DOCSLIB.ORG

Functional Analysis Lecture Notes Chapter 2. Operators on Hilbert Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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. Further, the operator norm is a norm on the space (X; Y ) of all bounded linear operators from X to Y , and we have the composition propertyBthat if L (X; Y ) and K (Y; Z), then KL (X; Z), with KL K L . 2 B 2 B 2 B k k ≤ k k k k Date: February 20, 2006. These notes closely follow and expand on the text by John B. Conway, \A Course in Functional Analysis," Second Edition, Springer, 1990. 1 2 CHRISTOPHER HEIL Exercise 1.2. Suppose that L: X Y is a bounded map of a Banach space X into a Banach space Y . Prove that if there !exists a c > 0 such that Lf c f for every f X, then range(L) is a closed subspace of Y . k k ≥ k k 2 n n Exercise 1.3. Let Cb(R ) be the set of all bounded, continuous functions f : R F. Let n n ! C0(R ) be the set of all continuous functions f : R F such that limjxj!1 f(x) = 0 (i.e., for every " > 0 there exists a compact set K such that! f(x) < " for all x = K). Prove that these are closed subspaces of L1(Rn) (under the L1j-norm;j note that for2a continuous function we have f 1 = sup f(x) ). Define δ : C (Rnk) k F by j j b ! δ(f) = f(0): n 0 Prove that δ is a bounded linear functional on Cb(R ), i.e., δ (Cb) , and find δ . This linear functional is the delta distribution (see also Exercise 1.262below). k k Example 1.4. In finite dimensions, all linear operators are given by matrices, this is just standard finite-dimensional linear algebra. Suppose that X is an n-dimensional complex normed vector space and Y is an m- dimensional complex normed vector space. By definition of dimension, this means that there exists a basis = x ; : : : ; x for X and a basis = y ; : : : ; y for Y . If x X, BX f 1 ng BY f 1 mg 2 then x = c1x1 + + cnxn for a unique choice of scalars ci. Define the coordinates of x with respect to the basis· · · to be BX c1 . Cn [x]BX = . : 2 3 2 cn The vector x is completely determined by its4 coordinates,5 and conversely each vector in Cn is the coordinates of a unique x X. The mapping x [x]BX is a linear mapping of X onto Cn 2Cm 7! . We similarly define [y]BY for vectors y Y . Let A: X Y be a linear map2 (it is automatically2 bounded since X is finite-dimensional). Then A transforms! vectors x X into vectors Ax Y . The vector x is determined by its 2 2 coordinates [x]BX and likewise Ax is determined by its coordinates [Ax]BY . The vectors x and Ax are related through the linear map A; we will show that the coordinate vectors [x]BX and [Ax] are related by multiplication by an m n matrix determined by A. We call this BY × matrix the standard matrix of A with respect to X and Y , and denote it by [A]BX ;BY . That is, the standard matrix should satisfy B B [Ax] = [A] [x] ; x X: BY BX ;BY BX 2 We claim that the standard matrix is the matrix whose columns are the coordinates of the vectors Axk, i.e., [A]B ;B = [Ax1] [Axn]B : X Y 2 BY · · · Y 3 4 5 CHAPTER 2. OPERATORS ON HILBERT SPACES 3 To see this, choose any x X and let x = c1x1 + + cnxn be its unique representation with respect to the basis 2 . Then · · · BX c1 . [A]BX ;BY [x]BX = [Ax1]B [Axn]BY . 2 Y · · · 3 2 3 cn = c4 [Ax ] + + c [Ax ]5 4 5 1 1 BY · · · n n BY = [c Ax + + c Ax ] 1 1 · · · n n BY = [A(c x + + c x )] 1 1 · · · n n BY = [Ax]BY : Exercise 1.5. Extend the idea of the preceding example to show that that any linear mapping L: `2(N) `2(N) (and more generally, L: H K with H, K separable) can be realized in terms of!multiplication by an (infinite but coun!table) matrix. Exercise 1.6. Let A be an m n complex matrix, which we view as a linear transformation A: Cn Cm. The operator×norm of A depends on the choice of norm for Cn and Cm. Compute!an explicit formula for A , in terms of the entries of A, when the norm on Cn and Cm is taken to be the `1 norm. kThenk do the same for the `1 norm. Compare your results to the version of Schur's Lemma given in Theorem 1.23. The following example is one that we will return to many times. Example 1.7. Let en n2N be an orthonormal basis for a separable Hilbert space H. Then we know that every ff gH can be written 2 1 f = f; e e : h ni n n=1 X Fix any sequence of scalars λ = (λn)n2N, and formally define 1 Lf = λ f; e e : (1.1) n h ni n n=1 X This is a \formal" definition because we do not know a priori that the series above will converge|in other words, equation (1.1) may not make sense for every f. 2 Note that if H = ` (N) and e N is the standard basis, then L is given by the formula f ngn2 Lx = (λ x ; λ x ; : : : ); x = (x ; x ; : : : ) `2(N): 1 1 2 2 1 2 2 We will show the following (the `1-norm of the sequence λ is λ = sup λ ). k k1 n j nj (a) The series defining Lf in (1.1) converges for each f H if and only if λ `1. In this case L is a bounded linear mapping of H into itself,2 and L = λ .2 k k k k1 4 CHRISTOPHER HEIL (b) If λ = `1, then L defines an unbounded linear mapping from the domain 2 1 domain(L) = f H : λ f; e 2 < (1.2) 2 j n h nij 1 n=1 n X o (which is dense in H) into H. Proof. (a) Suppose that λ `1, i.e., λ is a bounded sequence. Then for any f we have 2 1 1 λ f; e 2 λ 2 f; e 2 = λ 2 f 2 < ; j n h nij ≤ k k1 h nij k k1 k k 1 n=1 n=1 X X so the series defining Lf converges (because en is an orthonormal sequence). Moreover, the preceding calculation also shows that Lff 2g= 1 λ f; e 2 λ 2 f 2, so we k k n=1 j n h nij ≤ k k1 k k see that L λ 1. On the other hand, by orthonormality we have Len = λnen (i.e., each k k ≤ k k P en is an eigenvector for L with eigenvalue λn). Since en = 1 and Len = λn en = λn we conclude that k k k k j j k k j j L = sup Lf sup Len = sup λn = λ 1: k k kfk=1 k k ≥ n2N k k n2N j j k k The converse direction will be covered by the proof of part (b). (b) Suppose that λ = `1, i.e., λ is not a bounded sequence. Then we can find a subsequence 2 1 N (λnk )k2 such that λnk k for each k. Let cnk = k and define all other cn to be zero. Then 2 1 j j ≥ c = 2 < , so f = c e converges (and c = f; e ). But the formal series n j nj k k 1 n n n n h ni Lf = λncnen does not converge, because P n P P 1 1 1 P k2 c λ 2 = c λ 2 = : j n nj j nk nk j ≥ k2 1 n=1 X Xk=1 Xk=1 In fact, the series defining Lf in (1.1) only converges for those f which lie in the domain defined in (1.2).
Load more

Recommended publications
  • 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]
  • 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]
  • Volterra-Choquet Nonlinear Operators
    Volterra-Choquet nonlinear operators Sorin G. Gal Department of Mathematics and Computer Science, University of Oradea, Universitatii Street No.1, 410087, Oradea, Romania E-mail: [email protected] and Academy of Romanian Scientists, Splaiul Independentei nr. 54 050094 Bucharest, Romania Abstract In this paper we study to what extend some properties of the classical linear Volterra operators could be transferred to the nonlinear Volterra- Choquet operators, obtained by replacing the classical linear integral with respect to the Lebesgue measure, by the nonlinear Choquet integral with respect to a nonadditive set function. Compactness, Lipschitz and cyclic- ity properties are studied. MSC(2010): 47H30, 28A12, 28A25. Keywords and phrases: Choquet integral, monotone, submodular and continuous from below set function, Choquet Lp-space, distorted Lebesgue mea- sures, Volterra-Choquet nonlinear operator, compactness, Lipschitz properties, cyclicity. 1 Introduction Inspired by the electrostatic capacity, G. Choquet has introduced in [5] (see also arXiv:2003.00004v1 [math.CA] 28 Feb 2020 [6]) a concept of integral with respect to a non-additive set function which, in the case when the underlying set function is a σ-additive measure, coincides with the Lebesgue integral. Choquet integral is proved to be a powerful and useful tool in decision making under risk and uncertainty, finance, economics, insurance, pattern recognition, etc (see, e.g., [37] and [38] as well as the references therein). Many new interesting results were obtained as analogs in the framework of Choquet integral of certain known results for the Lebesgue integral. In this sense, we can mention here, for example, the contributions to function spaces theory in [4], to potential theory in [1], to approximation theory in [13]-[17] and to integral equations theory in [18], [19].
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [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]
  • THE SCHR¨ODINGER EQUATION AS a VOLTERRA PROBLEM a Thesis
    THE SCHRODINGER¨ EQUATION AS A VOLTERRA PROBLEM A Thesis by FERNANDO DANIEL MERA Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE May 2011 Major Subject: Mathematics THE SCHRODINGER¨ EQUATION AS A VOLTERRA PROBLEM A Thesis by FERNANDO DANIEL MERA Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Approved by: Chair of Committee, Stephen Fulling Committee Members, Peter Kuchment Christopher Pope Head of Department, Al Boggess May 2011 Major Subject: Mathematics iii ABSTRACT The Schr¨odingerEquation as a Volterra Problem. (May 2011) Fernando Daniel Mera, B.S., Texas A&M University Chair of Advisory Committee: Stephen Fulling The objective of the thesis is to treat the Schr¨odinger equation in parallel with a standard treatment of the heat equation. In the books of the Rubensteins and Kress, the heat equation initial value problem is converted into a Volterra integral equation of the second kind, and then the Picard algorithm is used to find the exact solution of the integral equation. Similarly, the Schr¨odingerequation boundary initial value problem can be turned into a Volterra integral equation. We follow the books of the Rubinsteins and Kress to show for the Schr¨odinger equation similar results to those for the heat equation. The thesis proves that the Schr¨odingerequation with a source function does indeed have a unique solution. The Poisson integral formula with the Schr¨odingerkernel is shown to hold in the Abel summable sense.
    [Show full text]
  • 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.
    [Show full text]
  • A Appendix: Linear Algebra and Functional Analysis
    A Appendix: Linear Algebra and Functional Analysis In this appendix, we have collected some results of functional analysis and matrix algebra. Since the computational aspects are in the foreground in this book, we give separate proofs for the linear algebraic and functional analyses, although finite-dimensional spaces are special cases of Hilbert spaces. A.1 Linear Algebra A general reference concerning the results in this section is [47]. The following theorem gives the singular value decomposition of an arbitrary real matrix. Theorem A.1. Every matrix A ∈ Rm×n allows a decomposition A = UΛV T, where U ∈ Rm×m and V ∈ Rn×n are orthogonal matrices and Λ ∈ Rm×n is diagonal with nonnegative diagonal elements λj such that λ1 ≥ λ2 ≥ ··· ≥ λmin(m,n) ≥ 0. Proof: Before going to the proof, we recall that a diagonal matrix is of the form ⎡ ⎤ λ1 0 ... 00... 0 ⎢ ⎥ ⎢ . ⎥ ⎢ 0 λ2 . ⎥ Λ = = [diag(λ ,...,λm), 0] , ⎢ . ⎥ 1 ⎣ . .. ⎦ 0 ... λm 0 ... 0 if m ≤ n, and 0 denotes a zero matrix of size m×(n−m). Similarly, if m>n, Λ is of the form 312 A Appendix: Linear Algebra and Functional Analysis ⎡ ⎤ λ1 0 ... 0 ⎢ ⎥ ⎢ . ⎥ ⎢ 0 λ2 . ⎥ ⎢ ⎥ ⎢ . .. ⎥ ⎢ . ⎥ diag(λ ,...,λn) Λ = ⎢ ⎥ = 1 , ⎢ λn ⎥ ⎢ ⎥ 0 ⎢ 0 ... 0 ⎥ ⎢ . ⎥ ⎣ . ⎦ 0 ... 0 where 0 is a zero matrix of the size (m − n) × n. Briefly, we write Λ = diag(λ1,...,λmin(m,n)). n Let A = λ1, and we assume that λ1 =0.Let x ∈ R be a unit vector m with Ax = A ,andy =(1/λ1)Ax ∈ R , i.e., y is also a unit vector. We n m pick vectors v2,...,vn ∈ R and u2,...,um ∈ R such that {x, v2,...,vn} n is an orthonormal basis in R and {y,u2,...,um} is an orthonormal basis in Rm, respectively.
    [Show full text]
  • Quantum Mechanics, Schrodinger Operators and Spectral Theory
    Quantum mechanics, SchrÄodingeroperators and spectral theory. Spectral theory of SchrÄodinger operators has been my original ¯eld of expertise. It is a wonderful mix of functional analy- sis, PDE and Fourier analysis. In quantum mechanics, every physical observable is described by a self-adjoint operator - an in¯nite dimensional symmetric matrix. For example, ¡¢ is a self-adjoint operator in L2(Rd) when de¯ned on an appropriate domain (Sobolev space H2). In quantum mechanics it corresponds to a free particle - just traveling in space. What does it mean, exactly? Well, in quantum mechanics, position of a particle is described by a wave 2 d function, Á(x; t) 2 L (R ): The physical meaningR of the wave function is that the probability 2 to ¯nd it in a region ­ at time t is equal to ­ jÁ(x; t)j dx: You can't know for sure where the particle is for sure. If initially the wave function is given by Á(x; 0) = Á0(x); the SchrÄodinger ¡i¢t equation says that its evolution is given by Á(x; t) = e Á0: But how to compute what is e¡i¢t? This is where Fourier transform comes in handy. Di®erentiation becomes multiplication on the Fourier transform side, and so Z ¡i¢t ikx¡ijkj2t ^ e Á0(x) = e Á0(k) dk; Rd R ^ ¡ikx where Á0(k) = Rd e Á0(x) dx is the Fourier transform of Á0: I omitted some ¼'s since they do not change the picture. Stationary phase methods lead to very precise estimates on free SchrÄodingerevolution.
    [Show full text]
  • AMATH 731: Applied Functional Analysis Lecture Notes
    AMATH 731: Applied Functional Analysis Lecture Notes Sumeet Khatri November 24, 2014 Table of Contents List of Tables ................................................... v List of Theorems ................................................ ix List of Definitions ................................................ xii Preface ....................................................... xiii 1 Review of Real Analysis .......................................... 1 1.1 Convergence and Cauchy Sequences...............................1 1.2 Convergence of Sequences and Cauchy Sequences.......................1 2 Measure Theory ............................................... 2 2.1 The Concept of Measurability...................................3 2.1.1 Simple Functions...................................... 10 2.2 Elementary Properties of Measures................................ 11 2.2.1 Arithmetic in [0, ] .................................... 12 1 2.3 Integration of Positive Functions.................................. 13 2.4 Integration of Complex Functions................................. 14 2.5 Sets of Measure Zero......................................... 14 2.6 Positive Borel Measures....................................... 14 2.6.1 Vector Spaces and Topological Preliminaries...................... 14 2.6.2 The Riesz Representation Theorem........................... 14 2.6.3 Regularity Properties of Borel Measures........................ 14 2.6.4 Lesbesgue Measure..................................... 14 2.6.5 Continuity Properties of Measurable Functions...................
    [Show full text]
  • Spectral Properties of Volterra-Type Integral Operators on Fock–Sobolev Spaces
    J. Korean Math. Soc. 54 (2017), No. 6, pp. 1801{1816 https://doi.org/10.4134/JKMS.j160671 pISSN: 0304-9914 / eISSN: 2234-3008 SPECTRAL PROPERTIES OF VOLTERRA-TYPE INTEGRAL OPERATORS ON FOCK{SOBOLEV SPACES Tesfa Mengestie Abstract. We study some spectral properties of Volterra-type integral operators Vg and Ig with holomorphic symbol g on the Fock{Sobolev p p spaces F . We showed that Vg is bounded on F if and only if g m m is a complex polynomial of degree not exceeding two, while compactness of Vg is described by degree of g being not bigger than one. We also identified all those positive numbers p for which the operator Vg belongs to the Schatten Sp classes. Finally, we characterize the spectrum of Vg in terms of a closed disk of radius twice the coefficient of the highest degree term in a polynomial expansion of g. 1. Introduction The boundedness and compactness properties of integral operators stand among the very well studied objects in operator related function-theories. They have been studied for a broad class of operators on various spaces of holomor- phic functions including the Hardy spaces [1, 2, 18], Bergman spaces [19{21], Fock spaces [6, 7, 11, 13, 14, 16, 17], Dirichlet spaces [3, 9, 10], Model spaces [15], and logarithmic Bloch spaces [24]. Yet, they still constitute an active area of research because of their multifaceted implications. Typical examples of op- erators subjected to this phenomena are the Volterra-type integral operator Vg and its companion Ig, defined by Z z Z z 0 0 Vgf(z) = f(w)g (w)dw and Igf(z) = f (w)g(w)dw; 0 0 where g is a holomorphic symbol.
    [Show full text]