Chapter 8 Bounded Linear Operators on a Hilbert Space
In this chapter we describe some important classes of bounded linear operators on Hilbert spaces, including projections, unitary operators, and self-adjoint operators. We also prove the Riesz representation theorem, which characterizes the bounded linear functionals on a Hilbert space, and discuss weak convergence in Hilbert spaces.
8.1 Orthogonal projections
We begin by describing some algebraic properties of projections. If M and N are subspaces of a linear space X such that every x X can be written uniquely as ∈ x = y + z with y M and z N, then we say that X = M N is the direct sum of ∈ ∈ ⊕ M and N, and we call N a complementary subspace of M in X. The decomposition x = y + z with y M and z N is unique if and only if M N = 0 . A given ∈ ∈ ∩ { } subspace M has many complementary subspaces. For example, if X = R3 and M is a plane through the origin, then any line through the origin that does not lie in M is a complementary subspace. Every complementary subspace of M has the same dimension, and the dimension of a complementary subspace is called the codimension of M in X. If X = M N, then we define the projection P : X X of X onto M along N ⊕ → by P x = y, where x = y + z with y M and z N. This projection is linear, with ∈ ∈ ran P = M and ker P = N, and satisfies P 2 = P . As we will show, this property characterizes projections, so we make the following definition.
Definition 8.1 A projection on a linear space X is a linear map P : X X such → that
P 2 = P. (8.1)
Any projection is associated with a direct sum decomposition.
Theorem 8.2 Let X be a linear space.
187 188 Bounded Linear Operators on a Hilbert Space
(a) If P : X X is a projection, then X = ran P ker P . → ⊕ (b) If X = M N, where M and N are linear subpaces of X, then there is a ⊕ projection P : X X with ran P = M and ker P = N. → Proof. To prove (a), we first show that x ran P if and only if x = P x. If ∈ x = P x, then clearly x ran P . If x ran P , then x = P y for some y X, and ∈ ∈ ∈ since P 2 = P , it follows that P x = P 2y = P y = x. If x ran P ker P then x = P x and P x = 0, so ran P ker P = 0 . If x X, ∈ ∩ ∩ { } ∈ then we have x = P x + (x P x), − where P x ran P and (x P x) ker P , since ∈ − ∈ P (x P x) = P x P 2x = P x P x = 0. − − − Thus X = ran P ker P . ⊕ To prove (b), we observe that if X = M N, then x X has the unique ⊕ ∈ decomposition x = y + z with y M and z N, and P x = y defines the required ∈ ∈ projection. When using Hilbert spaces, we are particularly interested in orthogonal sub- spaces. Suppose that is a closed subspace of a Hilbert space . Then, by M H Corollary 6.15, we have = ⊥. We call the projection of onto along H M ⊕ M H M ⊥ the orthogonal projection of onto . If x = y + z and x0 = y0 + z0, where M H M y, y0 and z, z0 ⊥, then the orthogonality of and ⊥ implies that ∈ M ∈ M M M P x, x0 = y, y0 + z0 = y, y0 = y + z, y0 = x, P x0 . (8.2) h i h i h i h i h i This equation states that an orthogonal projection is self-adjoint (see Section 8.4). As we will show, the properties (8.1) and (8.2) characterize orthogonal projections. We therefore make the following definition. Definition 8.3 An orthogonal projection on a Hilbert space is a linear map H P : that satisfies H → H P 2 = P, P x, y = x, P y for all x, y . h i h i ∈ H An orthogonal projection is necessarily bounded. Proposition 8.4 If P is a nonzero orthogonal projection, then P = 1. k k Proof. If x and P x = 0, then the use of the Cauchy-Schwarz inequality ∈ H 6 implies that P x, P x x, P 2x x, P x P x = h i = h i = h i x . k k P x P x P x ≤ k k k k k k k k Therefore P 1. If P = 0, then there is an x with P x = 0, and P (P x) = k k ≤ 6 ∈ H 6 k k P x , so that P 1. k k k k ≥ Orthogonal projections 189
There is a one-to-one correspondence between orthogonal projections P and closed subspaces of such that ran P = . The kernel of the orthogonal M H M projection is the orthogonal complement of . M Theorem 8.5 Let be a Hilbert space. H (a) If P is an orthogonal projection on , then ran P is closed, and H = ran P ker P H ⊕ is the orthogonal direct sum of ran P and ker P . (b) If is a closed subspace of , then there is an orthogonal projection P M H on with ran P = and ker P = ⊥. H M M Proof. To prove (a), suppose that P is an orthogonal projection on . Then, by H Theorem 8.2, we have = ran P ker P . If x = P y ran P and z ker P , then H ⊕ ∈ ∈ x, z = P y, z = y, P z = 0, h i h i h i so ran P ker P . Hence, we see that is the orthogonal direct sum of ran P and ⊥ H ker P . It follows that ran P = (ker P )⊥, so ran P is closed. To prove (b), suppose that is a closed subspace of . Then Corollary 6.15 M H implies that = ⊥. We define a projection P : by H M ⊕ M H → H P x = y, where x = y + z with y and z ⊥. ∈ M ∈ M Then ran P = , and ker P = ⊥. The orthogonality of P was shown in (8.2) M M above. If P is an orthogonal projection on , with range and associated orthogonal H M direct sum = , then I P is the orthogonal projection with range and H M ⊕ N − N associated orthogonal direct sum = . H N ⊕ M Example 8.6 The space L2(R) is the orthogonal direct sum of the space of M even functions and the space of odd functions. The orthogonal projections P N and Q of onto and , respectively, are given by H M N f(x) + f( x) f(x) f( x) P f(x) = − , Qf(x) = − − . 2 2 Note that I P = Q. − Example 8.7 Suppose that A is a measurable subset of R — for example, an interval — with characteristic function 1 if x A, χ (x) = ∈ A 0 if x A. 6∈ Then
PAf(x) = χA(x)f(x) 190 Bounded Linear Operators on a Hilbert Space is an orthogonal projection of L2(R) onto the subspace of functions with support contained in A.
A frequently encountered case is that of projections onto a one-dimensional subspace of a Hilbert space . For any vector u with u = 1, the map P H ∈ H k k u defined by P x = u, x u u h i projects a vector orthogonally onto its component in the direction u. Mathemati- cians use the tensor product notation u u to denote this projection. Physicists, ⊗ on the other hand, often use the “bra-ket” notation introduced by Dirac. In this notation, an element x of a Hilbert space is denoted by a “bra” x or a “ket” x , h | | i and the inner product of x and y is denoted by x y . The orthogonal projection h | i in the direction u is then denoted by u u , so that | ih | ( u u ) x = u x u . | ih | | i h | i| i Example 8.8 If = Rn, the orthogonal projection Pu in the direction of a unit H vector u has the rank one matrix uuT . The component of a vector x in the direction u is Pux = (uT x)u.
Example 8.9 If = l2(Z), and u = e , where H n ∞ en = (δk,n)k=−∞, and x = (xk), then Pen x = xnen.
Example 8.10 If = L2(T) is the space of 2π-periodic functions and u = 1/√2π H is the constant function with norm one, then the orthogonal projection Pu maps a function to its mean: P f = f , where u h i 1 2π f = f(x) dx. h i 2π Z0 The corresponding orthogonal decomposition, f(x) = f + f 0(x), h i decomposes a function into a constant mean part f and a fluctuating part f 0 with h i zero mean.
8.2 The dual of a Hilbert space
A linear functional on a complex Hilbert space is a linear map from to C. A H H linear functional ϕ is bounded, or continuous, if there exists a constant M such that ϕ(x) M x for all x . (8.3) | | ≤ k k ∈ H The dual of a Hilbert space 191
The norm of a bounded linear functional ϕ is
ϕ = sup ϕ(x) . (8.4) k k kxk=1 | | If y , then ∈ H ϕ (x) = y, x (8.5) y h i is a bounded linear functional on , with ϕ = y . H k yk k k Example 8.11 Suppose that = L2(T). Then, for each n Z, the functional H ∈ ϕ : L2(T) C, n → 1 −inx ϕn(f) = f(x)e dx, √2π T Z that maps a function to its nth Fourier coefficient is a bounded linear functional. We have ϕ = 1 for every n Z. k nk ∈ One of the fundamental facts about Hilbert spaces is that all bounded linear functionals are of the form (8.5).
Theorem 8.12 (Riesz representation) If ϕ is a bounded linear functional on a Hilbert space , then there is a unique vector y such that H ∈ H ϕ(x) = y, x for all x . (8.6) h i ∈ H Proof. If ϕ = 0, then y = 0, so we suppose that ϕ = 0. In that case, ker ϕ is 6 a proper closed subspace of , and Theorem 6.13 implies that there is a nonzero H vector z such that z ker ϕ. We define a linear map P : by ∈ H ⊥ H → H ϕ(x) P x = z. ϕ(z) Then P 2 = P , so Theorem 8.2 implies that = ran P ker P . Moreover, H ⊕ ran P = αz α C , ker P = ker ϕ, { | ∈ } so that ran P ker P . It follows that P is an orthogonal projection, and ⊥ = αz α C ker ϕ H { | ∈ } ⊕ is an orthogonal direct sum. We can therefore write x as ∈ H x = αz + n, α C and n ker ϕ. ∈ ∈ Taking the inner product of this decomposition with z, we get z, x α = h i, z 2 k k 192 Bounded Linear Operators on a Hilbert Space and evaluating ϕ on x = αz + n, we find that
ϕ(x) = αϕ(z).
The elimination of α from these equations, and a rearrangement of the result, yields
ϕ(x) = y, x , h i where ϕ(z) y = z. z 2 k k Thus, every bounded linear functional is given by the inner product with a fixed vector. We have already seen that ϕ (x) = y, x defines a bounded linear functional on y h i for every y . To prove that there is a unique y in associated with a given H ∈ H H linear functional, suppose that ϕy1 = ϕy2 . Then ϕy1 (y) = ϕy2 (y) when y = y1 y2, 2 − which implies that y1 y2 = 0, so y1 = y2. k − k The map J : ∗ given by Jy = ϕ therefore identifies a Hilbert space H → H y H with its dual space ∗. The norm of ϕ is equal to the norm of y (see Exercise 8.7), H y so J is an isometry. In the case of complex Hilbert spaces, J is antilinear, rather than linear, because ϕ = λϕ . Thus, Hilbert spaces are self-dual, meaning that and λy y H ∗ are isomorphic as Banach spaces, and anti-isomorphic as Hilbert spaces. Hilbert H spaces are special in this respect. The dual space of an infinite-dimensional Banach space, such as an Lp-space with p = 2 or C([a, b]), is in general not isomorphic to 6 the original space.
Example 8.13 In quantum mechanics, the observables of a system are represented by a space of linear operators on a Hilbert space . A state ω of a quantum A H mechanical system is a linear functional ω on the space of observables with the A following two properties:
ω(A∗A) 0 for all A , (8.7) ≥ ∈ A ω(I) = 1. (8.8)
The number ω(A) is the expected value of the observable A when the system is in the state ω. Condition (8.7) is called positivity, and condition (8.8) is called normalization. To be specific, suppose that = Cn and is the space of all n n H A × complex matrices. Then is a Hilbert space with the inner product given by A A, B = tr A∗B. h i By the Riesz representation theorem, for each state ω there is a unique ρ such ∈ A that
ω(A) = tr ρ∗A for all A . ∈ A The adjoint of an operator 193
The conditions (8.7) and (8.8) translate into ρ 0, and tr ρ = 1, respectively. ≥ Another application of the Riesz representation theorem is given in Section 12.11, where we use it to prove the existence and uniqueness of weak solutions of Laplace’s equation.
8.3 The adjoint of an operator
An important consequence of the Riesz representation theorem is the existence of the adjoint of a bounded operator on a Hilbert space. The defining property of the adjoint A∗ ( ) of an operator A ( ) is that ∈ B H ∈ B H x, Ay = A∗x, y for all x, y . (8.9) h i h i ∈ H The uniqueness of A∗ follows from Exercise 8.14. The definition implies that
(A∗)∗ = A, (AB)∗ = B∗A∗.
To prove that A∗ exists, we have to show that for every x , there is a vector ∈ H z , depending linearly on x, such that ∈ H z, y = x, Ay for all y . (8.10) h i h i ∈ H
For fixed x, the map ϕx defined by
ϕ (y) = x, Ay x h i is a bounded linear functional on , with ϕ A x . By the Riesz represen- H k xk ≤ k kk k tation theorem, there is a unique z such that ϕ (y) = z, y . This z satisfies ∈ H x h i (8.10), so we set A∗x = z. The linearity of A∗ follows from the uniqueness in the Riesz representation theorem and the linearity of the inner product.
Example 8.14 The matrix of the adjoint of a linear map on Rn with matrix A is AT , since
x (Ay) = AT x y. · · In component notation, we have
n n n n x a y = a x y . i ij j ij i j i=1 j=1 j=1 i=1 ! X X X X The matrix of the adjoint of a linear map on Cn with complex matrix A is the Hermitian conjugate matrix,
A∗ = AT . 194 Bounded Linear Operators on a Hilbert Space
Example 8.15 Suppose that S and T are the right and left shift operators on the sequence space `2(N), defined by
S(x1, x2, x3, . . .) = (0, x1, x2, x3, . . .), T (x1, x2, x3, . . .) = (x2, x3, x4, . . .).
Then T = S∗, since
x, Sy = x2y1 + x3y2 + x4y3 + . . . = T x, y . h i h i Example 8.16 Let K : L2([0, 1]) L2([0, 1]) be an integral operator of the form → 1 Kf(x) = k(x, y)f(y) dy, Z0 where k : [0, 1] [0, 1] C. Then the adjoint operator × → 1 K∗f(x) = k(y, x)f(y) dy Z0 is the integral operator with the complex conjugate, transpose kernel.
The adjoint plays a crucial role in studying the solvability of a linear equation
Ax = y, (8.11) where A : is a bounded linear operator. Let z be any solution of the H → H ∈ H homogeneous adjoint equation,
A∗z = 0.
We take the inner product of (8.11) with z. The inner product on the left-hand side vanishes because
Ax, z = x, A∗z = 0. h i h i Hence, a necessary condition for a solution x of (8.11) to exist is that y, z = 0 h i for all z ker A∗, meaning that y (ker A∗)⊥. This condition on y is not always ∈ ∈ sufficient to guarantee the solvability of (8.11); the most we can say for general bounded operators is the following result.
Theorem 8.17 If A : is a bounded linear operator, then H → H ran A = (ker A∗)⊥ , ker A = (ran A∗)⊥ . (8.12)
Proof. If x ran A, there is a y such that x = Ay. For any z ker A∗, we ∈ ∈ H ∈ then have
x, z = Ay, z = y, A∗z = 0. h i h i h i The adjoint of an operator 195
This proves that ran A (ker A∗)⊥. Since (ker A∗)⊥ is closed, it follows that ⊂ ran A (ker A∗)⊥. On the other hand, if x (ran A)⊥, then for all y we have ⊂ ∈ ∈ H 0 = Ay, x = y, A∗x . h i h i Therefore A∗x = 0. This means that (ran A)⊥ ker A∗. By taking the orthogonal ⊂ complement of this relation, we get
(ker A∗)⊥ (ran A)⊥⊥ = ran A, ⊂ which proves the first part of (8.12). To prove the second part, we apply the first part to A∗, instead of A, use A∗∗ = A, and take orthogonal complements. An equivalent formulation of this theorem is that if A is a bounded linear oper- ator on , then is the orthogonal direct sum H H = ran A ker A∗. H ⊕ If A has closed range, then we obtain the following necessary and sufficient condition for the solvability of (8.11).
Theorem 8.18 Suppose that A : is a bounded linear operator on a Hilbert H → H space with closed range. Then the equation Ax = y has a solution for x if and H only if y is orthogonal to ker A∗.
This theorem provides a useful general method of proving existence from unique- ness: if A has closed range, and the solution of the adjoint problem A∗x = y is unique, then ker A∗ = 0 , so every y is orthogonal to ker A∗. Hence, a solution of { } Ax = y exists for every y . The condition that A has closed range is implied by ∈ H an estimate of the form c x Ax , as shown in Proposition 5.30. k k ≤ k k A commonly occurring dichotomy for the solvability of a linear equation is sum- marized in the following Fredholm alternative.
Definition 8.19 A bounded linear operator A : on a Hilbert space H → H H satisfies the Fredholm alternative if one of the following two alternatives holds:
(a) either Ax = 0, A∗x = 0 have only the zero solution, and the equations Ax = y, A∗x = y have a unique solution x for every y ; ∈ H ∈ H (b) or Ax = 0, A∗x = 0 have nontrivial, finite-dimensional solution spaces of the same dimension, Ax = y has a (nonunique) solution if and only if y z ⊥ for every solution z of A∗z = 0, and A∗x = y has a (nonunique) solution if and only if y z for every solution z of Az = 0. ⊥ Any linear operator A : Cn Cn on a finite-dimensional space, associated with → an n n system of linear equations Ax = y, satisfies the Fredholm alternative. × The ranges of A and A∗ are closed because they are finite-dimensional. From linear algebra, the rank of A∗ is equal to the rank of A, and therefore the nullity 196 Bounded Linear Operators on a Hilbert Space of A is equal to the nullity of A∗. The Fredholm alternative then follows from Theorem 8.18. Two things can go wrong with the Fredholm alternative in Definition 8.19 for bounded operators A on an infinite-dimensional space. First, ran A need not be closed; and second, even if ran A is closed, it is not true, in general, that ker A and ker A∗ have the same dimension. As a result, the equation Ax = y may be solvable for all y even though A is not one-to-one, or Ax = y may not be solvable for ∈ H all y even though A is one-to-one. We illustrate these possibilities with some ∈ H examples.
Example 8.20 Consider the multiplication operator M : L2([0, 1]) L2([0, 1]) → defined by Mf(x) = xf(x). Then M ∗ = M, and M is one-to-one, so every g L2([0, 1]) is orthogonal to ∈ ker M ∗; but the range of M is a proper dense subspace of L2([0, 1]), so Mf = g is not solvable for every g L2([0, 1]) (see Example 9.5 for more details). ∈ Example 8.21 The range of the right shift operator S : `2(N) `2(N), defined → 2 in Example 8.15, is closed since it consists of y = (y1, y2, y3, . . .) ` (N) such ∗ ∈ that y1 = 0. The left shift operator T = S is singular since its kernel is the one-dimensional space with basis (1, 0, 0, . . .) . The equation Sx = y, or { } (0, x1, x2, . . .) = (y1, y2, y3, . . .), is solvable if and only if y1 = 0, or y ker T , which verifies Theorem 8.18 in this ⊥ case. If a solution exists, then it is unique. On the other hand, the equation T x = y is solvable for every y `2(N), even though T is not one-to-one, and the solution is ∈ not unique.
These examples motivate the following definition.
Definition 8.22 A bounded linear operator A on a Hilbert space is a Fredholm operator if:
(a) ran A is closed; (b) ker A and ker A∗ are finite-dimensional.
The index of a Fredholm operator A, ind A, is the integer ind A = dim ker A dim ker A∗. − For example, a linear operator on a finite-dimensional Hilbert space and the identity operator on an infinite-dimensional Hilbert space are Fredholm operators with index zero. The right and left shift operators S and T in Example 8.21 are Fredholm, but their indices are nonzero. Since dim ker S = 0, dim ker T = 1, and Self-adjoint and unitary operators 197
S = T ∗, we have ind S = 1 and ind T = 1. The multiplication operator in − Example 8.20 is not Fredholm because it does not have closed range. It is possible to prove that if A is Fredholm and K is compact, then A + K is Fredholm, and ind (A + K) = ind A. Thus the index of a Fredholm operator is unchanged by compact perturbations. In particular, compact perturbations of the identity are Fredholm operators with index zero, so they satisfy the Fredholm alternative in Definition 8.19. We will prove a special case of this result, for compact, self-adjoint perturbations of the identity, in Theorem 9.26.
8.4 Self-adjoint and unitary operators
Two of the most important classes of operators on a Hilbert space are the classes of self-adjoint and unitary operators. We begin by defining self-adjoint operators.
Definition 8.23 A bounded linear operator A : on a Hilbert space is H → H H self-adjoint if A∗ = A.
Equivalently, a bounded linear operator A on is self-adjoint if and only if H x, Ay = Ax, y for all x, y . h i h i ∈ H Example 8.24 From Example 8.14, a linear map on Rn with matrix A is self- adjoint if and only if A is symmetric, meaning that A = AT , where AT is the transpose of A. A linear map on Cn with matrix A is self-adjoint if and only if A is Hermitian, meaning that A = A∗.
Example 8.25 From Example 8.16, an integral operator K : L2([0, 1]) L2([0, 1]), → 1 Kf(x) = k(x, y)f(y) dy, Z0 is self-adjoint if and only if k(x, y) = k(y, x).
Given a linear operator A : , we may define a sesquilinear form H → H a : C H × H → by a(x, y) = x, Ay . If A is self-adjoint, then this form is Hermitian symmetric, or h i symmetric, meaning that
a(x, y) = a(y, x).
It follows that the associated quadratic form q(x) = a(x, x), or
q(x) = x, Ax , (8.13) h i 198 Bounded Linear Operators on a Hilbert Space is real-valued. We say that A is nonnegative if it is self-adjoint and x, Ax 0 for h i ≥ all x . We say that A is positive, or positive definite, if it is self-adjoint and ∈ H x, Ax > 0 for every nonzero x . If A is a positive, bounded operator, then h i ∈ H (x, y) = x, Ay h i defines an inner product on . If, in addition, there is a constant c > 0 such that H x, Ax c x 2 for all x , h i ≥ k k ∈ H then we say that A is bounded from below, and the norm associated with ( , ) is · · equivalent to the norm associated with , . h· ·i The quadratic form associated with a self-adjoint operator determines the norm of the operator.
Lemma 8.26 If A is a bounded self-adjoint operator on a Hilbert space , then H A = sup x, Ax . k k kxk=1 |h i|
Proof. Let
α = sup x, Ax . kxk=1 |h i|
The inequality α A is immediate, since ≤ k k x, Ax Ax x A x 2. |h i| ≤ k k k k ≤ k k k k To prove the reverse inequality, we use the definition of the norm,
A = sup Ax . k k kxk=1 k k For any z , we have ∈ H z = sup y, z . k k kyk=1 |h i| It follows that
A = sup y, Ax x = 1, y = 1 . (8.14) k k {|h i| | k k k k } The polarization formula (6.5) implies that 1 y, Ax = x + y, A(x + y) x y, A(x y) h i 4 {h i − h − − i i x + iy, A(x + iy) + i x iy, A(x iy) . − h i h − − i} Since A is self-adjoint, the first two terms are real, and the last two are imaginary. We replace y by eiϕy, where ϕ R is chosen so that eiϕy, Ax is real. Then the ∈ h i Self-adjoint and unitary operators 199
imaginary terms vanish, and we find that 1 y, Ax 2 = x + y, A(x + y) x y, A(x y) 2 |h i| 16|h i − h − − i| 1 α2( x + y 2 + x y 2)2 ≤ 16 k k k − k 1 = α2( x 2 + y 2)2, 4 k k k k where we have used the definition of α and the parallelogram law. Using this result in (8.14), we conclude that A α. k k ≤ As a corollary, we have the following result.
Corollary 8.27 If A is a bounded operator on a Hilbert space then A∗A = A 2. k k k k If A is self-adjoint, then A2 = A 2. k k k k Proof. The definition of A , and an application Lemma 8.26 to the self-adjoint k k operator A∗A, imply that A 2 = sup Ax, Ax = sup x, A∗Ax = A∗A . k k kxk=1 |h i| kxk=1 |h i| k k Hence, if A is self-adjoint, then A 2 = A2 . k k k k Next, we define orthogonal or unitary operators, on real or complex spaces, respectively.
Definition 8.28 A linear map U : 1 2 between real or complex Hilbert H → H spaces 1 and 2 is said to be orthogonal or unitary, respectively, if it is invertible H H and if
Ux, Uy = x, y for all x, y 1. h iH2 h iH1 ∈ H Two Hilbert spaces 1 and 2 are isomorphic as Hilbert spaces if there is a unitary H H linear map between them.
Thus, a unitary operator is one-to-one and onto, and preserves the inner product. A map U : is unitary if and only if U ∗U = UU ∗ = I. H → H Example 8.29 An n n real matrix Q is orthogonal if QT = Q−1. An n n × × complex matrix U is unitary if U ∗ = U −1.
Example 8.30 If A is a bounded self-adjoint operator, then ∞ 1 eiA = (iA)n n! n=0 X is unitary, since ∗ −1 eiA = e−iA = eiA . 200 Bounded Linear Operators on a Hilbert Space
A bounded operator S is skew-adjoint if S∗ = S. Any skew-adjoint operator S − on a complex Hilbert space may be written as S = iA where A is a self-adjoint operator. The commutator [A, B] = AB BA is a Lie bracket on the space of − bounded, skew-adjoint operators, and we say that this space is the Lie algebra of the Lie group of unitary operators.
Example 8.31 Let be a finite dimensional Hilbert space. If e1, e2, . . . , e is H { n} an orthonormal basis of , then U : Cn defined by H → H
U (z1, z2, . . . , zn) = z1e1 + z2e2 + . . . + znen is unitary. Thus, any n-dimensional, complex Hilbert space is isomorphic to Cn.
Example 8.32 Suppose that 1 and 2 are two Hilbert spaces of the same, pos- H H sibly infinite, dimension. Let u be an orthonormal basis of 1 and v an { α} H { α} orthonormal basis of 2. Any x 1 can be written uniquely as H ∈ H
x = cαuα, α X with coefficients c C. We define U : 1 2 by α ∈ H → H
U cαuα = cαvα. α ! α X X Then U is unitary. Thus, Hilbert spaces of the same dimension are isomorphic. iϕ More generally, if λ = e α are complex numbers with λ = 1, then U : 1 α | α| H → 2 defined by H Ux = λ u , x v αh α i α α X is unitary. For example, the periodic Hilbert transform H : L2(T) L2(T) is → defined by
∞ ∞ inx inx H fˆne = i (sgn n) fˆne , n=−∞ ! n=−∞ X X where sgn is the sign function, defined in (5.8). The Hilbert transform is not a uni- tary mapping on L2(T) because H(1) = 0; however, Parseval’s theorem implies that it is a unitary mapping on the subspace of square-integrable periodic functions H with zero mean,
= f L2(T) f(x) dx = 0 . H ∈ T Z
The mean ergodic theorem 201
Example 8.33 The operator U : L2(T) `2(Z) that maps a function to its Fourier → coefficents is unitary. Explicitly, we have
2π 1 −inx Uf = (cn)n∈Z, cn = f(x)e dx. √2π 0 Z Thus, the Hilbert space of square integrable functions on the circle is isomorphic to the Hilbert space of sequences on Z. As this example illustrates, isomorphic Hilbert spaces may be given concretely in forms that, at first sight, do not appear to be the same.
Example 8.34 For a T, we define the translation operator T : L2(T) L2(T) ∈ a → by
(T f)(x) = f(x a). a −
Then Ta is unitary, and
Ta+b = TaTb. We say that T a T is a unitary representation of the additive group R/(2πZ) { a | ∈ } on the linear space L2(T).
An operator T : is said to be normal if it commutes with its adjoint, H → H meaning that T T ∗ = T ∗T . Both self-adjoint and unitary operators are normal. An important feature of normal operators is that they have a nice spectral theory. We will discuss the spectral theory of compact, self-adjoint operators in detail in the next chapter.
8.5 The mean ergodic theorem
Ergodic theorems equate time averages with probabilistic averages, and they are important, for example, in understanding the statistical behavior of deterministic dynamical systems. The proof of the following ergodic theorem, due to von Neumann, is a good example of Hilbert space methods.
Theorem 8.35 (von Neumann ergodic) Suppose that U is a unitary operator on a Hilbert space . Let = x Ux = x be the subspace of vectors that H M { ∈ H | } are invariant under U, and P the orthogonal projection onto . Then, for all M x , we have ∈ H 1 N lim U nx = P x. (8.15) N→∞ N + 1 n=0 X That is, the averages of U n converge strongly to P . 202 Bounded Linear Operators on a Hilbert Space
Proof. It is sufficient to prove (8.15) for x ker P and x ran P , because then ∈ ∈ the orthogonal decomposition = ker P ran P implies that (8.15) holds for all H ⊕ x . Equation (8.15) is trivial when x ran P = , since then Ux = x and ∈ H ∈ M P x = x. To complete the proof, we show that (8.15) holds when x ker P . From the ∈ definition of P , we have ran P = ker(I U). If U is unitary, then Ux = x if and − only if U ∗x = x. Hence, using Theorem 8.17, we find that
ker P = ker(I U)⊥ = ker(I U ∗)⊥ = ran (I U). − − − Therefore every x ker P may be approximated by vectors of the form (I U)y. ∈ − If x = (I U)y, then − N N 1 1 U nx = U n U n+1 y N + 1 N + 1 − n=0 n=0 X X 1 = y U N+1y N + 1 − 0 as N . → → ∞ If x ker P , then there is a sequence of elements x = (I U)y with x x. ∈ k − k k → Hence,
N N 1 n 1 n lim U x lim sup U (x xk) N→∞ N + 1 ≤ N + 1 − n=0 N→∞ n=0 X X N 1 n + lim sup U xk N→∞ N + 1 n=0 X x xk . ≤ k − k Since k is arbitrary and x x, it follows that (8.15) holds for every x ker P . k → ∈ Next, we explain the implications of this result in probability theory. Suppose that P is a probability measure on a probability space Ω, as described in Section 6.4. A one-to-one, onto, measurable map T : Ω Ω is said to be measure preserving if → P T −1(A) = P (A) for all measurable subsets A of Ω. Here,