem 1.1. Given normed spaces X and Y and a linear map T : X ! Y; then T is continuous on X if and only if 9 c > 0 such that kT (x)k ≤ ckxk; 8 x 2 X: (1) Proof. Assume that T is continuous. In particular, T is continuous at the origin. By the definition of continuity, since T (0) = 0; there exits a neigh- bourhood U at the origin in X such that T (U) ⊆ B(Y ): But then 9r > 0 such that Sr(0) ⊂ U: This gives T (Sr(0)) ⊆ T (U) ⊆ B(Y ): (2) x Let x(6= 0) 2 X; for otherwise, (1) is trivially satisfied. Then kxk 2 Sr(0); so rx that by (2), T 2kxk 2 B(Y ): In other words, 2 kT (x)k ≤ ckxk where c = r which gives (1). Conversely, assume that (1) is true. To show that T is continuous on X; let 1 n n x 2 X and assume that xn −! x in X: Then xn − x −! 0 in X: Thus, 8 > 0; 9 N 3: 8 n > N; kT (xn) − T (x)k = kT (xn − x)k ≤ ckxn − xk ≤ , n which holds for all n > N: In other words, T (xn) −! T (x) in Y and therefore, T is continuous at x 2 X: Since x 2 X was arbitrarily chosen, it follows that T is continuous on X: Definition 1.2.
    [Show full text]
  • C*-Algebraic Spectral Sets, Twisted Groupoids and Operators
    C∗-Algebraic Spectral Sets, Twisted Groupoids and Operators Marius M˘antoiu ∗ July 7, 2020 Abstract We treat spectral problems by twisted groupoid methods. To Hausdorff locally compact groupoids endowed with a continuous 2-cocycle one associates the reduced twisted groupoid C∗-algebra. Elements (or multipliers) of this algebra admit natural Hilbert space representations. We show the relevance of the orbit closure structure of the unit space of the groupoid in dealing with spectra, norms, numerical ranges and ǫ-pseudospectra of the resulting operators. As an ex- ample, we treat a class of pseudo-differential operators introduced recently, associated to group actions. We also prove a Decomposition Principle for bounded operators connected to groupoids, showing that several relevant spectral quantities of these operators coincide with those of certain non-invariant restrictions. This is applied to Toeplitz-like operators with variable coefficients and to band dominated operators on discrete metric spaces. Introduction Many applications of C∗-algebras to spectral analysis rely on a very simple principle: a morphism between two C∗-algebras is contractive and reduces the spectrum of elements. In addition, if it is injective, it is isometric and preserves spectra. While working on this project, we became aware of the fact that such a principle works ef- fectively for much more than spectra (and essential spectra). We call C∗-algebraic spectral set a function Σ sending, for any unital C∗-algebra E , the elements E of this one to closed (usually com- pact) subsets Σ(E |E ) of C , having the above mentioned behavior under C∗-morphisms. Definition 1.1 states this formally.
    [Show full text]
  • Eberlein-Šmulian Theorem and Some of Its Applications
    Eberlein-Šmulian theorem and some of its applications Kristina Qarri Supervisors Trond Abrahamsen Associate professor, PhD University of Agder Norway Olav Nygaard Professor, PhD University of Agder Norway This master’s thesis is carried out as a part of the education at the University of Agder and is therefore approved as a part of this education. However, this does not imply that the University answers for the methods that are used or the conclusions that are drawn. University of Agder, 2014 Faculty of Engineering and Science Department of Mathematics Contents Abstract 1 1 Introduction 2 1.1 Notation and terminology . 4 1.2 Cornerstones in Functional Analysis . 4 2 Basics of weak and weak* topologies 6 2.1 The weak topology . 7 2.2 Weak* topology . 16 3 Schauder Basis Theory 21 3.1 First Properties . 21 3.2 Constructing basic sequences . 37 4 Proof of the Eberlein Šmulian theorem due to Whitley 50 5 The weak topology and the topology of pointwise convergence on C(K) 58 6 A generalization of the Ebrlein-Šmulian theorem 64 7 Some applications to Tauberian operator theory 69 Summary 73 i Abstract The thesis is about Eberlein-Šmulian and some its applications. The goal is to investigate and explain different proofs of the Eberlein-Šmulian theorem. First we introduce the general theory of weak and weak* topology defined on a normed space X. Next we present the definition of a basis and a Schauder basis of a given Banach space. We give some examples and prove the main theorems which are needed to enjoy the proof of the Eberlein-Šmulian theorem given by Pelchynski in 1964.
    [Show full text]
  • Arxiv:1909.02676V3 [Math.DG] 25 Jul 2021 Elsetu Jcb Arcshv Ipera Pcrm.Mor a of Spectrum)
    AN ATLAS ADAPTED TO THE TODA FLOW DAVID MART´INEZ TORRES AND CARLOS TOMEI Abstract. We describe an atlas adapted to the Toda flow on the mani- fold of full flags of any non-compact real semisimple Lie algebra, and on its Hessenberg-type submanifolds. We show that in these local coordinates the Toda flow becomes linear. The local coordinates are used to show that the Toda flow on the manifold of full flags is Morse-Smale, which generalizes the re- sult for traceless matrices in [27] to arbitrary non-compact real semisimple Lie algebras. As a byproduct we describe new features of classical constructions in matrix theory. 1. Introduction The non-periodic Toda lattice is a Hamiltonian model for a wave propagation along n particles in a line proposed by Toda [30]. A change of variables introduced by Flaschka [14] transforms the original O.D.E. into the matrix differential equation X′ = [X, πkX]= T (X), (1) where X runs over Jacobi matrices and πk is the first projection associated to the decomposition of a matrix into its antisymmetric and upper triangular summands. From a mathematical viewpoint (1) is a vector field everywhere defined on the Lie algebra of real traceless matrices. Since it is in Lax form it is tangent to every adjoint orbit and, in particular, to the orbit made of traceless matrices of any fixed simple real spectrum (Jacobi matrices have simple real spectrum). Moreover, formula (1) implies that the Toda vector field T is tangent to any vector subspace which is stable upon taking Lie bracket with antisymmetric matrices.
    [Show full text]
  • On the Origin and Early History of Functional Analysis
    U.U.D.M. Project Report 2008:1 On the origin and early history of functional analysis Jens Lindström Examensarbete i matematik, 30 hp Handledare och examinator: Sten Kaijser Januari 2008 Department of Mathematics Uppsala University Abstract In this report we will study the origins and history of functional analysis up until 1918. We begin by studying ordinary and partial differential equations in the 18th and 19th century to see why there was a need to develop the concepts of functions and limits. We will see how a general theory of infinite systems of equations and determinants by Helge von Koch were used in Ivar Fredholm’s 1900 paper on the integral equation b Z ϕ(s) = f(s) + λ K(s, t)f(t)dt (1) a which resulted in a vast study of integral equations. One of the most enthusiastic followers of Fredholm and integral equation theory was David Hilbert, and we will see how he further developed the theory of integral equations and spectral theory. The concept introduced by Fredholm to study sets of transformations, or operators, made Maurice Fr´echet realize that the focus should be shifted from particular objects to sets of objects and the algebraic properties of these sets. This led him to introduce abstract spaces and we will see how he introduced the axioms that defines them. Finally, we will investigate how the Lebesgue theory of integration were used by Frigyes Riesz who was able to connect all theory of Fredholm, Fr´echet and Lebesgue to form a general theory, and a new discipline of mathematics, now known as functional analysis.
    [Show full text]