DOCSLIB.ORG

MTL 411: Functional Analysis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
MTL 411: Functional Analysis MTL 411: Functional Analysis Lecture D: Spectral theory April 18, 2020 Let T : V ! V be a linear operator on a vector space V over C. We say that a scalar λ 2 C is said to be an eigenvalue of T if there exists a non-zero vector x 2 V such that T x = λx. If V is finite dimensional space, then T has atleast one eigenvalue (over C). But in the infinite dimensional case, this may not be true. For example, consider the right shift operator 2 T (x1; x2;:::) = (0; x1; x2;:::) on ` . It is easily verified that if T x = λx, then x = 0: Hence T has no eigenvalues. So the notion of eigenvalue should be treated in more general way which is called spectrum. Exercise. Show that the Volterra operator T : C[0; 1] ! C[0; 1] is defined by x Z (T f)(x) = f(t)dt; x 2 [0; 1] 0 has no eigenvalues. Remark. A scalar λ is an eigenvalue of an operator T iff N (T ) 6= f0g. Recall, if λ is an eigenvalue of an operator T on a finite dimensional space V . Then λI −T is not invertible on V . Alternatively, if λ is not an eigenvalue of T , then λI − T is invertible on V since that 1-1 implies onto in the finite dimensional space. Moreover, (λI − T )−1 is bounded linear operator because of every linear operator is bounded in the finite dimensional space. Of course, this may not be true in the infinite dimensional space. Throughout this section we assume that X is a Banach space and T 2 B[X]: Definition 0.1 (Resolvent). The resolvent set ρ(T ) of T is defined by −1 ρ(T ) = fλ 2 C :(λI − T ) exists and bounded operator g: Here I denotes the identity operator on X. Definition 0.2 (Spectrum). The spectrum of T (denoted by σ(T )) is defined by σ(T ) = C n ρ(T ) = fλ 2 C : N (λI − T ) 6= f0g or R(λI − T ) 6= Xg: Lemma 0.3. If jjT jj < 1, then I − T is invertible, and 1 X (I − T )−1 = T n: (Neumann series) n=0 1 Pn k Proof. Consider the partial sum Sn = k=0 T . Then we have n X k jjSn − Smjj = jj T jj (n > m) k=m+1 n X ≤ jjT jjk: k=m+1 Pn k Since jjT jj < 1, we get the sequence of operators (Sn) is Cauchy in B[X]. Hence Sn = k=0 T P1 k converges in B[X], say S = k=0 T . It is easy to verify that (I − T )S = S(I − T ) = I. Theorem 0.4 (Properties). (a) If jλj > jjT jj, then λ 2 ρ(T ): 1 (b) If T is invertible and jjSjj < jjT −1jj , then T − S has the bounded inverse. (c) The resolvent set ρ(T ) is open, and the spectrum σ(T ) is compact. T T Proof. (a) If jλj > jjT jj, then jj λ jj < 1. Hence by above lemma, we get I − λ has the bounded inverse =) (λI − T )−1 2 B[X], i.e., λ 2 ρ(T ): (b) Since jjST −1jj ≤ jjSjj jjT −1jj < 1, we get I − ST −1 has the bounded inverse. Then T − S = (I − ST −1)T . So T − S has the bounded inverse. (c) Let λ 2 ρ(T ). We have to show λ is an interior point of ρ(T ), i.e., there exists a δ > 0 such that λ − µ 2 ρ(T ) whenever jµj < δ: 1 Choose δ = jj(λI−T )−1jj : Then by above result (b), we get (λ−µ)I −T has the bounded inverse whenever jµj < δ: Thus λ − µ 2 ρ(T ) whenever jµj < δ: Hence ρ(T ) is open in C and it's complement σ(T ) is closed and also bounded (why?). Therefore, σ(T ) is a compact subset of C: Remark. The spectrum σ(T ) is nonempty. The proof involves complex analysis techniques, we omit the proof here. Examples. 2 (1) Right shift operator: T (x1; x2;:::) = (0; x1; x2;:::), x 2 ` . Since jjT jj = 1, the spectrum σ(T ) ⊆ fλ 2 C : jλj ≤ 1g: For jλj ≤ 1; (λI − T )x = (λx1; λx2 − x1; λx3 − x2;:::): 1 1 1 2 Observe that if w = ( λ ; λ2 ; λ3 ;:::) then (λI − T )w = (1; 0; 0;:::) and w2 = ` . So λI − T is not onto for jλj ≤ 1. Hence σ(T ) = fλ 2 C : jλj ≤ 1g: x2 x3 2 (2) Diagonal operator: T (x1; x2;:::) = (x1; 2 ; 3 ;:::), x 2 ` . 1 1 1 Since ( n I −T )en = 0, we get N (( n I −T )) 6= f0g for every n 2 N. Hence n 2 σ(T ): For λ = 0, 1 T does not have bounded inverse (why?). On the other hand, for λ2 = f n : n 2 Ng [ f0g, we 1 can show that λ2 = σ(T ): Therefore, σ(T ) = f0g [ f n : n 2 Ng: Exercise. Consider the Multiplication operator: T : f 7! mf from the space (C[a; b]; jj · jj1) to itself, where m 2 C[a; b]: Show that the spectrum σ(T ) = m([a; b]): 2.
Load more

Recommended publications
  • 1 Bounded and Unbounded Operators
    1 1 Bounded and unbounded operators 1. Let X, Y be Banach spaces and D 2 X a linear space, not necessarily closed. 2. A linear operator is any linear map T : D ! Y . 3. D is the domain of T , sometimes written Dom (T ), or D (T ). 4. T (D) is the Range of T , Ran(T ). 5. The graph of T is Γ(T ) = f(x; T x)jx 2 D (T )g 6. The kernel of T is Ker(T ) = fx 2 D (T ): T x = 0g 1.1 Operations 1. aT1 + bT2 is defined on D (T1) \D (T2). 2. if T1 : D (T1) ⊂ X ! Y and T2 : D (T2) ⊂ Y ! Z then T2T1 : fx 2 D (T1): T1(x) 2 D (T2). In particular, inductively, D (T n) = fx 2 D (T n−1): T (x) 2 D (T )g. The domain may become trivial. 3. Inverse. The inverse is defined if Ker(T ) = f0g. This implies T is bijective. Then T −1 : Ran(T ) !D (T ) is defined as the usual function inverse, and is clearly linear. @ is not invertible on C1[0; 1]: Ker@ = C. 4. Closable operators. It is natural to extend functions by continuity, when possible. If xn ! x and T xn ! y we want to see whether we can define T x = y. Clearly, we must have xn ! 0 and T xn ! y ) y = 0; (1) since T (0) = 0 = y. Conversely, (1) implies the extension T x := y when- ever xn ! x and T xn ! y is consistent and defines a linear operator.
    [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]
  • 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]
  • 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]
  • Curl, Divergence and Laplacian
    Curl, Divergence and Laplacian What to know: 1. The definition of curl and it two properties, that is, theorem 1, and be able to predict qualitatively how the curl of a vector field behaves from a picture. 2. The definition of divergence and it two properties, that is, if div F~ 6= 0 then F~ can't be written as the curl of another field, and be able to tell a vector field of clearly nonzero,positive or negative divergence from the picture. 3. Know the definition of the Laplace operator 4. Know what kind of objects those operator take as input and what they give as output. The curl operator Let's look at two plots of vector fields: Figure 1: The vector field Figure 2: The vector field h−y; x; 0i: h1; 1; 0i We can observe that the second one looks like it is rotating around the z axis. We'd like to be able to predict this kind of behavior without having to look at a picture. We also promised to find a criterion that checks whether a vector field is conservative in R3. Both of those goals are accomplished using a tool called the curl operator, even though neither of those two properties is exactly obvious from the definition we'll give. Definition 1. Let F~ = hP; Q; Ri be a vector field in R3, where P , Q and R are continuously differentiable. We define the curl operator: @R @Q @P @R @Q @P curl F~ = − ~i + − ~j + − ~k: (1) @y @z @z @x @x @y Remarks: 1.
    [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]
  • Numerical Operator Calculus in Higher Dimensions
    Numerical operator calculus in higher dimensions Gregory Beylkin* and Martin J. Mohlenkamp Applied Mathematics, University of Colorado, Boulder, CO 80309 Communicated by Ronald R. Coifman, Yale University, New Haven, CT, May 31, 2002 (received for review July 31, 2001) When an algorithm in dimension one is extended to dimension d, equation using only one-dimensional operations and thus avoid- in nearly every case its computational cost is taken to the power d. ing the exponential dependence on d. However, if the best This fundamental difficulty is the single greatest impediment to approximate solution of the form (Eq. 1) is not good enough, solving many important problems and has been dubbed the curse there is no way to improve the accuracy. of dimensionality. For numerical analysis in dimension d,we The natural extension of Eq. 1 is the form propose to use a representation for vectors and matrices that generalizes separation of variables while allowing controlled ac- r ͑ ͒ ϭ ͸ ␾l ͑ ͒ ␾l ͑ ͒ curacy. Basic linear algebra operations can be performed in this f x1,...,xd sl 1 x1 ··· d xd . [2] representation using one-dimensional operations, thus bypassing lϭ1 the exponential scaling with respect to the dimension. Although not all operators and algorithms may be compatible with this The key quantity in Eq. 2 is r, which we call the separation rank. representation, we believe that many of the most important ones By increasing r, the approximate solution can be made as are. We prove that the multiparticle Schro¨dinger operator, as well accurate as desired.
    [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]
  • 23. Kernel, Rank, Range
    23. Kernel, Rank, Range We now study linear transformations in more detail. First, we establish some important vocabulary. The range of a linear transformation f : V ! W is the set of vectors the linear transformation maps to. This set is also often called the image of f, written ran(f) = Im(f) = L(V ) = fL(v)jv 2 V g ⊂ W: The domain of a linear transformation is often called the pre-image of f. We can also talk about the pre-image of any subset of vectors U 2 W : L−1(U) = fv 2 V jL(v) 2 Ug ⊂ V: A linear transformation f is one-to-one if for any x 6= y 2 V , f(x) 6= f(y). In other words, different vector in V always map to different vectors in W . One-to-one transformations are also known as injective transformations. Notice that injectivity is a condition on the pre-image of f. A linear transformation f is onto if for every w 2 W , there exists an x 2 V such that f(x) = w. In other words, every vector in W is the image of some vector in V . An onto transformation is also known as an surjective transformation. Notice that surjectivity is a condition on the image of f. 1 Suppose L : V ! W is not injective. Then we can find v1 6= v2 such that Lv1 = Lv2. Then v1 − v2 6= 0, but L(v1 − v2) = 0: Definition Let L : V ! W be a linear transformation. The set of all vectors v such that Lv = 0W is called the kernel of L: ker L = fv 2 V jLv = 0g: 1 The notions of one-to-one and onto can be generalized to arbitrary functions on sets.
    [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]