Fredholm Operators and the Family Index
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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. -
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. -
18.102 Introduction to Functional Analysis Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.102 Introduction to Functional Analysis Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 108 LECTURE NOTES FOR 18.102, SPRING 2009 Lecture 19. Thursday, April 16 I am heading towards the spectral theory of self-adjoint compact operators. This is rather similar to the spectral theory of self-adjoint matrices and has many useful applications. There is a very effective spectral theory of general bounded but self- adjoint operators but I do not expect to have time to do this. There is also a pretty satisfactory spectral theory of non-selfadjoint compact operators, which it is more likely I will get to. There is no satisfactory spectral theory for general non-compact and non-self-adjoint operators as you can easily see from examples (such as the shift operator). In some sense compact operators are ‘small’ and rather like finite rank operators. If you accept this, then you will want to say that an operator such as (19.1) Id −K; K 2 K(H) is ‘big’. We are quite interested in this operator because of spectral theory. To say that λ 2 C is an eigenvalue of K is to say that there is a non-trivial solution of (19.2) Ku − λu = 0 where non-trivial means other than than the solution u = 0 which always exists. If λ =6 0 we can divide by λ and we are looking for solutions of −1 (19.3) (Id −λ K)u = 0 −1 which is just (19.1) for another compact operator, namely λ K: What are properties of Id −K which migh show it to be ‘big? Here are three: Proposition 26. -
5 Toeplitz Operators
2008.10.07.08 5 Toeplitz Operators There are two signal spaces which will be important for us. • Semi-infinite signals: Functions x ∈ ℓ2(Z+, R). They have a Fourier transform g = F x, where g ∈ H2; that is, g : D → C is analytic on the open unit disk, so it has no poles there. • Bi-infinite signals: Functions x ∈ ℓ2(Z, R). They have a Fourier transform g = F x, where g ∈ L2(T). Then g : T → C, and g may have poles both inside and outside the disk. 5.1 Causality and Time-invariance Suppose G is a bounded linear map G : ℓ2(Z) → ℓ2(Z) given by yi = Gijuj j∈Z X where Gij are the coefficients in its matrix representation. The map G is called time- invariant or shift-invariant if it is Toeplitz , which means Gi−1,j = Gi,j+1 that is G is constant along diagonals from top-left to bottom right. Such matrices are convolution operators, because they have the form ... a0 a−1 a−2 a1 a0 a−1 a−2 G = a2 a1 a0 a−1 a2 a1 a0 ... Here the box indicates the 0, 0 element, since the matrix is indexed from −∞ to ∞. With this matrix, we have y = Gu if and only if yi = ai−juj k∈Z X We say G is causal if the matrix G is lower triangular. For example, the matrix ... a0 a1 a0 G = a2 a1 a0 a3 a2 a1 a0 ... 1 5 Toeplitz Operators 2008.10.07.08 is both causal and time-invariant. -
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. -
L∞ , Let T : L ∞ → L∞ Be Defined by Tx = ( X(1), X(2) 2 , X(3) 3 ,... ) Pr
ASSIGNMENT II MTL 411 FUNCTIONAL ANALYSIS 1. For x = (x(1); x(2);::: ) 2 l1, let T : l1 ! l1 be defined by ( ) x(2) x(3) T x = x(1); ; ;::: 2 3 Prove that (i) T is a bounded linear operator (ii) T is injective (iii) Range of T is not a closed subspace of l1. 2. If T : X ! Y is a linear operator such that there exists c > 0 and 0 =6 x0 2 X satisfying kT xk ≤ ckxk 8 x 2 X; kT x0k = ckx0k; then show that T 2 B(X; Y ) and kT k = c. 3. For x = (x(1); x(2);::: ) 2 l2, consider the right shift operator S : l2 ! l2 defined by Sx = (0; x(1); x(2);::: ) and the left shift operator T : l2 ! l2 defined by T x = (x(2); x(3);::: ) Prove that (i) S is a abounded linear operator and kSk = 1. (ii) S is injective. (iii) S is, in fact, an isometry. (iv) S is not surjective. (v) T is a bounded linear operator and kT k = 1 (vi) T is not injective. (vii) T is not an isometry. (viii) T is surjective. (ix) TS = I and ST =6 I. That is, neither S nor T is invertible, however, S has a left inverse and T has a right inverse. Note that item (ix) illustrates the fact that the Banach algebra B(X) is not in general commutative. 4. Show with an example that for T 2 B(X; Y ) and S 2 B(Y; Z), the equality in the submulti- plicativity of the norms kS ◦ T k ≤ kSkkT k may not hold. -
Floer Homology on Symplectic Manifolds
Floer Homology on Symplectic Manifolds KWONG, Kwok Kun A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Philosophy in Mathematics c The Chinese University of Hong Kong August 2008 The Chinese University of Hong Kong holds the copyright of this thesis. Any person(s) intending to use a part or whole of the materials in the thesis in a proposed publication must seek copyright release from the Dean of the Graduate School. Thesis/Assessment Committee Professor Wan Yau Heng Tom (Chair) Professor Au Kwok Keung Thomas (Thesis Supervisor) Professor Tam Luen Fai (Committee Member) Professor Dusa McDuff (External Examiner) Floer Homology on Symplectic Manifolds i Abstract The Floer homology was invented by A. Floer to solve the famous Arnold conjecture, which gives the lower bound of the fixed points of a Hamiltonian symplectomorphism. Floer’s theory can be regarded as an infinite dimensional version of Morse theory. The aim of this dissertation is to give an exposition on Floer homology on symplectic manifolds. We will investigate the similarities and differences between the classical Morse theory and Floer’s theory. We will also explain the relation between the Floer homology and the topology of the underlying manifold. Floer Homology on Symplectic Manifolds ii ``` ½(ÝArnold øî×bÛñøÝFÝóê ×Íì§A. FloerxñÝFloer!§¡XÝ9 Floer!§¡Ú Morse§¡Ý×ÍP§îÌÍÍ¡Zº EøîÝFloer!®×Í+&ƺD¡BÎMorse§¡ Floer§¡Ý8«õ! ¬ÙÕFloer!ÍXòøc RÝn; Floer Homology on Symplectic Manifolds iii Acknowledgements I would like to thank my advisor Prof. Thomas Au Kwok Keung for his encouragement in writing this thesis. I am grateful to all my teachers. -
Operators, Functions, and Systems: an Easy Reading
http://dx.doi.org/10.1090/surv/092 Selected Titles in This Series 92 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 1: Hardy, Hankel, and Toeplitz, 2002 91 Richard Montgomery, A tour of subriemannian geometries, their geodesies and applications, 2002 90 Christian Gerard and Izabella Laba, Multiparticle quantum scattering in constant magnetic fields, 2002 89 Michel Ledoux, The concentration of measure phenomenon, 2001 88 Edward Frenkel and David Ben-Zvi, Vertex algebras and algebraic curves, 2001 87 Bruno Poizat, Stable groups, 2001 86 Stanley N. Burris, Number theoretic density and logical limit laws, 2001 85 V. A. Kozlov, V. G. Maz'ya, and J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations, 2001 84 Laszlo Fuchs and Luigi Salce, Modules over non-Noetherian domains, 2001 83 Sigurdur Helgason, Groups and geometric analysis: Integral geometry, invariant differential operators, and spherical functions, 2000 82 Goro Shimura, Arithmeticity in the theory of automorphic forms, 2000 81 Michael E. Taylor, Tools for PDE: Pseudodifferential operators, paradifferential operators, and layer potentials, 2000 80 Lindsay N. Childs, Taming wild extensions: Hopf algebras and local Galois module theory, 2000 79 Joseph A. Cima and William T. Ross, The backward shift on the Hardy space, 2000 78 Boris A. Kupershmidt, KP or mKP: Noncommutative mathematics of Lagrangian, Hamiltonian, and integrable systems, 2000 77 Fumio Hiai and Denes Petz, The semicircle law, free random variables and entropy, 2000 76 Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichmiiller theory, 2000 75 Greg Hjorth, Classification and orbit equivalence relations, 2000 74 Daniel W. -
Basic Theory of Fredholm Operators Annali Della Scuola Normale Superiore Di Pisa, Classe Di Scienze 3E Série, Tome 21, No 2 (1967), P
ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze MARTIN SCHECHTER Basic theory of Fredholm operators Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3e série, tome 21, no 2 (1967), p. 261-280 <http://www.numdam.org/item?id=ASNSP_1967_3_21_2_261_0> © Scuola Normale Superiore, Pisa, 1967, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ BASIC THEORY OF FREDHOLM OPERATORS (*) MARTIN SOHECHTER 1. Introduction. " A linear operator A from a Banach space X to a Banach space Y is called a Fredholm operator if 1. A is closed 2. the domain D (A) of A is dense in X 3. a (A), the dimension of the null space N (A) of A, is finite 4. .R (A), the range of A, is closed in Y 5. ~ (A), the codimension of R (A) in Y, is finite. The terminology stems from the classical Fredholm theory of integral equations. Special types of Fredholm operators were considered by many authors since that time, but systematic treatments were not given until the work of Atkinson [1]~ Gohberg [2, 3, 4] and Yood [5]. These papers conside- red bounded operators. -
Set Theory and Operator Algebras
SET THEORY AND OPERATOR ALGEBRAS ILIJAS FARAH AND ERIC WOFSEY These notes are based on the six-hour Appalachian Set Theory workshop given by Ilijas Farah on February 9th, 2008 at Carnegie Mellon Univer- sity. The first half of the workshop (Sections 1-3) consisted of a review of Hilbert space theory and an introduction to C∗-algebras, and the second half (Sections 4–6) outlined a few set-theoretic problems relating to C∗-algebras. The number and variety of topics covered in the workshop was unfortunately limited by the available time. Good general references on Hilbert spaces and C∗-algebras include [8], [10], [14], [27], and [35]. An introduction to spectral theory is given in [9]. Most of the omitted proofs can be found in most of these references. For a survey of applications of set theory to operator algebras, see [36]. Acknowledgments. We would like to thank Nik Weaver for giving us a kind permission to include some of his unpublished results. I.F. would like to thank George Elliott, N. Christopher Phillips, Efren Ruiz, Juris Stepr¯ans, Nik Weaver and the second author for many conversations related to the topics presented here. I.F. is partially supported by NSERC. Contents 1. Hilbert spaces and operators 2 1.1. Normal operators and the spectral theorem 3 1.2. The spectrum of an operator 6 2. C∗-algebras 7 2.1. Some examples of C∗-algebras 8 2.2. L∞(X, µ) 9 2.3. Automatic continuity and the Gelfand transform 10 3. Positivity, states and the GNS construction 12 3.1. -
Notex on Fredholm (And Compact) Operators
Notex on Fredholm (and compact) operators October 5, 2009 Abstract In these separate notes, we give an exposition on Fredholm operators between Banach spaces. In particular, we prove the theorems stated in the last section of the first lecture 1. Contents 1 Fredholm operators: basic properties 2 2 Compact operators: basic properties 3 3 Compact operators: the Fredholm alternative 4 4 The relation between Fredholm and compact operators 7 1emphasize that some of the extra-material is just for your curiosity and is not needed for the promised proofs. It is a good exercise for you to cross out the parts which are not needed 1 1 Fredholm operators: basic properties Let E and F be two Banach spaces. We denote by L(E, F) the space of bounded linear operators from E to F. Definition 1.1 A bounded operator T : E −→ F is called Fredholm if Ker(A) and Coker(A) are finite dimensional. We denote by F(E, F) the space of all Fredholm operators from E to F. The index of a Fredholm operator A is defined by Index(A) := dim(Ker(A)) − dim(Coker(A)). Note that a consequence of the Fredholmness is the fact that R(A) = Im(A) is closed. Here are the first properties of Fredholm operators. Theorem 1.2 Let E, F, G be Banach spaces. (i) If B : E −→ F and A : F −→ G are bounded, and two out of the three operators A, B and AB are Fredholm, then so is the third, and Index(A ◦ B) = Index(A) + Index(B). -
Fredholm Operators and Spectral Flow, Canad
Fredholm Operators and Spectral Flow Nils Waterstraat arXiv:1603.02009v1 [math.FA] 7 Mar 2016 2 Contents 1 Linear Operators 7 1.1 BoundedOperatorsandSubspaces . ...... 7 1.2 ClosedOperators................................. ... 10 1.3 SpectralTheory.................................. ... 15 2 Selfadjoint Operators 21 2.1 DefinitionsandBasicProperties . ....... 21 2.2 Spectral Theoryof Selfadjoint Operators . ........... 26 3 The Gap Topology 29 3.1 DefinitionandProperties . ..... 29 3.2 StabilityofSpectra.............................. ..... 34 3.3 Spaces ofSelfadjoint FredholmOperators . .......... 36 4 The Spectral Flow 39 4.1 DefinitionoftheSpectralFlow . ...... 39 4.2 PropertiesandUniqueness. ...... 42 4.3 CrossingForms ................................... .. 46 5 A Simple Example and a Glimpse at the Literature 49 5.1 ASimpleExample .................................. 49 5.2 AGlimpseattheLiterature. ..... 51 3 4 CONTENTS Introduction Fredholm operators are one of the most important classes of linear operators in mathematics. They were introduced around 1900 in the study of integral operators and by definition they share many properties with linear operators between finite dimensional spaces. They appear naturally in global analysis which is a branch of pure mathematics concerned with the global and topological properties of systems of differential equations on manifolds. One of the basic important facts says that every linear elliptic differential operator acting on sections of a vector bundle over a closed manifold induces a Fredholm operator on a suitable Banach space comple- tion of bundle sections. Every Fredholm operator has an integer-valued index, which is invariant under deformations of the operator, and the most fundamental theorem in global analysis is the Atiyah-Singer index theorem [AS68] which gives an explicit formula for the Fredholm index of an elliptic operator on a closed manifold in terms of topological data.