Linear Operators in Hilbert Spaces

11 Key excerpts on "Linear Operators in Hilbert Spaces"

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  Hilbert Space Methods in Quantum Mechanics

  • Werner O. Amrein(Author)
  • 2009(Publication Date)
  • EPFL PRESS

    (Publisher)

  CHAPTER 2 Linear Operators Throughout this text we are concerned with linear mappings from a Hilbert space into itself or into another Hilbert space, called linear operators. In the present chapter we introduce and discuss various types of operators that frequently occur in quan- tum mechanics. In Section 2.2 we consider projections, isometric and unitary opera- tors. Section 2.3 is devoted to compact and Hilbert-Schmidt operators and Section 2.5 to multiplication operators in L 2 spaces. Often quantum-mechanical operators are un- bounded. Section 2.4 is concerned with some of the subtleties that one meets in such situations, especially with the notion of closedness of an operator. In Section 2.6 we define the resolvent and the spectrum of closed operators. A very important class of linear operators are the self-adjoint operators, used to describe observables in quantum mechanics. For unbounded operators the property of being self-adjoint is rather subtle. Some preliminary explanations, pointing out the difference between symmetric and self-adjoint operators, are given in Section 2.4, and Chapter 3 will be entirely devoted to a detailed study of the relations between symmetric and self-adjoint operators. The present chapter ends with some criteria for the preservation of self-adjointness under perturbations (Section 2.7); in particular we prove the self-adjointness of some Hamiltonians of non-relativistic quantum mechan- ics as perturbations of the self-adjoint free Hamiltonian vector P 2 /2m. 2.1 The algebra B(H) 2.1.1. Let H be a Hilbert space. We denote by B(H) the set of all bounded linear operators on H. An element of B(H) is a mapping A that associates with each vector f ∈ H another vector Af belonging to H, and this mapping is linear and bounded, i.e. A(αf + g) = αAf + Ag and there is a finite constant M (depending on A) such that bardblAf bardbl ≤ M bardblf bardbl for each vector f in H. The infimum of all possible numbers M is

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  The Calculus of Complex Functions

  • William Johnston(Author)
  • 2022(Publication Date)
  • American Mathematical Society

    (Publisher)

  The abstraction is useful. For example, as a consequence, the nineteenth century results of Joseph Fourier, Friedrich Bessel and Marc-Antoine Parseval on trigonometric series became a straightforward part of the Hilbert space theory, establishing their mathematical footings in such fields as quan-tum physics, electrical engineering, and analytic chemistry. Why are Hilbert spaces important? Applications such as quantum mechanics are naturally formulated as a mathematical model that sits in an infinite-dimensional Hil-bert space. Quantum theory is effectively cast in terms of ? 2 (−∞, ∞) (or, more gen-erally, on ? 2 (ℝ ? ) ). Hilbert space theory reveals facts about linear operators, resolving 392 Chapter 6. Linear Algebra and Operator Theory questions about functions. Such advances generalized the study of Fourier series in terms of orthonormal bases, creating general strategies to understand applied prob-lems. The orthogonality of basis elements to form the Euclidean geometry is the key to this ability. For example, a vibrating string can be decomposed into its vibrations in distinct overtones (an infinite number of them). The overtones are each given by the projection of the vibrating string—thought of as a single infinite-dimensional point in the Hilbert space—onto each coordinate basis axis in the space. Barbara MacCluer in 2001 [photo by Michael Dritschel] Though modern work in function theory began more than a century ago, it is as if much of the discovery is only at its introductory level. Perhaps that indicates how advanced and challenging the research work is in these fields, or perhaps it indicates that there is relatively a small percentage of research mathematicians who have chosen to work in these areas. For example, the understandingof particular categories of bounded operators is recent and ongoing. Composition operators provide one such exam-ple. Two superb function theorists who have led the way are Barbara D.

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  Advanced Modern Physics

  Theoretical Foundations

  • John Dirk Walecka(Author)
  • 2010(Publication Date)
  • WSPC

    (Publisher)

  2.4.3 Linear Hermitian Operators In Vol. I, quantum mechanics was introduced in coordinate space, where the momentum p is given by p = ( planckover2pi1 /i ) ∂/∂x . It was observed in ProbI. 4.8 that one could equally well work in momentum space, where the position x is given by x = i planckover2pi1 ∂/∂p . It was also observed there that the commutation relation [ p,x ] = planckover2pi1 /i is independent of the particular representation. Our goal in this section is to similarly abstract the Schr¨ odinger equation and free it from any particular component representation. 20 Advanced Modern Physics 2.4.3.1 Eigenstates A linear hermitian operator L op takes one abstract vector | ψ angbracketright into another L op | ψ angbracketright .

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  Variational Methods for Eigenvalue Problems

  An Introduction to the Weinstein Method of Intermediate Problems (Second Edition)

  • S. H. Gould(Author)
  • 2019(Publication Date)
  • University of Toronto Press

    (Publisher)

  CHAPTER EIGHT LINEAR OPERATORS IN HILBERT SPACE 1. Purpose of the chapter. In the preceding chapters, where we have applied the Rayleigh-Ritz and Weinstein methods to differential problems, we have had to leave unanswered certain important questions about the existence of limiting functions and the convergence of successive approxi-mations. In the next chapter, following Aronszajn, we shall apply the Weinstein method to a closely related problem, namely the calculation of eigenvalues of a linear operator in Hubert space. For such a problem we shall find that the relevant questions of existence and convergence can be answered satisfactorily when we deal with operators that are completely continuous (see below). It is then natural to ask whether we can transform our differential problems into eigenvalue problems for completely continuous operators in Hubert space in such a way that the questions of existence and convergence are thereby answered for the differential problems as well. We shall find that the differential problems discussed up to now are among those which can be so transformed. In a subsequent chapter we actually make the transformation for the problem of the vibrating plate and give a numerical example. The present chapter contains the necessary basic information about linear operators in abstract Hubert space. 2. Restatement of the properties of Hubert space. Their consistency and categoricalness. For convenience we restate the five properties of Hubert space discussed in Chapter III. Definition. A set § of elements u, v,..., which we shall also caü points or vectors, is a (real) Hubert space if it has the following five properties.

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  Quantum Mechanics in Hilbert Space

  • Eduard Prugovecki(Author)
  • 1982(Publication Date)
  • Academic Press

    (Publisher)

  2. Linear Operators in Hilbert Spaces 193 Despite these difficulties we shall have to study unbounded operators in some detail since practically all of the operators which appear in physics are unbounded, such as the position and momentum operators or the Schroedinger operator (see Exercise 2.4). 2.7. NONEXISTENCE OF UNBOUNDED EVERYWHERE-DEFINED SELF-ADJOINT OPERATORS We shall proceed to prove that no symmetric unbounded operator has a self-adjoint extension to the entire Hilbert space. For this proof we need some preliminary results. If +1 , +2 ,... is a sequence of continuous linear functionals on Z, which is bounded for each fixed f E Z, i.e., *Lemma 2,2. I dn(f)l G Cf , f E z, then #1, d2 ,... are uniformly bounded within the unit sphere S, = {h: (1 h 1) < I}, i.e., g E s, , I Cn(g)l < c, It = 1,2,..., or equivalently I dn(f)l < c Ilf 11, = 1, 2,..., for all f E Z. Proof. We shall show first that q$(f), +2(f),... is uniformly bounded inside at least one sphere {h: 11 h - h,, )I < Y } in &. Assume that the above statement were not true! Then we would have I +n,(fl)I > I for at least some integer n1 and some fl E Z. By the continuity of +,J f ), we also have on a sphere S(l) C X with the centre at fl . Since +1( f ), +2( f ), ... are not uniformly bounded in any sphere, including S1), there must be an fi E 5l) such that I & ( f2)l > 2 for some integer n2 . Due to the continuity of+nl(f 1, I +n,(f)l > 2, f E s(2), for some sphere S2) with its center at f 2 . By continuing this reasoning, we establish the existence of a sequence S(1) 3 23'2) 3 -.- of spheres in which I d W k ( f ) I > k, f S(k), 194 111. Theory of Linear Operators in Hilbert Spaces m for some n k , K = 1,2, ... . Since 2 is complete, Stk) contains at least one element f, , for which we have according to the above construc- tion I +n,(fo)l > k, k = 1,2,... * Thus, $,(f0), $,(fO),... is not a bounded sequence, contrary to the assump- tion of the theorem.

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  Introduction to Mathematical Physics

  • Michael T. Vaughn(Author)
  • 2008(Publication Date)
  • Wiley-VCH

    (Publisher)

  The standard textbook by Byron and Fuller cited in Chapter 2 emphasizes linear vector spaces. There are many modern books on quantum mechanics that discuss the essential connection between linear vector spaces and quantum mechanics. Two good introductory books are David J. Griffiths, Introduction to Quantum Mechanics (2nd edition), Prentice-Hall (2004). Ramamurti Shankar, Principles of Quantum Mechanics (2nd edition), Springer (2005) Slightly more advanced but still introductory is Eugen Merzbacher, Quantum Mechanics (3rd edition), Wiley (1997). Two classics that strongly reflect the original viewpoints of their authors are Paul A. M. Dirac, The Principles of Quantum Mechanics ( 4 th edition), Clarendon Press, Oxford (1958) John von Neumann, Mathematical Foundations of Quantum Mechanics , Princeton University Press (1955). Dirac’s work describes quantum mechanics as the mathematics flows from his own physi-cal insight, while von Neumann presents an axiomatic formulation based on his deep under-standing of Hilbert space theory. Both are important works for the student of the historical development of the quantum theory. Problems 345 Problems 1. Let { φ n } ( n = 1 , 2 , . . . ) be a complete orthonormal system in the (infinite-dimensional) Hilbert space H . Consider the operators U k defined by U k φ n = φ n + k ( k = 1 , 2 , . . . ). (i) Give an explicit form for U † k . (ii) Find the eigenvalues and eigenvectors of U k and U † k . (iii) Discuss the convergence of the sequences { U k } , { U † k } , { U † k U k } , { U k U † k } . 2. Let { φ n } ( n = 1 , 2 , . . . ) be a complete orthonormal system in H , and define the linear operator T by T φ n ≡ nφ n +1 ( n = 1 , 2 , . . . ) (i) What is the domain of T ? (ii) How does T † act on { φ 1 , φ 2 , . . . } ? What is the domain of T † ? (iii) Find the eigenvalues and eigenvectors of T . (iv) Find the eigenvalues and eigenvectors of T † . 3.

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  Functional Analysis

  • L. V. Kantorovich, G. P. Akilov(Authors)
  • 2014(Publication Date)
  • Pergamon

    (Publisher)

  § 3. Linear functionals and operators on Hilbert space 3.1. An operator* on a separable Hilbert space admits a matrix representation similar to that for operators on finite-dimensional spaces (see 2.8). In fact, let U be an operator on a separable (infinite-dimensional) Hilbert space H. Choose any complete orthonormal system x u x 2 ,. . . , x„,. . . . Each element x e H can be expressed in the form 00 X = L C k X k'> k= 1 where c k = (x, x k ) is the Fourier coefficient of x. Therefore, as U is continuous, oo y=Ux= X c k Ux k . k= 1 Hence, comparing the Fourier coefficients on the left- and right-hand sides, we find that oo d j= Σ a jkCk, (!) k = 1 where the dj are the Fourier coefficients of y and the a jk are those of Ux k . Thus the sequence of Fourier coefficients of y = Ux is obtained from the sequence of * When we are dealing with Hilbert space, the term operator will always mean continuous linear operator. Also we write Ux instead of U(x). 142 Functional Analysis Fourier coefficients of x by means of the transformation with matrix

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  A First Course in Functional Analysis

  • Orr Moshe Shalit(Author)
  • 2017(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Chapter 5 Bounded linear operators on Hilbert space Recall that a map T : V → W between two vector spaces V and W is said to be a linear transformation if T (α u + v) = α T (u) + T (v) for all α in the field and all u, v ∈ V. The theory of linear operators on infinite dimensional spaces becomes really interesting only when additional topological assumptions are made. In this book, we will only consider the boundedness and the compactness assumptions. We begin by discussing bounded operators on Hilbert spaces. 5.1  Bounded operators Definition 5.1.1. A linear transformation T : H → K mapping between two inner product spaces is said to be bounded if the operator norm ‖ T ‖ of T, defined as ‖ T ‖ = sup ‖ h ‖ = 1 ‖ T h ‖, satisfies ‖ T ‖ < ∞. An immediate observation is that for all h ∈ H we have ‖ T h ‖ ≤ ‖ T ‖ ‖ h ‖. A bounded linear transformation is usually referred to as a bounded operator, or simply as an operator. The set of bounded operators between two inner product spaces H and K is denoted by B (H, K), and the notation for the space B (H, H) is usually abbreviated to B (H). Example 5.1.2. Let H = ℂ n be a finite dimensional Hilbert space (with the standard inner product). Every linear transformation on H is of the form T A given by T A (x) = Ax, where A ∈ M n (ℂ) is an n × n matrix. By elementary analysis, T A is continuous, and by compactness of the unit sphere S = { x ∈ ℂ n : ‖ x ‖ = 1 }, T A is bounded. Alternatively, one can find an explicit bound for ‖ T A ‖ by using the definition. Let x ∈ H. Then ‖ A x ‖ 2 = ∑ i = 1 n | ∑ j = 1 n a i j x j | 2. By the Cauchy-Schwarz inequality, the right-hand side is less. than ∑ i = 1 n (∑ j = 1 n | a i j | 2) ‖ x ‖ 2 = ‖ x ‖ 2 ∑ i, j = 1 n | a i j | 2. Thus ‖ T A ‖ ≤ ∑ i, j = 1 n | a i j | 2. Exercise 5.1.3

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  Solid State and Quantum Theory for Optoelectronics

  • Michael A. Parker(Author)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  Operators and Hilbert Space 195 3.33 Using the operators in Problems 3.30 and 3.31 determine if ^ O 1 ¼ ^ L 1 ^ L 2 is the same as ^ O 2 ¼ ^ L 2 ^ L 1 . Write the matrices for ^ O 1 and ^ O 2 . 3.34 A Hilbert space V has basis { j f 1 i , j f 2 i }. Assume that the linear operator ^ L : V ! V has the matrix L ¼ 0 1 2 3 ! . Write the operator in the form ^ L ¼ P ij L ij j f i ih f j j . 3.35 Write an operator ^ L : V ! V in the form ^ L ¼ P L ab j f a ih f b j when ^ L maps the basis set { j f 1 i , j f 2 i } into the basis set { j c 1 i , j c 2 i } according to the rule ^ L j f 1 i¼j c 1 i and ^ L j f 2 i¼j c 2 i . Assume that the two sets of basis vectors are related as follows: j c 1 i¼ 1 ffiffi 3 p j f 1 iþ ffiffi 2 3 r j f 2 i and j c 2 i¼ ffiffi 2 3 r j f 1 iþ 1 ffiffi 3 p j f 2 i 3.36 Let { j f 1 i , j f 2 i } be a basis set. Write the following operator in matrix notation ^ L ¼j f 1 ih f 1 jþ 2 j f 1 ih f 2 jþ 3 j f 2 ih f 2 j 3.37 Let { j f 1 i , j f 2 i } be a basis set. Write the following operator in matrix notation ^ L ¼j f 1 ih f 1 jþ ( 1 þ 2 j ) j f 1 ih f 2 jþ ( 1 2 j ) j f 2 ih f 1 jþ 3 j f 2 ih f 2 j 3.38 Suppose ^ H ¼ P n E n j n ih n j where E n 6 ¼ 0 for all n . What value of c n in ^ O ¼ P n C n j n ih n j makes ^ O the inverse of ^ H so that ^ H ^ O ¼ 1 ¼ ^ O ^ H . 3.39 If ^ H ¼ 1 j f 1 ih f 1 jþ 2 j f 2 ih f 2 j and j c (0) i¼ 0.86 j f 1 iþ 0.51 j f 2 i is the wave function for an electron at t ¼ 0. Find the average energy h c ( 0 ) j ^ H j c ( 0 ) i . 3.40 Prove that the required property h ^ A j a ^ B þ b ^ C i¼ a h ^ A j ^ B iþ b ^ A j ^ C for h ^ A j ^ B i¼ Tr ^ A þ ^ B to be an inner product. Use L ¼ ^ T : V ! V . 3.41 Prove h ^ A j ^ A i¼ 0 if and only if ^ A ¼ 0 for ^ A 2 L ¼ ^ T : V ! V , the set of linear operators. Hint: Consider the expansion of an operator in a basis set. 3.42 Show that the set of linear operators ^ T : V ! W mapping the vector space V into the vector space W forms a vector space.

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  Elementary Theory

  Fundamentals of the Theory of Operator Algebras

  • Richard V. Kadison, John R. Ringrose, Samuel Eilenberg, Hyman Bass(Authors)
  • 2016(Publication Date)
  • Academic Press

    (Publisher)

  138 2. BASICS OF HILBERT SPACE AND LINEAR OPERATORS Proof, (i) If there is a matrix [c jk ] with the stated properties, bilinearity of the mapping (x, y) -x (x) y implies that n n / n n n x j®yj = X J ® ( w*) = WJ ®^ j = l j = l k = l / j = l fc= 1 ( cjkXj)®yk = o. k=lj=l / k = Conversely, suppose that £ = x Xj ® ^ = 0. If ^,..., v r is an orthonormal basis of the linear subspace of Jf 2 generated by y 1 ,..., y n , we can choose an n x r matrix A = [a jk ] and an r x n matrix B = [b jk ] such that r X/= a ß v i U= h...,n) 9 »1= ^ * ( / = 1 -· · ^ ) · fc=l With [c jk ] the « x « matrix ^4i?, we have yj = ^ b ikyk ) = w * U = i,...,/i), i = l f c = l / fc=l and where n n / r r o = */ ® x/ = ·*; ® ( w ) = ®^ j = l j = l M = l / 1=1 Ui= (/= l,...,r). For each m = 1,..., r, r r 0 = < W i ® V l> U m ® O = < W *' W m > < ^ , l>m> = INmll 2 · / = 1 1=1 Thus «! = u 2 = ' ' · = u r = 0, and n ri r r C J**/ = fl Afc*/ = b ik u i = ° (*: = 1,...,«). (ii) Suppose that L is a bilinear mapping from 3#Ί ^f 2 into Jf. If Xi,..., x n G Jf i, j i , . . . , y n G Jf 2 » a n d = i */ ® yj = 0, we can choose a matrix 2.6. CONSTRUCTIONS WITH HILBERT SPACES 139 [c jk ] as in (i). The bilinearity of L then entails Suppose next that x l9 ... ,x„, u l9 .. .,u m eJ^ u y l9 ..., y n , v i9 ... ,v m e Jf 2 > and XJ Ä , xj (g) ^ = X7 = x w, ® Vj. Then the preceding paragraph shows that and therefore From this, it follows that the equation defines (unambiguously) a linear operator T from Jf 0 into M an d ( ® j) = L(x, y) (xeJHr i9 yeJP 2 )' 2.6.7. REMARK. The first part of Proposition 2.6.6 asserts, in effect, that the only finite families of simple tensors that have sum zero are those that are forced to have zero sum by the bilinearity of the mapping/? : (x, y) ^ x®y.

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  Fundamentals of the Theory of Operator Algebras. V1

  Elementary Theory

  • Richard V. Kadison, John R. Ringrose(Authors)
  • 1983(Publication Date)
  • Academic Press

    (Publisher)

  124 2. BASICS OF HILBERT SPACE A N D LINEAR OPERATORS and since the last inequality is satisfied for every finite subset IF of A, we have This shows that the family {x, - x:)}, as well as {x:)}, is in 10 ifa when n 2 n ( ~ ) . Accordingly, {x,} (= {x, - x:)} + {x:)}) is an element x of c@ Xu, and (2) asserts that JIx - x')(I < E whenever n 2 n(e). Thus (x()) converges to x; so I@ ifa is complete, and is therefore a Hilbert space. With b in A, %b is isomorphic to the closed subspace XL of COX, consisting of those families {x,} such that x, = 0 whenever a # b. We obtain a unitary transformation ub, from z b onto if;, by taking for ubx the family {x,} in which xb is x and x, = 0 (a # b). The subspaces X i (a E A) are pairwise orthogonal, and V 2; = c@ 2,. If { X,} is a family of mutually orthogonal subspaces of a Hilbert space X, and V X, = if, the corresponding projections form an orthogonal family {E,} with (strong-operator convergent) sum I. Just as in the case of finite direct sums, the equation U x = {E,x} defines an isomorphism U from 2 onto c@ X,, and we consider X as an internal direct sum of the family (2,). Moreover, U -' { x , } = E x , when {x,} EX@ X,; the sum converges since {x,} is an orthogonal set in X, and 111xaI12 < co. Suppose next that X,, X, are Hilbert spaces, and T, E a(#,, X,) for each a in A. If s~P{llTaIl:aEAl < a, the equation T{xa} = { Tax,} defines a bounded linear operator Tfrom c@ Za into c@ X,. We call T the direct sum 10 T, of the family { T,}. Just as in the case of finite direct sums, we have when S,, T,E~J(#,,X,) and R , E ~ ( X , , ~ ~ ) . reduces to the Hilbert space /,(A) of Example 2.1.12. When 2, is the one-dimensional Hilbert space @, for each a in A, c0 8, 2.6. CONSTRUCTIONS WITH HILBERT SPACES 125 Tensor products and the Hilbert-Schmidt class. The material in this subsection will be used in an essential way in later parts of the book (from Chapter 1 1 onward), but has a relatively minor role until that point.

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