Additional Notes on Adjoint and Unitary Operators

Home , Unitary operator, Volterra operator

AMATH 731: Applied Functional Analysis Fall 2017

Additional notes on adjoint and unitary operators

(to supplement Section 4.5 of the Course Notes)

Just to recall the basic facts: Let H be a Hilbert space and L : H H a bounded linear operator. → Then there exists a bounded linear operator L∗ : H H such that → Lx,y = x,L∗y . (1) h i h i L∗ is the adjoint of L. Here are a few additional properties not listed in Section 4.3 of the Course Notes. The proofs are left as an exercise.

L∗ is unique: If Lx,y = x,L∗y = x,L∗y for all x,y H, then L∗ = L∗. h i h 1 i h 2 i ∈ 1 2

I∗ = I, 0∗ = 0, (S + T )∗ = S∗ + T ∗, (αT )∗ = αT ∗, (ST )∗ = T ∗S∗.

Some additional examples of adjoint operators

4. Multiplication operator on L2( , ): Consider the operator −∞ ∞ F : x(t) f(t)x(t), (2) → where f(t) B < for all t. (Note that we do not require that f L2. After all, multiplication of | |≤ ∞ ∈ x(t) by a constant k R is a special case.) Then F is a bounded linear operator with ∈

F = sup f(t) = f ∞. k k t∈R | | k k Now consider the following,

∞ ∞ Fx,y = f(t)x(t)y(t) dt = x(t)f(t)y(t) dt h i Z−∞ Z−∞ = x, F ∗y . h i This implies that the associated adjoint operator is given by

F ∗ : y(t) f(t)y(t). (3) →

1 The condition that f(t) be strictly bounded may be replaced by the condition f(t) B < for | |≤ ∞ almost all t R. Then ∈ F = ess. sup. R f(t) = f , k k t∈ | | k k∞ where denotes the L∞ norm, i.e., f(t) f almost everywhere. k · k∞ | | ≤ k k∞ 5. An interesting example involving Volterra integral operators: Let I = [0, T ] and consider the linear operator K : C(I) C(I) defined as follows: y = Kx implies that → t y(t) = (Kx)(t)= k(t,s)x(s) ds. (4) Z0 In the usual case of the Volterra integral operator, k(t,s) needs only to be defined in the region s t, ≤ t [0, T ]. However, if we wish to define an adjoint operator, essentially replacing k(t,s) with k(s,t), ∈ we shall have to extend the definition of k. If we set k(t,s)=0 for s>t, then Eq. (4) becomes

y(t)= k(t,s)x(s) ds. (5) ZI Then, according to Example 3 in the Course Notes, p. 67, the adjoint is given by

T (K∗y)(t)= k(s,t)y(s) ds = k(s,t)y(s) ds. (6) ZI Zt Thus, the adjoint of a Volterra integral operator is also a Volterra integral operator. But if K depends on the “past” (i.e., y(t) = (Kx)(t) is determined by x(s) for 0 s t), then K∗ depends on the ≤ ≤ “future” (i.e., (K∗y(t) is determined by y(s) for t s T ). ≤ ≤ The Volterra operator in Eq. (4) is an example of a causal operator and the adjoint Volterra operator in Eq. (6) is an example of an anticausal operator.

6. Adjoint operators of (unbounded) differential operators: We take a little detour here and consider second-order (unbounded) differential operators on the linear subspace M of (piecewise) C2 functions u : [a, b] C satisfying the boundary conditions u(a)= u(b) = 0. In what follows, let g(x) → be a real-valued function. For an u M, define Lu as follows, ∈ (Lu)(x)= u′′(x)+ g(x)u(x). (7)

For u, v M, consider the inner product ∈ b Lu, v = v(x)[u′′(x)+ g(x)u(x)] dx. (8) h i Za Integrating the ﬁrst term by parts yields

b b ′′ ′ b ′ ′ v(x)u (x) dx = v(x)u (x) a v (x)u (x) dx. (9) Za | − Za The endpoint terms vanish because of the boundary conditions. Another integration by parts yields

b b ′ ′ ′′ b ′′ v (x)u (x) dx = v (x)u(x) a v (x)u(x) dx. (10) Za | − Za

2 Once again, the endpoint terms vanish because of the boundary conditions. Substitution of this result into (8) yields b Lu, v = [v′′(x)+ g(x)v(x)]u(x) dx = u, L∗v = u, Lv . (11) h i Za h i h i In other words, L∗ = L, implying that the diﬀerential operator L is self-adjoint. Note that the boundary conditions obeyed by functions in this space play an important role in making possible the self-adjointness of L. We shall see later that this self-adjointness property also applies to Sturm- Liouville diﬀerential operators.

If the diﬀerential operator L on M contains a ﬁrst order derivative, i.e.,

(Lu)(x)= a(x)u′′(x)+ b(x)u′(x)+ c(x)v(x), (12) where a, b, c are real-valued functions, then (Exercise)

(L∗v)(x) = (a(x)v(x))′′ (b(x)v(x))′ + c(x)v(x). (13) − Invariant subspaces

Deﬁnition 1 Let X be a vector space. If T : X X is a linear transformation and M is a linear → subspace of X such that T (M) M, then M is said to be invariant under T . ⊂ Example: Let X = L2( , ) and let I be an interval in R. Let −∞ ∞ M = x L2( , ) : x(t)=0 for t / I . (14) { ∈ −∞ ∞ ∈ } Then given the multiplication operator F deﬁned in (4) above, F (M) M. ⊂ This idea of invariant subspaces is used to prove the following result:

Theorem 1 If a bounded linear operator L is causal, then its adjoint L∗ is anticausal.

Proof: See Naylor and Sell, p. 356.

We present below some results that will be useful in future discussions in this course.

Theorem 2 Let L be a bounded (therefore continuous) linear transformation of a Hilbert space H into itself. A closed linear subspace M of H is invariant under L if and only if M ⊥ is invariant under L∗.

Proof: If M is invariant under L, then L(M) M. This means that x,Ly = 0 for all y M and ⊂ h i ∈ x M ⊥. But this means that L∗x,y = 0 for all y M and x M ⊥. Thus, L∗x M ⊥ for all ∈ h i ∈ ∈ ∈ x M ⊥, implying that L∗(M ⊥) M ⊥. If M ⊥ is invariant under L∗, the same type of argument ∈ ⊂ shows that M is invariant under L.

Let H be a Hilbert space and L : H H a bounded linear operator. Recall the following deﬁnitions: →

3 1. Range of L: R(L)= Lx : x H , { ∈ } 2. Null space of L: N(L)= x H : Lx = 0 . { ∈ } The following theorem is presented as a problem in Problem Set 6 of the Course Notes.

Theorem 3 Let H be a Hilbert space and L : H H a bounded linear operator with adjoint L∗. Let → N(L) and N(L∗) denote the null spaces of, respectively, L and L∗. Also let R(L) and R(L∗) denote the closures of the ranges of, respectively, L and L∗. Then

R(L) = N(L∗)⊥ R(L∗) = N(L)⊥. (15)

Unitary operators

Deﬁnition 2 Let U : H H be a bounded linear operator on a Hilbert space H. Then U is said to → be isometric if Ux,Uy = x,y , for all x,y H. (16) h i h i ∈ If R(U)= H, then U is called a unitary operator.

Theorem 4 Let U : H H be a bounded linear operator on a Hilbert space H. Then U is unitary → if and only if UU ∗ = U ∗U = I, i.e., U ∗ = U −1.

Proof: If U is unitary, then U ∗Ux,y = Ux,Uy = x,y , h i h i h i for all x,y H. Therefore U ∗U = I. Similarly, UU ∗ = I. ∈ Now assume that U ∗U = I. Then

x,y = U ∗Ux,y = Ux,Uy , h i h i h i for all x,y H, implying that U is unitary. ∈ Note: Setting x = y in Eq. (16) gives Ux 2 = x 2, implying that Ux = x , i.e., the operator k k k k k k k k norm of a unitary operator is 1.

Examples:

Delay or shift operator: Let H = L2( , ) and consider the delay or shift operator S : H H, −∞ ∞ τ → τ R, deﬁned as ∈ (S x)(t)= x(t τ), for t ( , ). τ − ∈ −∞ ∞ −1 Note that Sτ has an inverse: Sτ = S−τ . Sτ is also a unitary operator, since ∞ ∞ Sτ x,Sτ y = y(t τ)x(t τ) dt = y(t)x(t) dt, (17) h i Z−∞ − − Z−∞

4 for all x,y H. ∈ Note also that ∞ ∞ Sτ x,y = y(t)x(t τ) dt = y(t + τ)x(t) dt, (18) h i Z−∞ − Z−∞ for all x,y H. This implies that the adjoint of S is deﬁned by ∈ τ (S∗y)(t)= y(t + τ), t ( , ). (19) τ ∈ −∞ ∞

Therefore, as expected (since Sτ was found to be unitary),

∗ −1 Sτ = S−τ = Sτ . (20)

∗ If we interpret Sτ , for τ > 0, as a causal operator, then Sτ is anticausal.

Fourier transform on R: There are various definitions of the Fourier transform (depending upon the coefficients in front of the integrals and the sign of the exponents). Here we define the Fourier transform F (ω) of a function f L1(R) L2(R) as follows, ∈ ∩ 1 ∞ F (ω) = ( f)(ω)= e−iωxf(x) dx. (21) F √2π Z−∞

is a unitary mapping of L1(R) L2(R) onto itself. Its inverse is given by F ∩ 1 ∞ f(x) = ( −1F )(x)= eiωxF (ω) dω. (22) F √2π Z−∞ A discussion of these operators, including a proof of their unitarity, may be found in the book by Naylor and Sell, pp. 360-362. Here, we prove the following important result, known as Plancherel’s Formula, which is a consequence of the unitarity of and −1. F F

Theorem 5 (Plancherel) Let f, g L1(R) L2(R). Then ∈ ∩ f, g = F, G = f, g , (23) hF F i h i h i where , denotes the complex inner product on R, i.e., h· ·i ∞ ∞ f(t)g(t) dt = F (ω)G(ω) dω. (24) Z−∞ Z−∞ Before proving this theorem, we mention an important consequence: If f = g, then

f, f = F, F = f,f , (25) hF F i h i h i which implies that f = F . (26) k k2 k k2 In other words, the Fourier transform operator is (L2)-norm-preserving. Eq. (26) may be viewed as F the continuous version of Parseval’s equation studied earlier in the context of complete orthonormal

5 bases. Recall the following: Let H be a Hilbert space. If the orthonormal set e ∞ H is complete, { k}0 ⊂ then for any f H, ∈ ∞ f = c e , where c = f, e , (27) k k k h ki kX=0 and f 2 = c 2 , (28) k kL k kl i.e., ∞ f 2 = c 2 (Parseval’s equation). (29) k k | k| kX=0 On the other hand, Parseval’s equation may be viewed as a discrete version of Plancherel’s formula.

Proof of Plancherel’s Formula: We first express the function f(t) in terms of the inverse Fourier transform, i.e., 1 ∞ f(t)= F (ω)eiωtdω. (30) √2π Z−∞ Substitute for F (ω) using the definition of the Fourier transform, 1 ∞ ∞ f(t)= f(s)e−iωsds eiωtdω. (31) 2π Z−∞ Z−∞ Now take the inner product of f(t) with g(t): ∞ 1 ∞ ∞ f, g = f(x)e−iωsds eiωtdω g(t) dt. (32) h i Z−∞ 2π Z−∞ Z−∞ We now assume that f and g are sufficiently “nice” so that Fubini’s Theorem will allow us to rearrange the order of integration. The result is

∞ 1 ∞ 1 ∞ f, g = f(s)e−iωsds g(t)eiωtdt dω h i Z−∞ √2π Z−∞ √2π Z−∞ ∞ = F (ω)G(ω) dω, Z−∞ = F, G . (33) h i

The following result is important in the calculus of diﬀerential operators as well as being of fundamental importance in quantum mechanics. A proof of this result is to be found in the book of Naylor and Sell, pp. 362-363.

Theorem 6 Let P and Q be the linear operators deﬁned by du P : u(x) i → dx Q : u(x) xu(x), →

6 where the domains of the respective operators are

= u L2(R) : u is absolutely continuous and u′ L2(R) DP { ∈ ∈ } = u L2(R) : xu(x) L2(R) . DQ { ∈ ∈ } Then the Fourier transform F sets up a one-to-one correspondence between and in such a way DP DQ that P = F QF −1 and Q = F −1P F.

In quantum mechanics, the operator P corresponds (up to a constant) to the momentum operator in physical space coordinates. The operator Q corresponds to the position operator in physical space coordinates. But in momentum space, the operator P will correspond (up to a constant) to the position operator and Q will correspond to the momentum operator.

Normal and self-adjoint operators

Deﬁnition 3 Let H be a Hilbert space and let L : H H be a bounded linear transformation. L is → said to be normal if LL∗ = L∗L, i.e., if L commutes with its adjoint.

Deﬁnition 4 Let H be a Hilbert space and let L : H H be a bounded linear transformation. L is → said to be self-adjoint if L = L∗.

A trivial consequence of these deﬁnitions: If L is self-adjoint, then it is normal.

Examples of self-adjoint operators were given in Section 4.5 of the Course Notes, p. 67. In Example 3 of the notes, the complex valued integral operator K on L2[a, b] was deﬁned as

b g(x) = (Lf)(x)= k(x,y)f(y) dy, Za Its adjoint is given by b (L∗f)(x)= k(y,x)f(y) dy. Za L is self-adjoint if k(x,y)= k(y,x). (Once again, this is a continuous version of the “complex conjugate transpose” rule for complex matrices, cf. Example 2 of the Course Notes, p. 67.)

Theorem 7 If A and B are self-adjoint operators on a Hilbert space H, then AB is self-adjoint if and only if AB = BA.

If L is self-adjoint, then x,Lx = Lx,x = x,Lx , (34) h i h i h i for all x H. In other words, x,Lx is real-valued on H. In fact, we have the following ∈ h i Theorem 8 Let L : H H be a bounded linear operator on a Hilbert space H. L is self-adjoint if → and only if x,Lx is real-valued for all x H. h i ∈

7 Self-adjoint operators are important in quantum mechanics. (Actually, many of them, as diﬀer- ential operators, are unbounded.) The quantity x,Lx corresponds to the expectation value of the h i quantum mechanical linear operator L, where x H is the quantum mechanical wavefunction of the ∈ system. For L to represent an observable, it must be self-adjoint, so that the expectation value, which is potentially “observed”, is real.

The exponential of a self-adjoint operator is a unitary operator

In what follows, we consider a special class of unitary operators that are important in applications, in particular, quantum mechanics. Let us ﬁrst recall the following important result from our study of Banach spaces and linear operators on them: If A is a bounded linear operator on a Banach space X, we can deﬁne the following operator ∞ 1 eA = An. (35) nX=0 n! This operator, the exponential of A, is a bounded linear operator L:

L = eA implies that L = eA ekAk. (36) k k k k≤

Here is a result that we shall not prove:

Let A, B B(X, X)= B(X) (set of bounded linear operators on X). If AB = BA, then ∈ eAeB = eA+B. (37)

You know this result for n n matrices as operators in Rn. A couple of obvious consequences of this × result are (i) eAeA = e2A , (ii) eAe−A = I . (38) If AB = BA, which is often the case in quantum mechanics, then a more complicated relationship 6 applies, the so-called Baker-Campbell-Hausdorﬀ formula:

t2 et(A+B) = etB e 2 [A,B]etA, (39) where [A, B] = AB BA denotes the commutator of A and B: [A, B] = [B, A]. Actually, there − − are many BCH and related formulas. Here is another important one: 1 1 eABe−A = B + [A, B]+ [A, [A, B]] + [A, [A, [A, B]]] + . (40) 2! 3! · · · The importance of the matrix exponential in Rn is well known: Given the following initial value problem for the linear system of ODEs in Rn, dx = Ax, x(0) = x , (41) dt 0 its solution is given by tA x(t)= e x0. (42) tA The linear operator Lt = e is an evolution operator for this system satisfying the following properties: For s,t R, ∈

8 0 1. (Identity) L0 = e = I,

2. (Semigroup) LsLt = Lt+s,

−1 3. (Inverse) L−t = Lt .

Another important property:

λt tA If λ is an eigenvalue of A, then e is an eigenvalue of Lt = e .

In fact,

If v is a λ-eigenvector of A, i.e., Av = λv, then etAv = eλtv.

As you know from courses in ODEs, the asymptotic properties of solutions to Eq. (41) are determined by the eigenvalue spectrum of A.

Let us now consider bounded linear operators on a Hilbert space H. If A : H H is a bounded → linear operator, then its exponential L = eA, as deﬁned in Eq. (35), is also a bounded linear operator. ∗ Moreover, it is easy to show that the adjoint of L is L∗ = eA . And since L = L∗ , we have k k k k ∗ eA = eA . (43) k k k k

Now suppose that A is a bounded, self-adjoint operator on H, i.e., A = A∗. For t R, deﬁne ∈ ∞ itA 1 n Ut = e = (itA) . (44) nX=0 n! Then:

1. U satisﬁes the three properties listed earlier for the evolution operator etA.

∗ −1 ∗ −itA∗ −itA −1 2. Ut is unitary, i.e., Ut = Ut . This follows from Ut = e = e = U−t = Ut .

As such, we can imagine that U would serve as the evolution operator for the following system of DEs: dx = iAx, (45) dt where x H and A is a bounded, self-adjoint operator. ∈ Indeed, up to a constant, this is ALMOST the situation in quantum mechanics. The time- dependent Schr¨odinger equation for the wavefunction Ψ of a quantum mechanical system is given by dΨ i¯h = HΨ. (46) dt Here, H is the so-called Hamiltonian operator of the system, a self-adjoint linear operator representing the total energy. The constanth ¯ = h/2π, where h is Planck’s constant. Unfortunately, H is usually an

9 unbounded differential operator, but the results are qualitatively the same. (We have to worry about domains of definition.) The evolution operator associated with the Schrödinger equation has the form

U = e−itH/¯h, t R. (47) t ∈ Solutions of Eq. (46) have the form Ψ(t)= UtΨ(0). (48)

Because of the unitarity of Ut, we have

Ψ(t) 2 = Ψ(t) Ψ(t) = U Ψ(0) U Ψ(0) = Ψ(0) Ψ(0) = Ψ(0) 2 = 1. (49) k k h | i h t | t i h | i k k The norm of the wavefunction Ψ(t) is always normalized, in keeping with its probabilistic interpre- tation. To illustrate, if Ψ represents the wavefunction of a single quantum mechanical particle, then Ψ(x,t) 2dV represents the probability of ﬁnding it in a volume element dV situated at x at time t. k k At any time t, an integration of this probability over all of physical space should yield the value 1, corresponding to the probability of ﬁnding the particle somewhere.